Chapter 1 Resource Masters - Wikispaces1.pdf · Chapter 1 Resource Masters ... list of the key vocabulary terms you will learn in Chapter 1.As you study the chapter, ... McGraw-Hill

Chapter 1Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

11

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 1.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Geometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

acute angle

adjacent angles

uh·JAY·suhnt

angle

angle bisector

collinear

koh·LIN·ee·uhr

complementary angles

congruent

kuhn·GROO·uhnt

coplanar

koh·PLAY·nuhr

line segment

linear pair

(continued on the next page)

! " # " $

! " # " $

! " # " $

! " # " $

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

midpoint

obtuse angle

perimeter

perpendicular lines

polygon

PAHL·ee·gahn

ray

right angle

segment bisector

supplementary angles

vertical angles

! " # " $

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

11

Study Guide and InterventionPoints, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

© Glencoe/McGraw-Hill 1 Glencoe Geometry

Less

on

1-1

Name Points, Lines, and Planes In geometry, a point is a location, a line containspoints, and a plane is a flat surface that contains points and lines. If points are on the sameline, they are collinear. If points on are the same plane, they are coplanar.

Use the figure to name each of the following.

a. a line containing point A

The line can be named as !. Also, any two of the three points on the line can be used to name it.

AB!"#, AC!"#, or BC!"#

b. a plane containing point D

The plane can be named as plane N or can be named using three noncollinear points in the plane, such as plane ABD, plane ACD, and so on.

Refer to the figure.

1. Name a line that contains point A.

2. What is another name for line m ?

3. Name a point not on AC!"#.

4. Name the intersection of AC!"# and DB!"#.

5. Name a point not on line ! or line m .

Draw and label a plane Q for each relationship.

6. AB!"# is in plane Q.

7. ST!"# intersects AB!"# at P.

8. Point X is collinear with points A and P.

9. Point Y is not collinear with points T and P.

10. Line ! contains points X and Y.

!Q

AB

S

P

TY

X

!

Pm

AB

EC

D

!

N

AB

C

DExampleExample

ExercisesExercises


Points, Lines, and Planes in Space Space is a boundless, three-dimensional set ofall points. It contains lines and planes.

a. How many planes appear in the figure?

There are three planes: plane N , plane O, and plane P.

b. Are points A, B, and D coplanar?

Yes. They are contained in plane O.


1. Name a line that is not contained in plane N.

2. Name a plane that contains point B.

3. Name three collinear points.


4. How many planes are shown in the figure?

5. Are points B, E, G, and H coplanar? Explain.

6. Name a point coplanar with D, C, and E.

Draw and label a figure for each relationship.

7. Planes M andN intersect in HJ!"#.

8. Line r is in plane N , line s is in plane M, and lines r and sintersect at point J.

9. Line t contains point H and line t does not lie in plane M orplane N.

N

Ms

t

r

HJ

A B

CD

EF

G HI

J

N

A

C

B

DE

NO

P

AB

CD

Study Guide and Intervention (continued)

Points, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

ExampleExample

ExercisesExercises

Skills PracticePoints, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1


Less

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1-1


1. Name a line that contains point D.

2. Name a point contained in line n.

3. What is another name for line p?

4. Name the plane containing lines n and p.


5. Point K lies on RT!"#. 6. Plane J contains line s.

7. YP!"# lies in plane B and contains 8. Lines q and f intersect at point Zpoint C, but does not contain point H. in plane U.



10. How many of the planes contain points F and E?

11. Name four points that are coplanar.

12. Are points A, B, and C coplanar? Explain.

WA

B

E

C

DF

U

q f

Z

HP

B

CY

J

sTRK

G

n

A BD

C

p



1. Name a line that contains points T and P.

2. Name a line that intersects the plane containing points Q, N, and P.

3. Name the plane that contains TN!"# and QR!"#.


4. AK!"# and CG!"# intersect at point M 5. A line contains L(!4, !4) and M(2, 3). Line in plane T. q is in the same coordinate plane but does

not intersect LM!"#. Line q contains point N.




8. Are points N, R, S, and W coplanar? Explain.

VISUALIZATION Name the geometric term(s) modeled by each object.

9. 10. 11.

12. a car antenna 13. a library card

strings

tip of pinSTOP

A M N

PW SX R

QT

L

M

N

q

x

y

O

KT

M

C

GA

S

j

g

hT

M

N

Q

R

P

Practice Points, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

Reading to Learn MathematicsPoints, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1


Less

on

1-1

Pre-Activity Why do chairs sometimes wobble?

Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.

• Find three pencils of different lengths and hold them upright on yourdesk so that the three pencil points do not lie along a single line. Can youplace a flat sheet of paper or cardboard so that it touches all three pencilpoints?

• How many ways can you do this if you keep the pencil points in the sameposition?

• How will your answer change if there are four pencil points?

Reading the Lesson

1. Complete each sentence.

a. Points that lie on the same lie are called points.

b. Points that do not lie in the same plane are called points.

c. There is exactly one through any two points.

d. There is exactly one through any three noncollinear points.

2. Refer to the figure at the right. Indicate whether each statement is true or false.

a. Points A, B, and C are collinear.

b. The intersection of plane ABC and line m is point P.

c. Line ! and line m do not intersect.

d. Points A, P,and B can be used to name plane U.

e. Line ! lies in plane ACB.

3. Complete the figure at the right to show the following relationship: Lines !, m, and n are coplanar and lie in plane Q. Lines ! and m intersect at point P. Line nintersects line m at R, but does not intersect line !.

Helping You Remember

4. Recall or look in a dictionary to find the meaning of the prefix co-. What does this prefixmean? How can it help you remember the meaning of collinear?

Qn

m

!P

R

U

m

!PC

D

B

A


Points and Lines on a MatrixA matrix is a rectangular array of rows and columns. Points andlines on a matrix are not defined in the same way as in Euclideangeometry. A point on a matrix is a dot, which can be small orlarge. A line on a matrix is a path of dots that “line up.” Betweentwo points on a line there may or may not be other points. Threeexamples of lines are shown at the upper right. The broad line canbe thought of as a single line or as two narrow lines side by side.

Dot-matrix printers for computers used dots to form characters.The dots are often called pixels. The matrix at the right showshow a dot-matrix printer might print the letter P.

Draw points on each matrix to create the given figures.

1. Draw two intersecting lines that have 2. Draw two lines that cross but have four points in common. no common points.

3. Make the number 0 (zero) so that it 4. Make the capital letter O so that itextends to the top and bottom sides extends to each side of the matrix.of the matrix.

5. Using separate grid paper, make dot designs for several other letters. Which were theeasiest and which were the most difficult?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

Study Guide and InterventionLinear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

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1-2

Measure Line Segments A part of a line between two endpoints is called a linesegment. The lengths of M!N! and R!S! are written as MN and RS. When you measure asegment, the precision of the measurement is half of the smallest unit on the ruler.

Find the length of M!N!.

The long marks are centimeters, and theshorter marks are millimeters. The length of M!N! is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M!N! isbetween 3.35 centimeters and 3.45centimeters long.

1cm 2 3 4

M NFind the length of R!S!.

The long marks are inches and the short marks are quarter inches. The length of R!S!is about 1"

34" inches. The measurement is

accurate to within one half of a quarter inch,or "

18" inch, so R!S! is between 1"

58" inches and

1"78" inches long.

1 2in.

R S

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the length of each line segment or object.

1. 2.

3. 4.

Find the precision for each measurement.

5. 10 in. 6. 32 mm 7. 44 cm

8. 2 ft 9. 3.5 mm 10. 2"12" yd

1cm 2 31 2in.

1in.

S T

1cm 2 3

A B


Calculate Measures On PQ!"#, to say that point M is between points P and Q means P, Q, and M are collinear and PM # MQ $ PQ.

On AC!"#, AB $ BC $ 3 cm. We can say that the segments arecongruent, or A!B! " B!C!. Slashes on the figure indicate whichsegments are congruent.

AB

C

QMP


Linear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Find EF.

Calculate EF by adding ED and DF.

ED # DF $ EF1.2 # 1.9 $ EF

3.1 $ EF

Therefore, E!F! is 3.1 centimeters long.

E1.2 cm 1.9 cm

FD

Find x and AC.

B is between A and C.

AB # BC $ ACx # 2x $ 2x # 5

3x $ 2x # 5x $ 5

AC $ 2x # 5 $ 2(5) # 5 $ 15

A x 2x CB

2x ! 5


ExercisesExercises

Find the measurement of each segment. Assume that the art is not drawn to scale.

1. R!T! 2. B!C!

3. X!Z! 4. W!X!

Find x and RS if S is between R and T.

5. RS $ 5x, ST $ 3x, and RT $ 48. 6. RS $ 2x, ST $ 5x # 4, and RT $ 32.

7. RS $ 6x, ST $12, and RT $ 72. 8. RS $ 4x, R!S! " S!T!, and RT $ 24.

Use the figures to determine whether each pair of segments is congruent.

9. A!B! and C!D! 10. X!Y! and Y!Z!X

Y Z

3x ! 5 5x " 1

9x2

A D

CB

11 cm

11 cm

5 cm5 cm

W Y

6 cm

XX ZY

3–4 in.31–2 in.

A C

6 in.

23–4 in. BR T2.0 cm 2.5 cm

S

Skills PracticeLinear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

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1-2


1. 2.


3. 40 feet 4. 12 centimeters 5. 9"12" inches

Find the measurement of each segment.

6. N!Q! 7. A!C! 8. G!H!

Find the value of the variable and YZ if Y is between X and Z.

9. XY $ 5p, YZ $ p, and XY $ 25 10. XY $ 12, YZ $ 2g, and XZ $ 28

11. XY $ 4m, YZ $ 3m, and XZ $ 42 12. XY $ 2c # 1, YZ $ 6c, and XZ $ 81


13. B!E!, C!D! 14. M!P!, N!P! 15. W!X!, W!Z!

Y Z

WX

9 ft

5 ft 5 ft

N

P

M 10 yd

12 yd

12 ydE D

CB

5 m

2 m

3 m 3 m

F 9.7 mm HG

15 mmA4.9 cm 5.2 cm

CBQ1in. 11–

4 in.

NP

1 2in.

1cm 2 3 54



1. 2.


3. 120 meters 4. 7"14" inches 5. 30.0 millimeters

Find the measurement of each segment.

6. P!S! 7. A!D! 8. W!X!

Find the value of the variable and KL if K is between J and L.

9. JK $ 6r, KL $ 3r, and JL $ 27 10. JK $ 2s, KL $ s # 2, and JL $ 5s ! 10


11. T!U!, S!W! 12. A!D!, B!C! 13. G!F!, F!E!

14. CARPENTRY Jorge used the figure at the right to make a pattern for a mosaic he plans to inlay on a tabletop. Name all of the congruent segments in the figure.

D

A

BF

E C

G H

EF

5x

6xD C

BA

12.9 in.

12.7 in.

W

T S

U

2 ft

2 ft

3 ft

3 ft

W 89.6 cm YX

100 cmA

11–4 in.23–

8 in.

DCP4.7 cm18.4 cm

SQ

1cm 2 3 541 2in.

E F

Practice Linear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Reading to Learn MathematicsLinear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

on

1-2

Pre-Activity Why are units of measure important?


• The basic unit of length in the metric system is the meter. How manymeters are there in one kilometer?

• Do you think it would be easier to learn the relationships between thedifferent units of length in the customary system (used in the UnitedStates) or in the metric system? Explain your answer.

Reading the Lesson

1. Explain the difference between a line and a line segment and why one of these can bemeasured, while the other cannot.

2. What is the smallest length marked on a 12-inch ruler?What is the smallest length marked on a centimeter ruler?

3. Find the precision of each measurement.a. 15 cmb. 15.0 cm

4. Refer to the figure at the right. Which one of the following statements is true? Explain your answer.A!B! $ C!D! A!B! " C!D!

5. Suppose that S is a point on V!W! and S is not the same point as V or W. Tell whethereach of the following statements is always, sometimes, or never true.a. VS $ SWb. S is between V and W.c. VS # VW $ SW


6. A good way to remember terms used in mathematics is to relate them to everyday wordsyou know. Give three words that are used outside of mathematics that can help youremember that there are 100 centimeters in a meter.

A

B

4.5 cm 4.5 cmD

C


Points Equidistant from SegmentsThe distance from a point to a segment is zero if the point is on the segment. Otherwise, it is the length of the shortest segment from the point to the segment.

A figure is a locus if it is the set of all points that satisfy a set of conditions. The locus of all points that are "

14" inch

from the segment AB is shown by two dashed segments with semicircles at both ends.

1. Suppose A, B, C, and D are four different points, and consider the locus of all points x units from A!B! and x units from C!D!. Use any unit you findconvenient. The locus can take different forms. Sketch at least threepossibilities. List some of the things that seem to affect the form of the locus.

2. Conduct your own investigation of the locus of pointsequidistant from two segments. Describe your results on aseparate sheet of paper.

A BC D

P Q

R S

A

B

C

D

X

Y

A B

C DX Y

A B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Study Guide and InterventionDistance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

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1-3

Distance Between Two Points

Distance on a Number Line Distance in the Coordinate Plane

AB $ |b ! a | or |a ! b |x

y

OC(1, –1)

A(–2, –1)

B(1, 3)

Pythagorean Theorem:

a2 # b2 $ c2

Distance Formula:

d $ #(x2 !!x1)2 #! (y2 !! y1)2!

A B

a b

Find AB.

AB $ |(!4) ! 2|$ |! 6|$ 6

"5 "4 "3 "2 "1 0 1 2 3

A B

Find the distance between A(!2, !1) and B(1, 3).


Pythagorean Theorem

(AB)2 $ (AC)2 # (BC)2

(AB)2 $ (3)2 # (4)2

(AB)2 $ 25

AB $ #25!$ 5

Distance Formula

d $ #(x2 !!x1)2 #! (y2 !! y1)2!AB $ #(1 ! (!!2))2!# (3 !! (!1))!2!AB $ #(3)2 #! (4)2!

$ #25!$ 5

Use the number line to find each measure.

1. BD 2. DG

3. AF 4. EF

5. BG 6. AG

7. BE 8. DE

Use the Pythagorean Theorem to find the distance between each pair of points.

9. A(0, 0), B(6, 8) 10. R(!2, 3), S(3, 15)

11. M(1, !2), N(9, 13) 12. E(!12, 2), F(!9, 6)

Use the Distance Formula to find the distance between each pair of points.

13. A(0, 0), B(15, 20) 14. O(!12, 0), P(!8, 3)

15. C(11, !12), D(6, 2) 16. E(!2, 10), F(!4, 3)

–10 –8 –6 –4 –2 0 2 4 6 8

A B C D E F G

ExercisesExercises


Midpoint of a Segment

Midpoint on a If the coordinates of the endpoints of a segment are a and b,

Number Line then the coordinate of the midpoint of the segment is "a #2

b".

Midpoint on a If a segment has endpoints with coordinates (x1, y1) and (x2, y2),

Coordinate Plane then the coordinates of the midpoint of the segment are $"x1 #

2x2", "

y1 #

2y2"%.

Find the coordinate of the midpoint of P!Q!.

The coordinates of P and Q are !3 and 1.

If M is the midpoint of P!Q!, then the coordinate of M is "!32# 1" $ "

!22" or !1.

M is the midpoint of P!Q! for P("2, 4) and Q(4, 1). Find thecoordinates of M.

M $ $"x1 #

2x2", "

y1 #

2y2"% $ $"!2

2# 4", "

4 #2

1"% or (1, 2.5)

Use the number line to find the coordinate of the midpoint of each segment.

1. C!E! 2. D!G!

3. A!F! 4. E!G!

5. A!B! 6. B!G!

7. B!D! 8. D!E!

Find the coordinates of the midpoint of a segment having the given endpoints.

9. A(0, 0), B(12, 8) 10. R(!12, 8), S(6, 12)

11. M(11, !2), N(!9, 13) 12. E(!2, 6), F(!9, 3)

13. S(10, !22), T(9, 10) 14. M(!11, 2), N(!19, 6)

–10 –8 –6 –4 –2 0 2 4 6 8

A B C D E F G

–3 –2 –1 0 1 2

P Q


Distance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeDistance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

on

1-3


1. LN 2. JL

3. KN 4. MN


5. 6.

7. K(2, 3), F(4, 4) 8. C(!3, !1), Q(!2, 3)


9. Y(2, 0), P(2, 6) 10. W(!2, 2), R(5, 2)

11. A(!7, !3), B(5, 2) 12. C(!3, 1), Q(2, 6)


13. D!E! 14. B!C!

15. B!D! 16. A!D!


17. T(3, 1), U(5, 3) 18. J(!4, 2), F(5, !2)

Find the coordinates of the missing endpoint given that P is the midpoint of N!Q!.

19. N(2, 0), P(5, 2) 20. N(5, 4), P(6, 3) 21. Q(3, 9), P(!1, 5)

–6 –4 –2 0 2 4 6 8 10 12

A B C D E

x

y

O

D

S

x

y

O

F

G

–6 –4 –2 0 2 4 6 8 10

J K L M N



1. VW 2. TV

3. ST 4. SV


5. 6.


7. L(!7, 0), Y(5, 9) 8. U(1, 3), B(4, 6)


9. R!T! 10. Q!R!

11. S!T! 12. P!R!


13. K(!9, 3), H(5, 7) 14. W(!12, !7), T(!8, !4)

Find the coordinates of the missing endpoint given that E is the midpoint of D!F!.

15. F(5, 8), E(4, 3) 16. F(2, 9), E(!1, 6) 17. D(!3, !8), E(1, !2)

18. PERIMETER The coordinates of the vertices of a quadrilateral are R(!1, 3), S(3, 3),T(5, !1), and U(!2, !1). Find the perimeter of the quadrilateral. Round to the nearest tenth.

–6 –4–10 –8 –2 0 2 4 6

P Q R S T

x

y

O

E

S

x

y

OM

Z

–6 –4–10 –8 –2 0 2 4 6 8

S T U V W

Practice Distance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

Reading to Learn MathematicsDistance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

on

1-3

Pre-Activity How can you find the distance between two points without a ruler?


• Look at the triangle in the introduction to this lesson. What is the specialname for A!B! in this triangle?

• Find AB in this figure. Write your answer both as a radical and as adecimal number rounded to the nearest tenth.

Reading the Lesson

1. Match each formula or expression in the first column with one of the names in thesecond column.

a. d $ #(x2 !!x1)2 #! ( y2 !! y1)2! i. Pythagorean Theorem

b. "a #

2b

" ii. Distance Formula in the Coordinate Plane

c. XY $ |a ! b| iii. Midpoint of a Segment in the Coordinate Plane

d. c2 $ a2 # b2 iv. Distance Formula on a Number Line

e. $"x1 #

2x2

", "y1 #

2y2

"% v. Midpoint of a Segment on a Number Line

2. Fill in the steps to calculate the distance between the points M(4, !3) and N(!2, 7).

Let (x1, y1) $ (4, !3). Then (x2, y2) $ ( , ).

d $ #!!!!( ! )2 # ( ! )2

MN $ #!!!!( ! )2 # ( ! )2

MN $ #!!( )2 # ( )2

MN $ #!!#

MN $ #!Find a decimal approximation for MN to the nearest hundredth.


3. A good way to remember a new formula in mathematics is to relate it to one you alreadyknow. If you forget the Distance Formula, how can you use the Pythagorean Theorem tofind the distance d between two points on a coordinate plane?


Lengths on a GridEvenly-spaced horizontal and vertical lines form a grid.

You can easily find segment lengths on a grid if the endpoints are grid-lineintersections. For horizontal or verticalsegments, simply count squares. For diagonal segments, use the PythagoreanTheorem (proven in Chapter 7). This theorem states that in any right triangle,if the length of the longest side (the sideopposite the right angle) is c and the twoshorter sides have lengths a and b, then c2 $ a2 # b2.

Find the measure of E!F! on the grid at the right. Locate a right triangle with E!F! as its longest side.

EF $ #22 # 5!2! $ #29! & 5.4 units

Find each measure to the nearest tenth of a unit.

1. I!J! 2. M!N! 3. R!S! 4. Q!S!

5. I!K! 6. J!K! 7. L!M! 8. L!N!

Use the grid above. Find the perimeter of each triangle to the nearest tenth of a unit.

9. ! ABC 10. !QRS 11. ! DEF 12. ! LMN

13. Of all the segments shown on the 14. On the grid, 1 unit $ 0.5 cm. How can the grid, which is longest? What is its answers above be used to find the measures length? in centimeters?

15. Use your answer from exercise 8 to 16. Use a centimeter ruler to find the perimeter calculate the length of segment LN of triangle IJK to the nearest tenth of a in centimeters. Check by measuring centimeter.with a centimeter ruler.

E

2

5 F

A

B

F

I

J

K N M

C

D

E

S

R

Q

L

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

ExampleExample

Study Guide and InterventionAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Measure Angles If two noncollinear rays have a common endpoint, they form an angle. The rays are the sides of the angle.The common endpoint is the vertex. The angle at the right can be named as "A, "BAC, "CAB, or "1.

A right angle is an angle whose measure is 90. An acute anglehas measure less than 90. An obtuse angle has measure greater than 90 but less than 180.

A C

B

1

a. Name all angles that have R as avertex.Three angles are "1, "2, and "3. Forother angles, use three letters to namethem: "SRQ, "PRT, and "SRT.

b. Name the sides of !1.

RS""#, RP""#

S R T

PQ

1 2 3

Measure each angle andclassify it as right, acute, or obtuse.

a. !ABDUsing a protractor, m"ABD $ 50.50 % 90, so "ABD is an acute angle.

b. !DBCUsing a protractor, m"DBC $ 115.180 & 115 & 90, so "DBC is an obtuseangle.

c. !EBCUsing a protractor, m"EBC $ 90."EBC is a right angle.

BA

D E

C


ExercisesExercises


1. Name the vertex of "4.

2. Name the sides of "BDC.

3. Write another name for "DBC.

Measure each angle in the figure and classify it as right,acute, or obtuse.

4. "MPR

5. "RPN

6. "NPS

P

NM

R

S

C

BA

12

34

D


Congruent Angles Angles that have the same measure are congruent angles. A ray that divides an angle into two congruent angles is called an angle bisector. In the figure, PN""# is the angle bisector of "MPR. Point N lies in the interior of "MPR and "MPN " "NPR.

Refer to the figure above. If m!MPN # 2x ! 14 and m!NPR # x ! 34, find x and find m!MPR.Since PN""# bisects "MPR, "MPN " "NPR, or m"MPN $ m"NPR.

2x # 14 $ x # 34 m"NPR $ (2x # 14) # (x # 34)2x # 14 ! x $ x # 34 ! x $ 54 # 54

x # 14 $ 34 $ 108x # 14 ! 14 $ 34 ! 14

x $ 20

QS!!" bisects !PQT, and QP!!" and QR!!" are opposite rays.

1. If m"PQT $ 60 and m"PQS $ 4x # 14, find the value of x.

2. If m"PQS $ 3x # 13 and m"SQT $ 6x ! 2, find m"PQT.

BA!!" and BC!!" are opposite rays, BF!!" bisects !CBE, and BD!!" bisects !ABE.

3. If m"EBF $ 6x # 4 and m"CBF $ 7x ! 2, find m"EBC.

4. If m"1 $ 4x # 10 and m"2 $ 5x, find m"2.

5. If m"2 $ 6y # 2 and m"1 $ 8y ! 14, find m"ABE.

6. Is "DBF a right angle? Explain.

BA C

F

ED

12 3

4

QP R

TS

P R

NM


Angle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

ExampleExample

ExercisesExercises

Skills PracticeAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

For Exercises 1–12, use the figure at the right.

Name the vertex of each angle.

1. "4 2. "1

3. "2 4. "5

Name the sides of each angle.

5. "4 6. "5

7. "STV 8. "1

Write another name for each angle.

9. "3 10. "4

11. "WTS 12. "2

Measure each angle and classify it as right, acute,or obtuse.

13. "NMP 14. "OMN

15. "QMN 16. "QMO

ALGEBRA In the figure, BA!!" and BC!!" are opposite rays,BD!!" bisects !EBC, and BF!!" bisects !ABE.

17. If m"EBD $ 4x # 16 and m"DBC $ 6x # 4,find m"EBD.

18. If m"ABF $ 7x ! 8 and m"EBF $ 5x # 10,find m"EBF.

B CA

FE

D

M NL

Q O

P

U

T

VW

S5 3

21

4


For Exercises 1–10, use the figure at the right.

Name the vertex of each angle.

1. "5 2. "3

3. "8 4. "NMP

Name the sides of each angle.

5. "6 6. "2

7. "MOP 8. "OMN

Write another name for each angle.

9. "QPR 10. "1

Measure each angle and classify it as right, acute,or obtuse.

11. "UZW 12. "YZW

13. "TZW 14. "UZT

ALGEBRA In the figure, CB!!" and CD!!" are opposite rays,CE!!" bisects !DCF, and CG!!" bisects !FCB.

15. If m"DCE $ 4x # 15 and m"ECF $ 6x ! 5,find m"DCE.

16. If m"FCG $ 9x # 3 and m"GCB $ 13x ! 9,find m"GCB.

17. TRAFFIC SIGNS The diagram shows a sign used to warn drivers of a school zone or crossing. Measure and classify each numbered angle. 2

1

B

C

G

F

ED

Z YT

U

V W

X

N

O

P Q

R

M12

8

3

54

6

7

Practice Angle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

Reading to Learn MathematicsAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Pre-Activity How big is a degree?

Read the introduction to Lesson 1-4 at the top of page 29 in your textbook.• A semicircle is half a circle. How many degrees are there in a

semicircle?• How many degrees are there in a quarter circle?

Reading the Lesson1. Match each description in the first column with one of the terms in the second column.

Some terms in the second column may be used more than once or not at all.a. a figure made up of two noncollinear rays with a 1. vertex

common endpoint 2. angle bisectorb. angles whose degree measures are less than 90 3. opposite raysc. angles that have the same measure 4. angled. angles whose degree measures are between 90 and 180 5. obtuse anglese. a tool used to measure angles 6. congruent anglesf. the common endpoint of the rays that form an angle 7. right anglesg. a ray that divides an angle into two congruent angles 8. acute angles

9. compass10. protractor

2. Use the figure to name each of the following.a. a right angleb. an obtuse anglec. an acute angled. a point in the interior of "EBCe. a point in the exterior of "EBAf. the angle bisector of "EBCg. a point on "CBEh. the sides of "ABFi. a pair of opposite raysj. the common vertex of all angles shown in the figurek. a pair of congruent anglesl. the angle with the greatest measure

Helping You Remember3. A good way to remember related geometric ideas is to compare them and see how they

are alike and how they are different. Give some similarities and differences betweencongruent segments and congruent angles.

B G

28$28$

A

EF

CD


Angle RelationshipsAngles are measured in degrees ('). Each degree of an angle is divid-ed into 60 minutes ((), and each minute of an angle is divided into 60 seconds ()).

60( $ 1'

60) $ 1(

67"12"' $ 67'30(

70.4' $ 70°24(

90' $ 89°60(

Two angles are complementary if the sum of their measures is 90$.Find the complement of each of the following angles.

1. 35'15( 2. 27'16( 3. 15'54(

4. 29'18(22) 5. 34'29(45) 6. 87'2(3)

Two angles are supplementary if the sum of their measures is 180$.Find the supplement of each of the following angles.

7. 120'18( 8. 84'12( 9. 110'2(

10. 45'16(24) 11. 39'21(54) 12. 129'18(36)

13. 98'52(59) 14. 9'2(32) 15. 1'2(3)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

Study Guide and InterventionAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Pairs of Angles Adjacent angles are angles in the same plane that have a commonvertex and a common side, but no common interior points. Vertical angles are twononadjacent angles formed by two intersecting lines. A pair of adjacent angles whosenoncommon sides are opposite rays is called a linear pair.

Identify each pair of angles as adjacent angles, vertical angles,and/or as a linear pair.

ExampleExample

a.

"SRT and "TRU have a commonvertex and a common side, but nocommon interior points. They areadjacent angles.

c.

"6 and "5 are adjacent angles whosenoncommon sides are opposite rays.The angles form a linear pair.

D

CBA

5 6

RU

TS

b.

"1 and "3 are nonadjacent angles formedby two intersecting lines. They are verticalangles. "2 and "4 are also vertical angles.

d.

"A and "B are two angles whose measureshave a sum of 90. They are complementary."F and "G are two angles whose measureshave a sum of 180. They are supplementary.

AB

F G

30$

60$

60$120$

N

RP

SM1

43

2

ExercisesExercises

Identify each pair of angles as adjacent, vertical, and/or as a linear pair.

1. "1 and "2 2. "1 and "6

3. "1 and "5 4. "3 and "2

For Exercises 5–7, refer to the figure at the right.

5. Identify two obtuse vertical angles.

6. Identify two acute adjacent angles.

7. Identify an angle supplementary to "TNU.

8. Find the measures of two complementary angles if the difference in their measures is 18.

R S

NU

T

V

R

S

TU

V

P

Q5

432

1 6


Perpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. The right angle symbol indicates that the lines are perpendicular. In the figure at the right, AC!"# is perpendicular to BD!"#,or AC!"# ! BD!"#.

Find x so that D!Z! ! P!Z!.If D!Z! ! P!Z!, then m"DZP $ 90.

m"DZQ # m"QZP $ m"DZP Sum of parts $ whole

(9x # 5) # (3x # 1) $ 90 Substitution

12x # 6 $ 90 Simplify.12x $ 84 Subtract 6 from each side.

x $ 7 Divide each side by 12.

1. Find x and y so that NR!"# ! MQ!"#.

2. Find m"MSN.

3. m"EBF $ 3x # 10, m"DBE $ x, and BD""# ! BF""#. Find x.

4. If m"EBF $ 7y ! 3 and m"FBC $ 3y # 3, find y so that EB""# ! BC""#.

5. Find x, m"PQS, and m"SQR.

6. Find y, m"RPT, and m"TPW.

P

S

V

R

W

T(4y " 5)$

(2y ! 5)$

Q R

P S3x $

(8x ! 2)$

B CA

DE

F

M

N

R

S Q

Px $5x $

(9y ! 18)$

Z

D

P

Q(9x ! 5)$

(3x ! 1)$

B

CD

A


Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

ExampleExample

ExercisesExercises

Skills PracticeAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

For Exercises 1–6, use the figure at the right and a protractor.

1. Name two acute vertical angles.

2. Name two obtuse vertical angles.

3. Name a linear pair.

4. Name two acute adjacent angles.

5. Name an angle complementary to "EKH.

6. Name an angle supplementary to "FKG.

7. Find the measures of an angle and its complement if one angle measures 18 degreesmore than the other.

8. The measure of the supplement of an angle is 36 less than the measure of the angle.Find the measures of the angles.

ALGEBRA For Exercises 9–10, use the figure at the right.

9. If m"RTS $ 8x # 18, find x so that TR""# ! TS""#.

10. If m"PTQ $ 3y ! 10 and m"QTR $ y, find y so that "PTR is a right angle.

Determine whether each statement can be assumed from the figure. Explain.

11. "WZU is a right angle.

12. "YZU and "UZV are supplementary.

13. "VZU is adjacent to "YZX.

ZX

W

V

YU

TP

QR

S

K

G

J

FE

H


For Exercises 1–4, use the figure at the right and a protractor.


2. Name a linear pair whose vertex is B.

3. Name an angle not adjacent to but complementary to "FGC.

4. Name an angle adjacent and supplementary to "DCB.

5. Two angles are complementary. The measure of one angle is 21 more than twice themeasure of the other angle. Find the measures of the angles.

6. If a supplement of an angle has a measure 78 less than the measure of the angle, whatare the measures of the angles?

ALGEBRA For Exercises 7–8, use the figure at the right.

7. If m"FGE $ 5x # 10, find x so that FC!"# ! AE!"#.

8. If m"BGC $ 16x ! 4 and m"CGD $ 2x # 13,find x so that "BGD is a right angle.

Determine whether each statement can be assumed from the figure. Explain.

9. "NQO and "OQP are complementary.

10. "SRQ and "QRP is a linear pair.

11. "MQN and "MQR are vertical angles.

12. STREET MAPS Darren sketched a map of the cross streets nearest to his home for his friend Miguel. Describe two different anglerelationships between the streets. Olive

Beac

on

Main

Q

O

P

RS

M

N

GF

AB

E

D

C

DAE

HG

B C

F

Practice Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

Reading to Learn MathematicsAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Pre-Activity What kinds of angles are formed when streets intersect?

Read the introduction to Lesson 1-5 at the top of page 37 in your textbook.• How many separate angles are formed if three lines intersect at a common

point? (Do not use an angle whose interior includes part of another angle.)• How many separate angles are formed if n lines intersect at a common

point? (Do not count an angle whose interior includes part of another angle.)

Reading the Lesson1. Name each of the following in the figure at the right.

a. two pairs of congruent anglesb. a pair of acute vertical anglesc. a pair of obtuse vertical anglesd. four pairs of adjacent anglese. two pairs of vertical anglesf. four linear pairsg. four pairs of supplementary angles

2. Tell whether each statement is always, sometimes, or never true.a. If two angles are adjacent angles, they form a linear pair.b. If two angles form a linear pair, they are complementary.c. If two angles are supplementary, they are congruent.d. If two angles are complementary, they are adjacent.e. When two perpendicular lines intersect, four congruent angles are formed.f. Vertical angles are supplementary.g. Vertical angles are complementary.h. The two angles in a linear pair are both acute.i. If two angles form a linear pair, one is acute and the other is obtuse.

3. Complete each sentence.a. If two angles are supplementary and x is the measure of one of the angles, then the

measure of the other angle is .b. If two angles are complementary and x is the measure of one of the angles, then the

measure of the other angle is .

Helping You Remember4. Look up the nonmathematical meaning of supplementary in your dictionary. How can

this definition help you to remember the meaning of supplementary angles?

65$ 2 34

1


Curve StitchingThe star design at the right was created by a method known as curve stitching. Although the design appears to contain curves, it is made up entirely of line segments.

To begin the star design, draw a 60° angle. Mark eight equally-spaced points on each ray, and number the points as shown below. Then connect pairs of points that have the same number.

To make a complete star, make the same design in six 60° angles that have a common central vertex.

1. Complete the section of the star design above by connecting pairs of points that have the same number.

2. Complete the following design.

3. Create your own design. You may use several angles, and the angles may overlap.

1 2 3 4 5 6 7 8 9

11

11

12

13

14

15

16

17

18

19

1

2

3

4

5

6

7

8

9

12 13 14 15 16 17 18 19

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

Study Guide and InterventionPolygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Polygons A polygon is a closed figure formed by a finite number of coplanar linesegments. The sides that have a common endpoint must be noncollinear and each sideintersects exactly two other sides at their endpoints. A polygon is named according to itsnumber of sides. A regular polygon has congruent sides and congruent angles. A polygoncan be concave or convex.

Name each polygon by its number of sides. Then classify it asconcave or convex and regular or irregular.

ExampleExample

a.

The polygon has 4 sides, so it is a quadrilateral.It is concave because part of D!E! or E!F! lies in theinterior of the figure. Because it is concave, itcannot have all its angles congruent and so it isirregular.

c.

The polygon has 5 sides, so it is a pentagon. It isconvex. All sides are congruent and all angles arecongruent, so it is a regular pentagon.

D F

G

E b.

The figure is not closed, so it isnot a polygon.

d.

The figure has 8 congruent sidesand 8 congruent angles. It isconvex and is a regular octagon.

H

J K

LI

ExercisesExercises

Name each polygon by its number of sides. Then classify it as concave or convexand regular or irregular.

1. 2. 3.

4. 5. 6.


Perimeter The perimeter of a polygon is the sum of the lengths of all the sides of thepolygon. There are special formulas for the perimeter of a square or a rectangle.

Write an expression or formula for the perimeter of each polygon.Find the perimeter.


Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

ExampleExample

a.

P $ a # b # c$ 3 # 4 # 5$ 12 in.

b.

P $ 4s$ 4(5)$ 20 cm

c.

P $ 2! # 2w$ 2(3) # 2(2)$ 10 ft

3 ft

2 ft

!

!

w w

5 cm

5 cm 5 cm

5 cm

s

s

s s3 in.5 in.

4 in.

a

b

c

ExercisesExercises

Find the perimeter of each figure.

1. 2.

3. 4.

Find the length of each side of the polygon for the given perimeter.

5. P $ 96 6. P $ 48x

x

2x

x " 2

rectangle

2x

x

1 cm

24 yd

19 yd

12 yd 14 yd

27 yd

square

5.5 ft

3.5 cm

3 cm2.5 cm

Skills PracticePolygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Name each polygon by its number of sides and then classify it as convex orconcave and regular or irregular.

1. 2. 3.

4. 5. 6.


7. 8. 9.

COORDINATE GEOMETRY Find the perimeter of each polygon.

10. triangle ABC with vertices A(3, 5), B(3, 1), and C(0, 1)

11. quadrilateral QRST with vertices Q(!3, 2), R(1, 2), S(1, !4), and T(!3, !4)

12. quadrilateral LMNO with vertices L(!1, 4), M(3, 4), N(2, 1), and O(!2, 1)

ALGEBRA Find the length of each side of the polygon for the given perimeter.

13. P $ 104 millimeters 14. P $ 84 kilometers 15. P $ 88 feet4w " 1

w

10 in.

10 in.2 in. 2 in.

2 in.2 in.2 in. 2 in.

5 m2 m 3 m

6 m4 m

40 yd

20 yd

18 yd 20 yd


Name each polygon by its number of sides and then classify it as convex orconcave and regular or irregular.

1. 2. 3.


4. 5. 6.

COORDINATE GEOMETRY Find the perimeter of each polygon.

7. quadrilateral OPQR with vertices O(!3, 2), P(1, 5), Q(6, 4), and R(5, !2)

8. pentagon STUVW with vertices S(0, 0), T(3, !2), U(2, !5), V(!2, !5), and W(!3, !2)

ALGEBRA Find the length of each side of the polygon for the given perimeter.

9. P $ 26 inches 10. P $ 39 centimeters 11. P $ 89 feet

SEWING For Exercises 12–13, use the following information.Jasmine plans to sew fringe around the scarf shown in the diagram.

12. How many inches of fringe does she need to purchase?

13. If Jasmine doubles the width of the scarf, how many inches of fringe will she need?

16 in.

4 in. 4 in.

16 in.

2x ! 2

x ! 9

5x " 4

3x ! 52x " 3

6n " 8

n

14 cm

14 cm

4 cm

4 cm 6 cm6 cm

6 cm2 cm

32 mi

33 mi21 mi

7 mm

10 mm18 mm

18 mm

Practice Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Reading to Learn MathematicsPolygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Pre-Activity How are polygons related to toys?


Name four different shapes that can each be formed by four sticks connected toform a closed figure. Assume you have sticks with a good variety of lengths.

Reading the Lesson

1. Tell why each figure is not a polygon.a. b. c.

2. Name each polygon by its number of sides. Then classify it as convex or concave andregular or not regular.a. b. c.

3. What is another name for a regular quadrilateral?

4. Match each polygon in the first column with the formula in the second column that canbe used to find its perimeter. (s represents the length of each side of a regular polygon.)

a. regular dodecagon i. P $ 8s

b. square ii. P $ 6s

c. regular hexagon iii. P $ a # b # c

d. rectangle iv. P $ 12s

e. regular octagon v. P $ 2! # 2w

f. triangle vi. P $ 4s


5. One way to remember the meaning of a term is to explain it to another person.How would you explain to a friend what a regular polygon is?


Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Perimeter and Area of Irregular ShapesTwo formulas that are used frequently in mathematics are perimeter andarea of a rectangle.

Perimeter: P $ 2! # 2wArea: A $ !w, where ! is the length and w is the width

However, many figures are combinations of two or more rectangles creatingirregular shapes. To find the area of an irregular shape, it helps to separatethe shape into rectangles, calculate the formula for each rectangle, then findthe sum of the areas.

Find the area of thefigure at the right.Separate the figure into two rectangles.

A $ !wA1 $ 9 * 2 A2 $ 3 * 3

$ 18 $ 9

18 # 9 $ 27

The area of the irregular shape is 27 m2.

Find the area and perimeter of each irregular shape.

1. 2.

3.4.

For Exercises 5–8, find the perimeter of the figures in Exercises 1–4.

5. 6. 7. 8.

9. Describe the steps you used to find the perimeter in Exercise 1.

9 ft

2 ft

3 ft

7 ft

6 ft

4 ft

8 cm

4 cm

2 cm

2 cm

4 cm

4 cm

6 cm

4 cm

9 m

26 m

6 m13 m7 m

12 m

2 in.

4 in. 4 in.

1 in.

9 m

3 m

1

25 m

2 m

9 m

3 m

5 m

2 m

ExampleExample

Chapter 1 Test, Form 111


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of each question.

1. Name the geometric shape modeled by a pinhole in a wall.A. line segment B. plane C. line D. point

For Questions 2–4, use the figure given at the right.

2. Which is another name for line !?A. AB!"# B. BD!"#

C. C D. P

3. Name the intersection of lines ! and m.A. A B. BC. C D. P

4. Name three points coplanar with point A.A. B, C, F B. E, F, GC. B, C, E D. B, D, G

5. Find the length of R!S!.A. 33 mm B. 34 mmC. 35 mm D. 36 mm

6. Find the precision for a measurement of 72 centimeters.A. 0.5 cm B. 0.1 cmC. 1 mm D. 0.5 mm

7. Find the length of B!C!.A. 12 cm B. 13 cmC. 25 cm D. 38 cm

8. Use the number line to find MN.A. !5 B. 1C. 5 D. 10

For Questions 9 and 10, use the figure given at the right.

9. Find the distance between points P and Q.A. 5 B. 7C. 9 D. 25

10. Find the coordinates of the midpoint of P!Q!.

A. "2, 3"12"# B. "0, 3"

12"# C. (0, 3) D. "3"

12", 0#

x

y

O

P

Q

!5 !4 !3 !2 !1 0 1 2 3

M N

A C

25 cm

13 cm B

1cm 2 3 4

R S

!

P

m

AB

C

DE

G

F

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

NAME DATE PERIOD


Chapter 1 Test, Form 1 (continued)11

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

For Questions 11–13, use the figure at the right.

11. Which point is the vertex of all the angles in this figure?A. A B. BC. C D. E

12. What type of angle is !ABC?A. acute angle B. right angle C. obtuse angle D. straight angle

13. Which is true?A. m!EBF # 140 B. m!EBF # 90 C. m!EBF # 50 D. m!EBF # 40

14. For what value of x is !ATK $ !MJS if m!ATK # 5x $ 4 and m!MJS # 8x ! 11?A. 29 B. 15 C. 10 D. 5


15. Which pair of angles are vertical angles?A. !RST, !TSU B. !RSX, !TSUC. !TSU, !USV D. !RSX, !XSW

16. Which angle is supplementary to !USV ?A. !TSU B. !VSW C. !RSV D. !WSR

17. Find x and y.A. x # 10, y # 12 B. x # 20, y # 7 C. x # 10, y # 8 D. x # 50, y # 40

For Questions 18–20, use the figures below.

18. Which figure is not a polygon?A. Figure A B. Figure B C. Figure C D. Figure D

19. Find the perimeter of the convex pentagon.A. 46 cm B. 50 cm C. 61 cm D. 72 cm

20. Suppose the length and width of the rectangle are doubled. What is itsperimeter?A. 120 cm B. 92 cm C. 76 cm D. 46 cm

Bonus Each side of a square is 2x ! 6 yards long. If the perimeter of the square is 72 yards what is the value of x?

8 cm

8 cm

8 cm

4 cm

15 cm15 cm

18 cm

18 cm

25 cm

15 cm

8 cm8 cm15 cm

12 cm

15 cm12 cmFigure A Figure DFigure B Figure C

25 cm

SRX W V

U

T

5x "4x "

(10y # 10)"

B DA

C

EF

50"

B:

Chapter 1 Test, Form 2A11


Ass

essm

ents


1. How many planes can be drawn through any three noncollinear points?A. 0 B. 1 C. 2 D. 3

For Questions 2 and 3, use the figure at the right.

2. Which three points in the figure are collinear?A. A, B, D B. E, C, AC. A, B, C D. F, E, G

3. Name the intersection of the plane P and the plane that contains points B, C, and D.A. point B B. B!D!C. BC!"# D. triangle BCD

4. Find the length of R!S!.

A. 1"156" in. B. 1"

38" in.

C. 1"176" in. D. 1"

58" in.

5. Find the precision for a measurement of 18"12" feet.

A. "12" ft B. "

14" ft C. "

18" ft D. 1 in.

6. Find the length of P!Q!.A. 50.9 cm B. 46.3 cmC. 25.7 cm D. 21.3 cm

7. Find y if B is between A and C, AB is 2y, BC is 6y, and AC is 48.A. 24 B. 8 C. 6 D. 4

8. Find the distance between P(2, 8) and Q(5, 3).A. 9 B. %18! C. %34! D. %170!

9. Find the coordinates of the midpoint of L!B! if L(8, 5) and B(!6, 2).

A. "1, 3"12"# B. "2, 1"

12"# C. "7, 3"

12"# D. "7, 1"

12"#

10. Find the coordinates of T given that S is the midpoint of R!T!, R(!4, 2),and S(6, 8).A. (!14, !4) B. (16, 14) C. (2, 10) D. (1, 5)

P R

38.3 cm

12.6 cmQ

1 2in.

R S

PAB

CE

F

D

G

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 1 Test, Form 2A (continued)11

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.


11. What type of angle is !ABC?A. acute angle B. right angleC. obtuse angle D. straight angle

12. Use a protractor to measure the angles in the figure.Which segment is an angle bisector?A. G!E! B. B!C! C. E!D! D. E!F!


13. Find m!FBD if !FBD and !DBE are complementary and m!FBD is twice m!DBE.A. 30 B. 45C. 60 D. 90

14. Which pair of angles are supplementary?A. !ABE, !CBD B. !ABC, !ABD C. !ABC, !CBD D. !ABC, !EBD

15. Which angle is a vertical angle to !ABE?A. !DBE B. !CBD C. !ABC D. !EBA

16. If m!CBF # 6x $ 18, find x so that CB ! BF.A. 90 B. 45 C. 18 D. 12

17. Find m!ABC if m!ABC # 4x $ 9 and m!EBD # 7x ! 9.A. 6 B. 33 C. 45 D. 73


18. Which describes this figure?A. hexagon, concave, not regularB. pentagon, concave, regularC. hexagon, convex, not regularD. not a polygon

19. What is x for a perimeter of 108 kilometers?A. 53 B. 15 C. 18 D. 105

20. A rectangle has a length of 1.4 feet and a width of 1.2 feet. What is the effecton the perimeter of this rectangle if the length and width are doubled?A. The perimeter is doubled. B. The perimeter is increased by 8.C. The perimeter is multiplied by 4. D. The perimeter is tripled.

Bonus Find m!A if !A is complementary to !B, !B is supplementary to !C, m!B # 15x ! 2, and m!C # 25x $ 22.

(x # 3) km

F

DE

B

AC

A B

C

DE

F

G

B:

NAME DATE PERIOD

Chapter 1 Test, Form 2B11


Ass

essm

ents


1. Suppose A and B are points. How many lines contain both A and B?A. 0 B. 1C. 2 D. 3


2. Which three points in the figure are collinear?A. C, D, F B. B, C, DC. A, E, F D. A, D, E

3. Name the intersection of the plane that containspoints A, B, and D and the plane P.A. point D B. A!D!C. triangle BCD D. BD!"#

4. Find the length of X!Y!.

A. 1"1116" in. B. 1"

58" in.

C. 1"196" in. D. 1"

12" in.

5. Find the precision for a measurement of 34.0 centimeters.A. 0.5 cm B. 1 mm C. 0.5 mm D. 1 cm

6. Find the length of H!J!.A. 11.3 cm B. 12.3 cmC. 13.7 cm D. 45.9 cm

7. Find x if S is between R and T, RS is x $ 3, ST is 5x, and RT is 57.A. 9 B. 10 C. 10.8 D. 12

8. Find the distance between M(!2, 3) and N(8, 2).

A. 8 B. %61! C. 10 D. %101!

9. Find the coordinates of the midpoint of A!S! if A(!4, 7) and S(5, 3).

A. (1, 10) B. "!4"12", 2# C. ""

12", 5# D. "!"

12", 5#

10. Find the coordinates of T given that S is the midpoint of R!T!, R(2, 6),and S(!2, 0).A. (6, 12) B. (!6, !6) C. (0, 3) D. (!2, 3)

G J

29.1 cm

16.8 cm H

1 2in.

X Y

PCB

D

E

F

A

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 1 Test, Form 2B (continued)11

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.


11. What type of angle is !BAC?A. acute angle B. right angleC. obtuse angle D. straight angle

12. Use a protractor to measure the angles in the figure.Which segment is an angle bisector?A. A!B! B. C!D! C. C!B! D. A!E!


13. Find m!VSW if !WSR and !VSW are complementaryand m!WSR is four times m!VSW.A. 72 B. 36C. 22.5 D. 18

14. Which pair of angles are supplementary?A. !USV, !VSW B. !VSW, !WSR C. !TSV, !VSW D. !TSR, !USW

15. Which angle is a vertical angle to !UST?A. !VSW B. !USV C. !TSR D. !WSR

16. If m!VSR # 8x $ 18, find x so that U!S! ! V!S!.A. 9 B. 12.25 C. 72 D. 90

17. Find m!USW if m!USW # 7x ! 34 and m!TSR # 4x $ 29.A. 147 B. 113 C. 84 D. 21


18. Which describes this figure?A. hexagon, convex, regularB. pentagon, concave, regularC. pentagon, convex, not regularD. not a polygon

19. What is y for a perimeter of 100 feet?A. 5 B. 15 C. 17 D. 23

20. A square has sides with a length of 5.8 inches. What is the effect on theperimeter of this square if the sides are tripled?A. The perimeter stays the same. B. The perimeter is increased by 12.C. The perimeter is multiplied by 3. D. The perimeter is multiplied by 9.

Bonus Find m!A if !A is supplementary to !B, !B is supplementary to !C, m!B # 12x $ 8, and m!C # 8x ! 8.

y # 5

y

R

T

U

V W

S

FE

D

CA

B

B:

NAME DATE PERIOD

Chapter 1 Test, Form 2C11


Ass

essm

ents


1. What is another name for line !?

2. Name three points on plane P.

3. Name the intersection of planes P and N.

4. Name three noncoplanar points.

For Questions 5 and 6,use the figure at theright.

5. What is the length ofA!B!?

6. What is the precision of your measurement of A!B!?

7. Find the length of D!E! if D is between points C and E,CD # 6.5 centimeters, and CE # 13.8 centimeters.

8. Find the length of X!Z!.

9. Find x if R!S! $ S!T!.

For Questions 10–12, use the coordinate grid.

10. Find the distance between A and B.

11. Find the coordinates of the midpoint of C!D!.

12. Find the coordinates of a point E if Cis the midpoint of A!E!.

13. The vertices of a triangle are located at P(0, 0), Q(8, 6), andR(!3, 4). What is the perimeter of this triangle?

14. Find x and y if U!V! bisects T!W! and UV # 40.

ZT W

U

V

3y ! 1

2y # 6

3y # 13x ! 2

x

y

O

B

D

A

C

R T

52 in.

6x # 8 S

X Z

4x # 3

2x # 78 cm Y

1cm 2 3 4 5 6

A B

!

P

N

CA B

DE

F

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

NAME DATE PERIOD

SCORE


Chapter 1 Test, Form 2C (continued)11

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

15. Measure !PQR. Then classify !PQRas right, acute, or obtuse.

In the figure, EA!!" and EB!!" are opposite rays and EC!!" bisects !FEG.

16. Find x if m!FEG # 82, and m!FEC # 5x $ 11.

17. If m!AED # 16y $ 10, find y so that E!D! ! A!B!.


18. Find y.

19. Find m!1.

20. Find m!2.

21. Find x .

For Questions 22–25, use the polygons at the right.

22. Name polygon ABCDEF by its sides. Then classify it asconvex or concave and regular or not regular.

23. Find the perimeter of polygon ABCDEF for x # 4.

24. Find the length of each side of polygon RST.

25. Find the length of one side of a regular pentagon whoseperimeter is the same as the perimeter of RST.

Bonus Find the dimensions of a rectangle whose length is 3 more than twice its width and has a perimeter of 30 centimeters.

6x # 53y ! 1

2y # 11

A D

B C R

F E T S

40"

72"1

2

(11x ! 24)"

(8y ! 16)"

BAD

GC

F

E

R

P

Q

NAME DATE PERIOD

Chapter 1 Test, Form 2D11


Ass

essm

ents


1. What is another name for line m?

2. Name three points on plane B.

3. Name the intersection of planes A and B.

4. Name three noncollinear points.


5. What is the length of Q!R!?

6. What is the precision of your measurement of Q!R!?

7. Find the length of L!O! if O is between points L and M,LM # 18.6 centimeters, and OM # 12.9 centimeters.

8. Find the length of D!E!.

9. Find y if X!Y! $ Y!Z!.

For Questions 10–12, use the coordinate grid.

10. Find the distance between Land M.

11. Find the coordinates of the midpoint of M!N!.

12. Find the coordinates of a point Q if P is the midpoint of N!Q!.

13. The vertices of a triangle are located at P(0, 6), Q(8, 12), andR(3, !3). What is the perimeter of this triangle?

14. Find x if R!S! bisects A!B! and RS # 36.

TA B

R

S

18

2y ! 6

2y ! 525 ! 3x

x

y

O

M

NPL

X Z

5y # 11

23 in. Y

D F

24 cm

5x ! 133x # 5 E

1 2in.

Q R

A

B

m

T

R

SU

YZ

X

V

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

NAME DATE PERIOD

SCORE


Chapter 1 Test, Form 2D (continued)11

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

15. Measure !ABC. Then classify !ABC as right, acute, or obtuse.

In the figure, RC!!" and RD!!" are opposite rays and RQ!!" bisects !WRV.

16. Find y if m!WRQ # 48 and m!QRV # 7y $ 6.

17. Find x so that C!R! ! P!R!.


18. Find x.

19. Find m!1.

20. Find m!2.

21. Find y.

For Questions 22–25, use the polygons at the right.

22. Name polygon RSTUVby its sides. Then classify it as convex orconcave and regular or not regular.

23. Find the perimeter of polygon RSTUV for y # 9.

24. Find the length of each side of polygon ABCD.

25. Find the length of the sides of a regular triangle whoseperimeter is the same as the perimeter of ABCD.

Bonus Find the lengths of the sides of a triangle whose perimeter is 37. The measure of the first side of thetriangle is 8 less than the second side, and the second side is twice the length of the third side.

5y ! 8

8x # 3

27 # 4x

S

R T

U D C

A B

V

(9x ! 5)" 58"1

2(7y # 27)"

DR

P

C

WQ V

(13x # 12)"

C

A

B

NAME DATE PERIOD

Chapter 1 Test, Form 311


Ass

essm

ents


1. Name five planes shown in the figure.

2. Name a line that is coplanar with AD!"# and AB!"#.

3. Name the intersection of plane P and the plane that containspoints A, B, and E.

For Questions 4 and 5, usethe figure at the right.

4. Find the length of A!B!.

5. Find the precision for the measurement of A!B!.

6. Find two possible lengths for C!D! if C, D, and E are collinear,CE # 15.8 centimeters, and DE # 3.5 centimeters.

7. Find the length of R!S! if S is between R and T, the length of R!S! is "

13" the length of R!T!, RS # 3x ! 3, and ST # 2x $ 6.

8. Find y if AC # 3y $ 5, CB # 4y ! 1, AB # 9y ! 12, and pointC lies between A and B.

For Questions 9–11, use the coordinate grid at the right.

9. Find the distance between A and B.

10. Find two possible coordinates of a point D on a line containing A!B! so that AD # "

14"AB.

11. Find two values of y for C located at (1, y) and AC # 5.

x

y

O

B

A

6 7in.

A B

P

A

E

F

D

C

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 1 Test, Form 3 (continued)11

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

12. Find y if S is the midpoint of R!T!, T is the midpoint of R!U!,RS # 6x $ 5, ST # 8x ! 1, and TU # 11y $ 13.

13. Find all values of x that will make !A an obtuse angle givenm!A # 12x ! 6.

14. Find m!RST if ST""# bisects !RSU and SU""# bisects !TSV.

15. Find m!1 if !1 is complementary to !2, !2 is supplementaryto !3, and m!3 # 126.

16. Find y if XW""# ! XZ""#, Y is in the interior of !WXZ,m!WXY # 6y ! 3, and m!YXZ # 4y $ 13.

17. Find the length of L!M! if ON!"# is the bisector of L!M! and LN # 3x $ 2.

For Questions 18 and 19, use the coordinate grid.

18. Graph polygon ABCD with vertices A(4, 3), B(0, 3), C(!2, 2),and D(!5, 6). Then name polygon ABCD by its number ofsides and classify it as convex or concave and regular orirregular.

19. Find the perimeter of polygon ABCD.

20. Find the perimeter of regular triangle DEF if DE # 28 ! 3yand EF # 2y $ 3.

Bonus Suppose a regular quadrilateral and a regular triangle havethe same perimeter. The sides of the triangle are 3 incheslonger than the sides of the quadrilateral. Find the lengthsof the sides of the quadrilateral and the triangle.

N ML

O

7x ! 1

V

U

T

R

S

(x # 2y ! 1)"

(6x ! 9)"

(2y # 5)"

NAME DATE PERIOD

Chapter 1 Open-Ended Assessment11


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Draw and label a figure that shows that plane R contains both lines s andAC!"# that intersect at point B. Name three collinear points in plane R.

2. Draw a line on a coordinate plane so that you can determine at least twopoints on the graph. Label those two points D and G.

a. Find the distance between points D and G.

b. Find the coordinates of E, the midpoint of D!G!.

c. Find the coordinates of point H given that G is the midpoint of D!H!.

3. Rectangle WXYZ has a length that is 5 more than three times its width.

a. Draw and label a figure for rectangle WXYZ.

b. Write an algebraic expression for the perimeter of the rectangle.

c. Find the width if the perimeter is 58 millimeters. Explain how you cancheck that your answer is correct.

d. Use a ruler to draw and label P!Q!, which is congruent to the segmentrepresenting the length of rectangle WXYZ. What is the measure of P!Q!?

e. Explain how to find the precision of the measurement of P!Q!.

4. Draw an acute angle, !ABC. Let m!ABC # 6x ! 1.

a. Use a protractor to determine the measure of !ABC. Use thismeasure to determine the value of x.

b. Explain how you would determine the measure of an angle that iscomplementary to !ABC.

c. Explain how you would determine the measure of an angle that issupplementary to !ABC.

5. RS""# is in the interior of !TRU, m!TRS # 4x $ 6, and m!SRU # 8x ! 6.

a. Draw !TRU and RS""#.

b. Determine the value of x that will make RS""# an angle bisector. Explainyour steps.

c. Describe the relationship between RU""# and RT""# when x # 7.5.

NAME DATE PERIOD

SCORE


Chapter 1 Vocabulary Test/Review11

Choose from the terms above to complete each sentence.

1. Two lines are if they intersect to form a right angle.

2. Two angles are if their measures have a sum of 90°.

3. When two rays intersect with a common endpoint a(n) is formed.

4. The is the point located halfway between theendpoints of a segment.

5. are nonadjacent angles formed by the intersection oftwo lines.

6. A(n) divides an angle into two congruent angles.

7. Two angles are if their measures have a sum of 180°.

8. Two angles that lie in the same plane are called ifthey share a common side and a common vertex.

9. A(n) is an angle whose measure is less than 90°.

10. Two segments are if they have the same measure.

In your own words—

11. Explain how to find the precision of a measurement of 5"

12" inches on a ruler marked in half inches.

12. Describe what is meant by betweenness of points usingcollinear points M, P, and Q.

?

?

?

?

?

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

acute angleadjacent anglesangleangle bisectorbetweenness collinearcomplementary concavecongruentconstruction

convexcoplanardegreedistanceexteriorinteriorlineline segmentlinear pairmidpoint

n-gonobtuse angleopposite raysperimeterperpendicularplanepointpolygonprecisionray

regular polygonrelative errorright anglesegment bisectorsidesspacesupplementaryundefined termsvertexvertical angles

NAME DATE PERIOD

SCORE

Chapter 1 Quiz (Lessons 1–1 and 1–2)

11


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


1. What is another name for line !?

2. Name the intersection of lines ! and m.



4. Find the length of A!B!.

5. Find the precision of the measurement of A!B!.

6. Find the length of U!W! if W is between U and V,UV # 16.8 centimeters, and VW # 7.9 centimeters.

7. Find x if RS # 24 centimeters.

8. Find the length of L!O! if M is between L and O, LM # 7x ! 9,MO # 14 inches, and LO # 10x ! 7.

9. Find x if P!Q! $ R!S!, PQ # 9x ! 7, and RS # 29.

10. STANDARDIZED TEST PRACTICE Which of the following isnot an undefined term in geometry?A. plane B. point C. bisector D. line

R S

10 cm6x ! 4

T

1in.

A B

R T

SU

m

!V

Chapter 1 Quiz (Lesson 1–3)

11

1.

2.

3.

4.

5.

1. Find the coordinates of the midpoint of A!B! for A(2, 5) and B(6, 9).

2. Find the coordinates of D if E is the midpoint of C!D!, for C(!3, 4) and E(0, 1).

3. What is the length of F!H! if G is the midpoint, FG # 12x ! 5,and GH # 7x?

4. What is the length of U!V! if WX!"# is the segment bisector of U!V! at point Z?

5. STANDARDIZED TEST PRACTICE Find the distance betweenA(!2, 1) and B(4, !3).A. 52 B. %52! C. %20! D. %8!

U

V9x # 2

6x # 8Z

X

W

NAME DATE PERIOD

SCORE


Chapter 1 Quiz (Lessons 1–4 and 1–5)

11

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


1. Name the vertex of !1.

2. Classify !TSV as right, acute, or obtuse.

3. Name a point in the exterior of !RTS.

4. Find m!TSU if S!U! bisects !TSV,m!TSU # 4y $ 11, and m!USV # 6y $ 5.

5. Name a pair of adjacent angles.

6. Name a pair of vertical angles.

7. Find m!DBC if m!ABC # 5x $ 3 and !ABD $ !DBC.

For Questions 8 and 9, lines p and q intersect to formadjacent angles 1 and 2.

8. If m!1 # 7x $ 6 and m!2 # 8x ! 6, find x so that p isperpendicular to q.

9. If m!1 # 4x ! 3 and m!2 # 3x $ 8, find x so that !1 issupplementary to !2.

10. STANDARDIZED TEST PRACTICE The difference betweentwo complementary angles is 14. Which is the measure of oneof those angles?A. 14 B. 52 C. 83 D. 90

(3x ! 2)"B

A

DC

ST

U V

W

PR

1

NAME DATE PERIOD

SCORE

Chapter 1 Quiz (Lesson 1–6)

11

1.

2.

3.

4.

5.

1. Draw a concave pentagon.

2. Find the length of each side of a regular hexagon whoseperimeter is 84 meters.

3. If x # 5, find the perimeter of the rectangle whose length is 6x ! 4 and whose width is 3x $ 2.

4. The perimeter of a convex pentagon is 15 feet. What is theeffect on its perimeter if each side is doubled?

5. For what value of y is triangle ABC a regular triangle?

7y # 12

5y # 209y # 4

B

CA

NAME DATE PERIOD

SCORE

Chapter 1 Mid-Chapter Test (Lessons 1–1 through 1–3)

11


Ass

essm

ents


1. Which point is coplanar with points A and C?A. A B. BC. C D. D

2. Name the point of intersection of plane M and DE!"#.A. D B. E C. B D. M


3. What is the length of A!B!?

A. about 1"14" in. B. about 1"

12" in.

C. about 1"34" in. D. about 2 in.

4. What is the precision for the measurement of A!B!?

A. 1 in. B. "12" in. C. "

14" in. D. "

18" in.

5. What is the length of T!S!?A. 9.4 cm B. 8.9 cmC. 4.7 cm D. 4.2 cm

R S

8.9 cm

4.7 cm T

1 2in.

A B

MBCA

D

E

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part IIFor Questions 6–8, use the coordinate grid.

6. Find the distance between R and S.

7. Find the coordinates of the midpoint of T!U!.

8. Find the coordinates of a point M given that U is the midpoint of M!S!.

9. Find y if M is the midpoint of L!N!.

10. In the figure, WZ!"# bisects X!Y!. Find the length of X!Y!.

XY

6x ! 114x # 5

V

Z

W

L N

6y # 59y ! 4

M

x

y

O

R US

T

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 1 Cumulative Review(Chapter 1)

11

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.


1. Name three points that are collinear.(Lesson 1-1)

2. Name the intersection of AE!"# and CG!"#.(Lesson 1-1)

Find the measurement of each segment. Assume that theart is not drawn to scale. (Lesson 1-2)

3. A!B!

4. K!N!

5. Use the Pythagorean Theorem to find the distance betweenA(!12, 13) and B(!2, !11). (Lesson 1-3)

6. Find the coordinates of B if A has coordinates (3, 5) and Y(!2, 3) is the midpoint of A!B!. (Lesson 1-3)

For Questions 7 and 8, use the figure to name the vertex and sides of each angle. Then measure and classify each angle. (Lesson 1-4)

7. !JNK 8. !HNK

For Questions 9–11, use the figure at the right. (Lesson 1-5)

9. Name a pair of supplementarynonadjacent angles.


11. Name an angle complementary to !CFD.

12. If m!HJK # 7y ! 2 and m!PQR # 133, find y so that !HJKis supplementary to !PQR. (Lesson 1-5)

13. Name this polygon by its number of sides and then classify it as convex or concave and regularor irregular. (Lesson 1-6)

14. Find the perimeter of "ABC if A(1, 1), B(4, !3), and C(!3, !2).(Lesson 1-6)

15. Find the length of each side of a regular pentagon whoseperimeter is 90 centimeters. (Lesson 1-6)

A F DE

B C

G60" 30"

120"

N

K

M

J

H

L

157"

77"26"

J K L M N

4 mm 3 mm

A C5 cm

2 cmB

A

C

E

B

D

FG

H

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapter 1)


1. Which object models a line? (Lesson 1-1)

A. a fly B. a wall C. a meter stick D. a diskette

2. Which figure shows AB!"# and point G contained in plane R ? (Lesson 1-1)

E. F.

G. H.

3. What is the precision of a measurement of 49.5 centimeters on aruler with millimeter marks? (Lesson 1-2)

A. 49 cm to 50 cm B. 49.0 cm to 50.0 cmC. 490 mm to 500 mm D. 494.5 mm to 495.5 mm

4. When segments have the same measure, they are said to be. (Lesson 1-2)

E. accurate F. congruentG. precise H. constructed

5. Find the distance between A(!3, 5) and B(4, 2), to the nearesthundredth. (Lesson 1-3)

A. 6.75 B. 7.62 C. 8.06 D. 10

6. Find EF if E is the midpoint of D!F!, DE # 15 ! 3x, and EF # x $ 3. (Lesson 1-3)

E. 1 F. 3 G. 6 H. 9

For Questions 7–9, use the figure.

7. What is another name for !2? (Lesson 1-4)

A. !WYX B. !WXYC. !3 D. !Y

8. Which angles form a linear pair? (Lesson 1-5)

E. !1 and !3 F. !2 and !5 G. !2 and !3 H. !1 and !4

9. Name the angle that is vertical to !3. (Lesson 1-5)

A. !1 B. !2 C. !3 D. !4

10. Find the length of one side of a regular hexagon whose perimeteris 75 feet. (Lesson 1-6)

E. 25 ft F. 18.75 ft G. 15 ft H. 12.5 ft

WY

X14

235

?

A

B

G RA B

G

R

A BR

GA BG R

NAME DATE PERIOD

SCORE

Part 1: Multiple ChoiceInstructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents

11


Standardized Test Practice (continued)

11. How many points name a line? (Lesson 1-1)

12. What is the measure of A!C!? (Lesson 1-2)

For Questions 13–15, HL!!" bisects !KHI andHG!!" and HI!!" are opposite rays.

13. If !1 $ !2, m!KHG # 70, and m!1 # 3d $ 2,find d. (Lesson 1-4)

14. If m!2 # a $ 15 and m!3 # a $ 35, find a sothat HL""# ! HJ""#. (Lesson 1-5)

15. Find m!4, if m!GHL # 125. (Lesson 1-5)

G H I

J

KL

1 42 3

A B C3.7 5.2

NAME DATE PERIOD

Part 2: Grid InInstructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short ResponseInstructions: Show your work or explain in words how you found your answer.

11. 12.

13. 14.

15.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

16. Find the length of X!Z! if Y(!4, 4) is the midpoint of X!Z! and X has coordinates (2, !4). (Lesson 1-3)

17. Find the perimeter of this hexagon.(Lesson 1-6)

18. Find the measure of W!X!, if the perimeter of pentagon UVWXY is 48 units. (Lesson 1-6)

10 ! a

2aW X

UV Y

4a # 7

30 m

20 m

7 m 6 m

10 m8.5 m

16.

17.

18.

2 8 . 9

1 1

5 5

2 0

11

Standardized Test PracticeStudent Record Sheet (Use with pages 58–59 of the Student Edition.)

11

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9 DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 14 and 15, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

10 14 15

11

12

13

14 (grid in)

15 (grid in)

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 16–17 on the back of this paper.


Stu

dy

Gu

ide

and I

nte

rven

tion

Poin

ts,L

ines

,and

Pla

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill1

Gle

ncoe

Geo

met

ry

Lesson 1-1

Nam

e Po

ints

, Lin

es, a

nd

Pla

nes

In g

eom

etry

,a p

oin

tis

a lo

cati

on,a

lin

eco

ntai

nspo

ints

,and

a p

lan

eis

a f

lat

surf

ace

that

con

tain

s po

ints

and

line

s.If

poi

nts

are

on t

he s

ame

line,

they

are

col

lin

ear.

If p

oint

s on

are

the

sam

e pl

ane,

they

are

cop

lan

ar.

Use

th

e fi

gure

to

nam

e ea

ch o

f th

e fo

llow

ing.

a.a

lin

e co

nta

inin

g p

oin

t A

The

line

can

be

nam

ed a

s !.

Als

o,an

y tw

o of

the

thr

ee

poin

ts o

n th

e lin

e ca

n be

use

d to

nam

e it

.

AB

!"# ,

AC

!"# ,

or B

C!"

#

b.a

pla

ne

con

tain

ing

poi

nt

D

The

pla

ne c

an b

e na

med

as

plan

e N

or c

an b

e na

med

usi

ng t

hree

no

ncol

linea

r po

ints

in t

he p

lane

,suc

h as

pla

ne A

BD

,pla

ne A

CD

,and

so

on.

Ref

er t

o th

e fi

gure

.

1.N

ame

a lin

e th

at c

onta

ins

poin

t A

.A

B!"

# ,A

C!"

# ,B

C!"

# ,or

!

2.W

hat

is a

noth

er n

ame

for

line

m?

BD

!"#

3.N

ame

a po

int

not

on A

C!"

# .D

or E

4.N

ame

the

inte

rsec

tion

of

AC

!"#

and

DB

!"# .

B

5.N

ame

a po

int

not

on li

ne !

or li

ne m

.E

Dra

w a

nd

lab

el a

pla

ne

Qfo

r ea

ch r

elat

ion

ship

.

6.A

B!"

#is

in p

lane

Q.

7.S

T!"

#in

ters

ects

AB

!"#

at P

.

8.Po

int

Xis

col

linea

r w

ith

poin

ts A

and

P.

Ans

wer

s fo

r E

xerc

ises

6–1

0

9.Po

int

Yis

not

col

linea

r w

ith

poin

ts T

and

P.

10.L

ine

!co

ntai

ns p

oint

s X

and

Y.

!Q

AB

S P

TY

X

!

PmA

B EC

D

!

N

AB

C

DExam

ple

Exam

ple

Exercis

esExercis

es

©G

lenc

oe/M

cGra

w-H

ill2

Gle

ncoe

Geo

met

ry

Poin

ts, L

ines

, an

d P

lan

es in

Sp

ace

Sp

ace

is a

bou

ndle

ss,t

hree

-dim

ensi

onal

set

of

all p

oint

s.It

con

tain

s lin

es a

nd p

lane

s.

a.H

ow m

any

pla

nes

ap

pea

r in

th

e fi

gure

?

The

re a

re t

hree

pla

nes:

plan

e N

,pla

ne O

,and

pla

ne P

.

b.A

re p

oin

ts A

,B,a

nd

Dco

pla

nar

?

Yes.

The

y ar

e co

ntai

ned

in p

lane

O.

Ref

er t

o th

e fi

gure

.

1.N

ame

a lin

e th

at is

not

con

tain

ed in

pla

ne N

.A

B!"

#

2.N

ame

a pl

ane

that

con

tain

s po

int

B.

plan

e N

,pla

ne A

BC

,pla

ne A

BD

,pla

ne E

BC

,pl

ane

EB

D3.

Nam

e th

ree

colli

near

poi

nts.

A,B

,E

Ref

er t

o th

e fi

gure

.

4.H

ow m

any

plan

es a

re s

how

n in

the

fig

ure?

6

5.A

re p

oint

s B

,E,G

,and

Hco

plan

ar?

Exp

lain

.N

o;B

,G,a

nd H

lie in

pla

ne B

GH

,but

Edo

es n

ot.

6.N

ame

a po

int

copl

anar

wit

h D

,C,a

nd E

.F

or J

Dra

w a

nd

lab

el a

fig

ure

for

eac

h r

elat

ion

ship

.

7.P

lane

s M

andN

inte

rsec

t in

HJ

!"# .

8.L

ine

ris

in p

lane

N,l

ine

sis

in p

lane

M,a

nd li

nes

rand

sin

ters

ect

at p

oint

J.

9.L

ine

tco

ntai

ns p

oint

Han

d lin

e td

oes

not

lie in

pla

ne M

orpl

ane

N.

Ans

wer

s fo

r E

xerc

ises

7–9

N

Ms

t

r

HJ

AB C

D

EFG

HI

J

N

A

C

B

DE

NO

P

AB

CD

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Poin

ts,L

ines

,and

Pla

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-1

1-1

Exam

ple

Exam

ple

Exercis

esExercis

es

Answers (Lesson 1-1)


An

swer

s

Skil

ls P

ract

ice

Poin

ts,L

ines

,and

Pla

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill3

Gle

ncoe

Geo

met

ry

Lesson 1-1

Ref

er t

o th

e fi

gure

.

1.N

ame

a lin

e th

at c

onta

ins

poin

t D

.

por

CD

!"#

2.N

ame

a po

int

cont

aine

d in

line

n.

Aor

B

3.W

hat

is a

noth

er n

ame

for

line

p?C

D!"

#or

DC

!"#

4.N

ame

the

plan

e co

ntai

ning

line

s n

and

p.S

ampl

e an

swer

:pla

ne G

Dra

w a

nd

lab

el a

fig

ure

for

eac

h r

elat

ion

ship

.S

ampl

e an

swer

s ar

e gi

ven.

5.Po

int

Klie

s on

RT

!"# .

6.P

lane

Jco

ntai

ns li

ne s.

7.Y

P! "

#lie

s in

pla

ne B

and

cont

ains

8.

Lin

es q

and

fint

erse

ct a

t po

int

Zpo

int

C,b

ut d

oes

not

cont

ain

poin

t H

.in

pla

ne U

.

Ref

er t

o th

e fi

gure

.

9.H

ow m

any

plan

es a

re s

how

n in

the

fig

ure?

5

10.H

ow m

any

of t

he p

lane

s co

ntai

n po

ints

Fan

d E

?

2

11.N

ame

four

poi

nts

that

are

cop

lana

r.

A,B

,E,F

or B

,C,D

,Eor

A,C

,D,F

12.A

re p

oint

s A

,B,a

nd C

copl

anar

? E

xpla

in.

Yes;

poin

ts A

,B,a

nd C

lie in

pla

ne W

.

WA

BE

CDF

U

qf

Z

HP

B

CY

J

sT

RK

Gn

AB

D

C

p

©G

lenc

oe/M

cGra

w-H

ill4

Gle

ncoe

Geo

met

ry

Ref

er t

o th

e fi

gure

.

1.N

ame

a lin

e th

at c

onta

ins

poin

ts T

and

P.

g,TP!"

# ,TN!"

# ,N

P!"

#

2.N

ame

a lin

e th

at in

ters

ects

the

pla

ne c

onta

inin

g po

ints

Q,N

,and

P.

jor

MT

!"#

3.N

ame

the

plan

e th

at c

onta

ins

TN

!"#

and

QR

!"# .

Sam

ple

answ

er:p

lane

S

Dra

w a

nd

lab

el a

fig

ure

for

eac

h r

elat

ion

ship

.S

ampl

e an

swer

s ar

e gi

ven.

4.A

K!"

#an

d C

G!"

#in

ters

ect

at p

oint

M5.

A li

ne c

onta

ins

L(!

4,!

4) a

nd M

(2,3

).L

ine

in p

lane

T.

q is

in t

he s

ame

coor

dina

te p

lane

but

doe

sno

t in

ters

ect

LM

!"# .

Lin

e q

cont

ains

poi

nt N

.

Ref

er t

o th

e fi

gure

.

6.H

ow m

any

plan

es a

re s

how

n in

the

fig

ure?

6

7.N

ame

thre

e co

lline

ar p

oint

s.S

,X,M

8.A

re p

oint

s N

,R,S

,and

Wco

plan

ar?

Exp

lain

.N

o;sa

mpl

e an

swer

:poi

nts

N,R

,and

Slie

in

pla

ne A

,but

poi

nt W

doe

s no

t.

VIS

UA

LIZA

TIO

NN

ame

the

geom

etri

c te

rm(s

) m

odel

ed b

y ea

ch o

bjec

t.

9.10

.11

.

plan

e an

d lin

epo

int

lines

12.a

car

ant

enna

13.a

libr

ary

card

line

and

poin

tpl

ane

strin

gs

tip o

f pin

STOP

AM

N

PW

S XRQ

T

L

M

N

q

x

y

O

KT

M

C

GA

S

j

ghTM

NQ

R

P

Pra

ctic

e (A

vera

ge)

Poin

ts,L

ines

,and

Pla

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-1

1-1



Rea

din

g t

o L

earn

Math

emati

csPo

ints

,Lin

es,a

nd P

lane

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill5

Gle

ncoe

Geo

met

ry

Lesson 1-1

Pre-

Act

ivit

yW

hy

do

chai

rs s

omet

imes

wob

ble?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

1 at

the

top

of

page

6 in

you

r te

xtbo

ok.

•F

ind

thre

e pe

ncils

of

diff

eren

t le

ngth

s an

d ho

ld t

hem

upr

ight

on

your

desk

so

that

the

thr

ee p

enci

l poi

nts

do n

ot li

e al

ong

a si

ngle

line

.Can

you

plac

e a

flat

she

et o

f pa

per

or c

ardb

oard

so

that

it t

ouch

es a

ll th

ree

penc

ilpo

ints

?ye

s•

How

man

y w

ays

can

you

do t

his

if y

ou k

eep

the

penc

il po

ints

in t

he s

ame

posi

tion

?on

e•

How

will

you

r an

swer

cha

nge

if t

here

are

fou

r pe

ncil

poin

ts?

Sam

ple

answ

er:I

t m

ay n

ot b

e po

ssib

le t

o pl

ace

the

pape

r to

tou

chal

l fou

r po

ints

.

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sen

tenc

e.

a.Po

ints

tha

t lie

on

the

sam

e lie

are

cal

led

poin

ts.

b.Po

ints

tha

t do

not

lie

in t

he s

ame

plan

e ar

e ca

lled

poin

ts.

c.T

here

is e

xact

ly o

ne

thro

ugh

any

two

poin

ts.

d.T

here

is e

xact

ly o

ne

thro

ugh

any

thre

e no

ncol

linea

r po

ints

.

2.R

efer

to

the

figu

re a

t th

e ri

ght.

Indi

cate

whe

ther

eac

h st

atem

ent

is t

rue

or f

alse

.

a.Po

ints

A,B

,and

Car

e co

lline

ar.

fals

e

b.T

he in

ters

ecti

on o

f pl

ane

AB

Can

d lin

e m

is p

oint

P.

true

c.L

ine

!an

d lin

e m

do n

ot in

ters

ect.

fals

e

d.Po

ints

A,P

,and

Bca

n be

use

d to

nam

e pl

ane

U.

fals

e

e.L

ine

!lie

s in

pla

ne A

CB

.tr

ue

3.C

ompl

ete

the

figu

re a

t th

e ri

ght

to s

how

the

fol

low

ing

rela

tion

ship

:Lin

es !

,m,a

nd n

are

copl

anar

and

lie

in

plan

e Q.

Lin

es !

and

min

ters

ect

at p

oint

P.L

ine

nin

ters

ects

line

mat

R,b

ut d

oes

not

inte

rsec

t lin

e !.

Hel

pin

g Y

ou

Rem

emb

er

4.R

ecal

l or

look

in a

dic

tion

ary

to f

ind

the

mea

ning

of

the

pref

ix c

o-.W

hat

does

thi

s pr

efix

mea

n? H

ow c

an it

hel

p yo

u re

mem

ber

the

mea

ning

of c

olli

near

?S

ampl

e an

swer

:The

pre

fix c

o-m

eans

toge

ther

.The

wor

d co

lline

arco

ntai

ns t

he w

ord

line,

so c

ollin

ear

mea

ns to

geth

er o

n a

line.

Qn

m

!P

R

U

m

!P

C

D

B

A

plan

e

line

nonc

opla

nar

colli

near

©G

lenc

oe/M

cGra

w-H

ill6

Gle

ncoe

Geo

met

ry

Poin

ts a

nd L

ines

on

a M

atri

xA

mat

rix

is a

rec

tang

ular

arr

ay o

f ro

ws

and

colu

mns

.Poi

nts

and

lines

on

a m

atri

x ar

e no

t de

fine

d in

the

sam

e w

ay a

s in

Euc

lidea

nge

omet

ry.A

poi

nt

on a

mat

rix

is a

dot

,whi

ch c

an b

e sm

all o

rla

rge.

A l

ine

on a

mat

rix

is a

pat

h of

dot

s th

at “

line

up.”

Bet

wee

ntw

o po

ints

on

a lin

e th

ere

may

or

may

not

be

othe

r po

ints

.Thr

eeex

ampl

es o

f lin

es a

re s

how

n at

the

upp

er r

ight

.The

bro

ad li

ne c

anbe

tho

ught

of

as a

sin

gle

line

or a

s tw

o na

rrow

line

s si

de b

y si

de.

Dot

-mat

rix

prin

ters

for

com

pute

rs u

sed

dots

to

form

cha

ract

ers.

The

dot

s ar

e of

ten

calle

d p

ixel

s.T

he m

atri

x at

the

rig

ht s

how

sho

w a

dot

-mat

rix

prin

ter

mig

ht p

rint

the

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An

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Lesson 1-2

Pre-

Act

ivit

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hy

are

un

its

of m

easu

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mp

orta

nt?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

2 at

the

top

of

page

13

in y

our

text

book

.

•T

he b

asic

uni

t of

leng

th in

the

met

ric

syst

em is

the

met

er.H

ow m

any

met

ers

are

ther

e in

one

kilo

met

er?

1000

•D

o yo

u th

ink

it w

ould

be

easi

er t

o le

arn

the

rela

tion

ship

s be

twee

n th

edi

ffer

ent

unit

s of

leng

th in

the

cus

tom

ary

syst

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used

in t

he U

nite

dSt

ates

) or

in t

he m

etri

c sy

stem

? E

xpla

in y

our

answ

er.

Sam

ple

answ

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The

met

ric

syst

em is

eas

ier

beca

use

you

can

chan

gebe

twee

n th

e di

ffer

ent

units

by

just

mov

ing

the

deci

mal

poi

nt.

Rea

din

g t

he

Less

on

1.E

xpla

in t

he d

iffe

renc

e be

twee

n a

line

and

a li

ne s

egm

ent

and

why

one

of

thes

e ca

n be

mea

sure

d,w

hile

the

oth

er c

anno

t.S

ampl

e an

swer

:A li

ne is

infin

ite.S

ince

it h

as n

o en

dpoi

nts,

a lin

e do

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ve a

def

inite

leng

th a

nd c

anno

t be

mea

sure

d.A

line

seg

men

t ha

stw

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dpoi

nts,

so it

has

a d

efin

ite le

ngth

and

can

be

mea

sure

d.

2.W

hat

is t

he s

mal

lest

leng

th m

arke

d on

a 1

2-in

ch r

uler

?S

ampl

e an

swer

:!11 6!

in.

Wha

t is

the

sm

alle

st le

ngth

mar

ked

on a

cen

tim

eter

rul

er?

1 m

m

3.F

ind

the

prec

isio

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eac

h m

easu

rem

ent.

a.15

cm

0.5

cmb.

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cm

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4.R

efer

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the

figu

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ght.

Whi

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tru

e? E

xpla

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our

answ

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A !B!

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C !D!

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ampl

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tw

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re

cong

ruen

t be

caus

e th

ey h

ave

the

sam

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easu

re o

rle

ngth

.The

y ar

e no

t eq

ual b

ecau

se t

hey

are

not

the

sam

e se

gmen

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5.Su

ppos

e th

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is a

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n V!

W!an

d S

is n

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som

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s is

to

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to

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ow.G

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e us

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the

re a

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Geo

met

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Poin

ts E

quid

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om S

egm

ents

The

dis

tanc

e fr

om a

poi

nt t

o a

segm

ent

is z

ero

if t

he p

oint

is o

n th

e se

gmen

t.O

ther

wis

e,it

is t

he le

ngth

of

the

shor

test

seg

men

t fr

om t

he

poin

t to

the

seg

men

t.

A f

igur

e is

a l

ocu

sif

it is

the

set

of

all p

oint

s th

at s

atis

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a se

t of

con

diti

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locu

s of

all

poin

ts t

hat

are

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from

the

seg

men

t A

Bis

sho

wn

by t

wo

dash

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ents

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sem

icir

cles

at

both

end

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ppos

e A

,B,C

,and

Dar

e fo

ur d

iffe

rent

poi

nts,

and

cons

ider

the

locu

s of

all

poin

ts x

unit

s fr

om A!

B!an

dx

unit

s fr

om C!

D!.U

se a

ny u

nit

you

find

conv

enie

nt.T

he lo

cus

can

take

dif

fere

nt f

orm

s.Sk

etch

at

leas

t th

ree

poss

ibili

ties

.Lis

t so

me

of t

he t

hing

s th

at s

eem

to

affe

ct t

he f

orm

of

the

locu

s.S

ampl

e an

swer

s ar

e sh

own.

The

locu

s is

1Th

e lo

cus

is a

set

of

The

locu

s is

a p

air

segm

ent

X!Y! m

idw

ay2

poin

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and

Y.

of li

ne s

egm

ents

,be

twee

n A!

B!an

d C!

D!.

R!S!

and

P!Q!.

The

locu

s of

poi

nts

xun

its f

rom

A!B!

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C!D!

depe

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on t

he

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ance

xan

d ho

w A!

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d C!

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e si

tuat

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ive

to o

ne a

noth

er.

2.C

ondu

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our

own

inve

stig

atio

n of

the

locu

s of

poi

nts

equi

dist

ant

from

tw

o se

gmen

ts.D

escr

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your

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ults

on

ase

para

te s

heet

of

pape

r.S

ee s

tude

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wor

k.

AB

CD

PQ

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A

B

C

D

X

Y

AB

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AB

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rich

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t

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1-2

1-2



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Gu

ide

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nte

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Dis

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idpo

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ME

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1-3

1-3

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met

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Lesson 1-3

Dis

tan

ce B

etw

een

Tw

o P

oin

ts

Dis

tanc

e on

a N

umbe

r Li

neD

ista

nce

in t

he C

oord

inat

e P

lane

AB

$|b

!a

|or

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b|

x

y

OC

( 1, –

1)A

( –2,

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The

orem

:

a2#

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c2

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tanc

e F

orm

ula:

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AB

ab

Fin

d A

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AB

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d t

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dis

tan

ce b

etw

een

A

(!2,

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an

d B

(1,3

).

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ple1

Exam

ple1

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ple2

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ple2

Pyt

hag

orea

n T

heo

rem

(AB

)2$

(AC

)2#

(BC

)2

(AB

)2$

(3)2

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)2

(AB

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25

AB

$#

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5

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tan

ce F

orm

ula

d$

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2!

!x 1

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th

e n

um

ber

lin

e to

fin

d e

ach

mea

sure

.

1.B

D6

2.D

G9

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5.B

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G17

7.B

E7

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th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

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(0,0

),B

(6,8

)10

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5)13

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(1,!

2),N

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3)17

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F(!

9,6)

5

Use

th

e D

ista

nce

For

mu

la t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

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(0,0

),B

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20)

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02

46

8

AB

CD

EF

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Exercis

esExercis

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ncoe

Geo

met

ry

Mid

po

int

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a Se

gm

ent

Mid

poin

t on

aIf

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dina

tes

of th

e en

dpoi

nts

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seg

men

t are

aan

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Num

ber

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then

the

coor

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te o

f the

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t of t

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ent i

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poin

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1) a

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ind

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x 2"

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r (1

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e n

um

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lin

e to

fin

d t

he

coor

din

ate

of

the

mid

poi

nt

of e

ach

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d t

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din

ates

of

the

mid

poi

nt

of a

seg

men

t h

avin

g th

e gi

ven

en

dp

oin

ts.

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ide

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(con

tinu

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tanc

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idpo

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ME

____

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ATE

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RIO

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1-3

1-3

Exam

ple1

Exam

ple1

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ple2

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ple2

Exercis

esExercis

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An

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s

Skil

ls P

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Dis

tanc

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____

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1-3

1-3

©G

lenc

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ncoe

Geo

met

ry

Lesson 1-3

Use

th

e n

um

ber

lin

e to

fin

d e

ach

mea

sure

.

1.L

N6

2.JL

8

3.K

N11

4.M

N3

Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

5.5

6.10

7.K

(2,3

),F

(4,4

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C(!

3,!

1),Q

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3)

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$2.

2#

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4.1

Use

th

e D

ista

nce

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mu

la t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

9.Y

(2,0

),P

(2,6

)10

.W(!

2,2)

,R(5

,2)

67

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12.C

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)

13#

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7.1

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um

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lin

e to

fin

d t

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din

ate

of t

he

mid

poi

nt

of e

ach

seg

men

t.

13.D !

E!9

14.B!

C!1

15.B!

D!3

16.A!

D!1!

1 2!

Fin

d t

he

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din

ates

of

the

mid

poi

nt

of a

seg

men

t h

avin

g th

e gi

ven

en

dp

oin

ts.

17.T

(3,1

),U

(5,3

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4,2)

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0 &F

ind

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e co

ord

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es o

f th

e m

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ng

end

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nt

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n t

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th

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t of

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5)

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5,1)

–6–4

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24

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x

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Gle

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Geo

met

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Use

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um

ber

lin

e to

fin

d e

ach

mea

sure

.

1.V

W4

2.T

V5

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V8

Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

5.6.

#65!

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1#

113

!$

10.6

Use

th

e D

ista

nce

For

mu

la t

o fi

nd

th

e d

ista

nce

bet

wee

n e

ach

pai

r of

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nts

.

7.L

(!7,

0),Y

(5,9

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B(4

,6)

15#

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ber

lin

e to

fin

d t

he

coor

din

ate

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mid

poi

nt

of e

ach

seg

men

t.

9.R !

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4

11.S!

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1 2!12

.P!R!

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Fin

d t

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din

ates

of

the

mid

poi

nt

of a

seg

men

t h

avin

g th

e gi

ven

en

dp

oin

ts.

13.K

(!9,

3),H

(5,7

)14

.W(!

12,!

7),T

(!8,

!4)

(#2,

5)(#

10,#

5.5)

Fin

d t

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coor

din

ates

of

the

mis

sin

g en

dp

oin

t gi

ven

th

at E

is t

he

mid

poi

nt

of D!

F!.

15.F

(5,8

),E

(4,3

)16

.F(2

,9),

E(!

1,6)

17.D

(!3,

!8)

,E(1

,!2)

D(3

,#2)

D(#

4,3)

F(5

,4)

18.P

ERIM

ETER

The

coo

rdin

ates

of

the

vert

ices

of

a qu

adri

late

ral a

re R

(!1,

3),S

(3,3

),T

(5,!

1),a

nd U

(!2,

!1)

.Fin

d th

e pe

rim

eter

of

the

quad

rila

tera

l.R

ound

to

the

near

est

tent

h.19

.6 u

nits

–6–4

–10

–8–2

02

46

PQ

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T

x

y

O

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x

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OM

Z

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46

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W

Pra

ctic

e (A

vera

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Dis

tanc

e an

d M

idpo

ints

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ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

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___

1-3

1-3



Rea

din

g t

o L

earn

Math

emati

csD

ista

nce

and

Mid

poin

ts

NA

ME

____

____

____

____

____

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____

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____

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ATE

____

____

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RIO

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___

1-3

1-3

©G

lenc

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cGra

w-H

ill17

Gle

ncoe

Geo

met

ry

Lesson 1-3

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e d

ista

nce

bet

wee

n t

wo

poi

nts

wit

hou

t a

rule

r?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

3 at

the

top

of

page

21

in y

our

text

book

.

•L

ook

at t

he t

rian

gle

in t

he in

trod

ucti

on t

o th

is le

sson

.Wha

t is

the

spe

cial

nam

e fo

r A!

B!in

thi

s tr

iang

le?

hypo

tenu

se•

Fin

d A

Bin

thi

s fi

gure

.Wri

te y

our

answ

er b

oth

as a

rad

ical

and

as

ade

cim

al n

umbe

r ro

unde

d to

the

nea

rest

ten

th.

#61!

units

;7.8

uni

ts

Rea

din

g t

he

Less

on

1.M

atch

eac

h fo

rmul

a or

exp

ress

ion

in t

he f

irst

col

umn

wit

h on

e of

the

nam

es in

the

seco

nd c

olum

n.

a.d

$#

(x2

!!

x 1)2

#!

(y2

!!

y 1)2

!ii

i.P

ytha

gore

an T

heor

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# 2b

"v

ii.D

ista

nce

Form

ula

in t

he C

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e P

lane

c.X

Y$

| a!

b|iv

iii.

Mid

poin

t of

a S

egm

ent

in t

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e P

lane

d.c2

$a2

#b2

iiv

.D

ista

nce

Form

ula

on a

Num

ber

Lin

e

e.$"x 1

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%iii

v.M

idpo

int

of a

Seg

men

t on

a N

umbe

r L

ine

2.F

ill in

the

ste

ps t

o ca

lcul

ate

the

dist

ance

bet

wee

n th

e po

ints

M(4

,!3)

and

N(!

2,7)

.

Let

(x 1

,y1)

$(4

,!3)

.The

n (x

2,y 2

) $(

,).

d$

#!

!!

!(

!)2

#(

!)2

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$#!

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!)2

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$#!

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$#!

!#

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$#!

Fin

d a

deci

mal

app

roxi

mat

ion

for

MN

to t

he n

eare

st h

undr

edth

.11

.66

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a ne

w f

orm

ula

in m

athe

mat

ics

is t

o re

late

it t

o on

e yo

u al

read

ykn

ow.I

f yo

u fo

rget

the

Dis

tanc

e Fo

rmul

a,ho

w c

an y

ou u

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gore

an T

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em t

ofi

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ista

nce

dbe

twee

n tw

o po

ints

on

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ordi

nate

pla

ne?

Sam

ple

answ

er:I

f th

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gmen

t de

term

ined

by

the

poin

ts is

nei

ther

hor

izon

tal n

or v

ertic

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raw

a ri

ght

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s th

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t as

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oriz

onta

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2#

x 1|a

nd t

he v

ertic

al s

ide

will

hav

e le

ngth

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#y 1

|.By

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136

100

36

10#

6

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74

#2

y 1y 2

x 1x 2

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lenc

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w-H

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ncoe

Geo

met

ry

Leng

ths

on a

Gri

dE

venl

y-sp

aced

hor

izon

tal a

nd v

erti

cal l

ines

for

m a

gri

d.

You

can

easi

ly f

ind

segm

ent

leng

ths

on

a gr

id if

the

end

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ts a

re g

rid-

line

inte

rsec

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s.Fo

r ho

rizo

ntal

or

vert

ical

segm

ents

,sim

ply

coun

t sq

uare

s.Fo

r di

agon

al s

egm

ents

,use

the

Pyt

hago

rean

The

orem

(pr

oven

in C

hapt

er 7

).T

his

theo

rem

sta

tes

that

in a

ny r

ight

tri

angl

e,if

the

leng

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e lo

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the

side

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the

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can

d th

e tw

osh

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Fin

d t

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grid

at

the

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t.L

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s

Fin

d e

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the

nea

rest

ten

th o

f a

un

it.

1.I!J!

32.

M!N!

73.

R!S!

4.2

4.Q!

S!5.

8

5.I!K!

7.6

6.J!K!

57.

L!M!

4.1

8.L!N!

7.2

Use

th

e gr

id a

bove

.Fin

d t

he

per

imet

er o

f ea

ch t

rian

gle

to t

he

nea

rest

ten

th

of a

un

it.

9.!

AB

C20

.210

.!

QR

S18

11.

!D

EF

16.6

12.

!L

MN

18.3

Ans

wer

s sh

own

are

foun

d by

rou

ndin

g se

gmen

t le

ngth

s be

fore

add

ing.

13.O

f al

l the

seg

men

ts s

how

n on

the

14.O

n th

e gr

id,1

uni

t$0.

5 cm

.How

can

the

gr

id,w

hich

is lo

nges

t? W

hat

is it

san

swer

s ab

ove

be u

sed

to f

ind

the

mea

sure

s le

ngth

?B

C!

8.1

in c

enti

met

ers?

D

ivid

e by

2 o

r m

ultip

ly b

y 0.

5.

15.U

se y

our

answ

er f

rom

exe

rcis

e 8

to16

.Use

a c

enti

met

er r

uler

to

find

the

per

imet

er

calc

ulat

e th

e le

ngth

of

segm

ent

LN

of t

rian

gle

IJK

to t

he n

eare

st t

enth

of

a

in c

enti

met

ers.

Che

ck b

y m

easu

ring

cent

imet

er.

7.8

cmw

ith

a ce

ntim

eter

rul

er.

3.6

cm

E 2

5F

A B F

I J

KN

M

C

D

E

S

R Q L

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-3

1-3

Exam

ple

Exam

ple



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Ang

le M

easu

re

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill19

Gle

ncoe

Geo

met

ry

Lesson 1-4

Mea

sure

An

gle

sIf

tw

o no

ncol

linea

r ra

ysha

ve a

com

mon

en

dpoi

nt,t

hey

form

an

angl

e.T

he r

ays

are

the

sid

esof

the

ang

le.

The

com

mon

end

poin

t is

the

ver

tex.

The

ang

le a

t th

e ri

ght

can

be

nam

ed a

s "

A,"

BA

C,"

CA

B,o

r "

1.

A r

igh

t an

gle

is a

n an

gle

who

se m

easu

re is

90.

An

acu

te a

ngl

eha

s m

easu

re le

ss t

han

90.A

n ob

tuse

an

gle

has

mea

sure

gre

ater

th

an 9

0 bu

t le

ss t

han

180.

AC

B

1

a.N

ame

all

angl

es t

hat

hav

e R

as a

vert

ex.

Thr

ee a

ngle

s ar

e "

1,"

2,an

d "

3.Fo

rot

her

angl

es,u

se t

hree

lett

ers

to n

ame

them

:"S

RQ

,"P

RT

,and

"S

RT

.

b.N

ame

the

sid

es o

f !

1.

RS

" "# ,

RP

""#

SR

T

PQ

12

3

Mea

sure

eac

h a

ngl

e an

dcl

assi

fy i

t as

rig

ht,

acu

te,o

r ob

tuse

.

a.!

AB

DU

sing

a p

rotr

acto

r,m

"A

BD

$50

.50

%90

,so

"A

BD

is a

n ac

ute

angl

e.

b.!

DB

CU

sing

a p

rotr

acto

r,m

"D

BC

$11

5.18

0 &

115

&90

,so

"D

BC

is a

n ob

tuse

angl

e.

c.!

EB

CU

sing

a p

rotr

acto

r,m

"E

BC

$90

."

EB

Cis

a r

ight

ang

le.

BA

DE

C

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Ref

er t

o th

e fi

gure

.

1.N

ame

the

vert

ex o

f "4.

B

2.N

ame

the

side

s of

"B

DC

.D

B""

# ,D

C""#

3.W

rite

ano

ther

nam

e fo

r "

DB

C.

!3

or !

CB

D

Mea

sure

eac

h a

ngl

e in

th

e fi

gure

an

d c

lass

ify

it a

s ri

ght,

acu

te,o

r ob

tuse

.

4."

MP

R12

0;ob

tuse

5."

RP

N90

;rig

ht

6."

NP

S45

;acu

te

P

NM

R

S

C

BA

12

34

D

©G

lenc

oe/M

cGra

w-H

ill20

Gle

ncoe

Geo

met

ry

Co

ng

ruen

t A

ng

les

Ang

les

that

hav

e th

e sa

me

mea

sure

are

co

ngr

uen

t an

gles

.A r

ay t

hat

divi

des

an a

ngle

into

tw

o co

ngru

ent

angl

es is

cal

led

an a

ngl

e bi

sect

or.I

n th

e fi

gure

,PN

" "#

is t

he a

ngle

bi

sect

or o

f "M

PR

.Poi

nt N

lies

in t

he in

teri

or o

f "M

PR

and

"M

PN

""

NP

R. R

efer

to

the

figu

re a

bove

.If

m!

MP

N$

2x"

14 a

nd

m

!N

PR

$x

"34

,fin

d x

and

fin

d m

!M

PR

.Si

nce

PN

" "#

bise

cts

"M

PR

,"M

PN

""

NP

R,o

r m

"M

PN

$m

"N

PR

.

2x#

14 $

x#

34m

"N

PR

$(2

x#

14)

#(x

#34

)2x

#14

!x

$x

#34

!x

$54

#54

x#

14 $

34$

108

x#

14 !

14 $

34 !

14x

$20

QS

" "#

bise

cts

!P

QT

,an

d Q

P""

#an

d Q

R""

#ar

e op

pos

ite

rays

.

1.If

m"

PQ

T$

60 a

nd m

"P

QS

$4x

#14

,fin

d th

e va

lue

of x

.

4

2.If

m"

PQ

S$

3x#

13 a

nd m

"S

QT

$6x

!2,

find

m"

PQ

T.

56

BA

""#

and

BC

""#

are

opp

osit

e ra

ys,B

F""

#bi

sect

s !

CB

E,a

nd

B

D" "

#bi

sect

s !

AB

E.

3.If

m"

EB

F$

6x#

4 an

d m

"C

BF

$7x

!2,

find

m"

EB

C.

80

4.If

m"

1 $

4x#

10 a

nd m

"2

$5x

,fin

d m

"2.

50

5.If

m"

2 $

6y#

2 an

d m

"1

$8y

!14

,fin

d m

"A

BE

.

100

6.Is

"D

BF

a ri

ght

angl

e? E

xpla

in.

Yes;

sinc

e B

D""

#an

d B

F""

# are

bis

ecto

rs,m

!2

"m

!3

mus

t eq

ual h

alf

the

tota

l ang

le m

easu

re,a

nd h

alf

of 1

80 is

90.

BA

CF

ED

12

34

QP

R

TS

PR

NM

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Ang

le M

easu

re

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

Ang

le M

easu

re

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill21

Gle

ncoe

Geo

met

ry

Lesson 1-4

For

Exe

rcis

es 1

–12,

use

th

e fi

gure

at

the

righ

t.

Nam

e th

e ve

rtex

of

each

an

gle.

1."

4T

2."

1W

3."

2V

4."

5T

Nam

e th

e si

des

of

each

an

gle.

5."

46.

"5

TU""# ,

TS""#

TS""# ,

TW""#

7."

ST

V8.

"1

TS""# ,

TV""#

WT

""# ,

WV

""#

Wri

te a

not

her

nam

e fo

r ea

ch a

ngl

e.

9."

310

."4

!W

TV,!

VTW

!U

TS,!

STU

11."

WT

S12

."2

!S

TW,!

5!

WV

T,!

TVW

,!W

VU

,!U

VW

Mea

sure

eac

h a

ngl

e an

d c

lass

ify

it a

s ri

ght,

acu

te,

or o

btu

se.

13."

NM

P14

."O

MN

90%,

righ

t40

%,ac

ute

15."

QM

N16

."Q

MO

140%

,obt

use

100%

,obt

use

ALG

EBR

AIn

th

e fi

gure

,BA

""#

and

BC

""#

are

opp

osit

e ra

ys,

BD

" "#

bise

cts

!E

BC

,an

d B

F""

#bi

sect

s !

AB

E.

17.I

f m

"E

BD

$4x

#16

and

m"

DB

C$

6x#

4,fi

nd m

"E

BD

.40

18.I

f m

"A

BF

$7x

!8

and

m"

EB

F$

5x#

10,

find

m"

EB

F.

55

BC

A

FE

D

MN

L

QO

P

U T VW

S5

3 21

4

©G

lenc

oe/M

cGra

w-H

ill22

Gle

ncoe

Geo

met

ry

For

Exe

rcis

es 1

–10,

use

th

e fi

gure

at

the

righ

t.

Nam

e th

e ve

rtex

of

each

an

gle.

1."

5M

2."

3P

3."

8O

4."

NM

PM

Nam

e th

e si

des

of

each

an

gle.

5."

66.

"2

NM""

# ,N

O""

#or

NP

""#

or N

R""

#P

R""# ,

PM""

#

7."

MO

P8.

"O

MN

OM

""# ,

OP

""# o

r O

R""

#M

O""

# ,M

N""

#

Wri

te a

not

her

nam

e fo

r ea

ch a

ngl

e.

9."

QP

R10

."1

!3,

!R

PQ

!M

PO

,!O

PM

,!M

PN

,!N

PM

Mea

sure

eac

h a

ngl

e an

d c

lass

ify

it a

s ri

ght,

acu

te,

or o

btu

se.

11."

UZ

W12

."Y

ZW

90%,

righ

t70

%,ac

ute

13."

TZ

W14

."U

ZT

110%

,obt

use

20%,

acut

e

ALG

EBR

AIn

th

e fi

gure

,CB

""#

and

CD

""#

are

opp

osit

e ra

ys,

CE

" "#

bise

cts

!D

CF

,an

d C

G""

#bi

sect

s !

FC

B.

15.I

f m

"D

CE

$4x

#15

and

m"

EC

F$

6x!

5,fi

nd m

"D

CE

.55

16.I

f m

"F

CG

$9x

#3

and

m"

GC

B$

13x

!9,

find

m"

GC

B.

30

17.T

RA

FFIC

SIG

NS

The

dia

gram

sho

ws

a si

gn u

sed

to w

arn

driv

ers

of a

sch

ool z

one

or c

ross

ing.

Mea

sure

and

cla

ssif

y ea

ch n

umbe

red

angl

e.

m!

1 $

90,r

ight

ang

le;m

!2

$13

0,ob

tuse

2 1BC

G

F

EDZ

YTU

VW

X

N

O

PQ

R

M1 2

8

3

54

6

7

Pra

ctic

e (A

vera

ge)

Ang

le M

easu

re

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csA

ngle

Mea

sure

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill23

Gle

ncoe

Geo

met

ry

Lesson 1-4

Pre-

Act

ivit

yH

ow b

ig i

s a

deg

ree?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

4 at

the

top

of

page

29

in y

our

text

book

.•

A s

emic

ircl

e is

hal

f a

circ

le.H

ow m

any

degr

ees

are

ther

e in

a

sem

icir

cle?

180

•H

ow m

any

degr

ees

are

ther

e in

a q

uart

er c

ircl

e?90

Rea

din

g t

he

Less

on

1.M

atch

eac

h de

scri

ptio

n in

the

fir

st c

olum

n w

ith

one

of t

he t

erm

s in

the

sec

ond

colu

mn.

Som

e te

rms

in t

he s

econ

d co

lum

n m

ay b

e us

ed m

ore

than

onc

e or

not

at

all.

a.a

figu

re m

ade

up o

f tw

o no

ncol

linea

r ra

ys w

ith

a1.

vert

ex

com

mon

end

poin

t4

2.an

gle

bise

ctor

b.an

gles

who

se d

egre

e m

easu

res

are

less

tha

n 90

83.

oppo

site

ray

sc.

angl

es t

hat

have

the

sam

e m

easu

re6

4.an

gle

d.an

gles

who

se d

egre

e m

easu

res

are

betw

een

90 a

nd 1

805

5.ob

tuse

ang

les

e.a

tool

use

d to

mea

sure

ang

les

106.

cong

ruen

t an

gles

f.th

e co

mm

on e

ndpo

int

of t

he r

ays

that

for

m a

n an

gle

17.

righ

t an

gles

g.a

ray

that

div

ides

an

angl

e in

to t

wo

cong

ruen

t an

gles

28.

acut

e an

gles

9.co

mpa

ss10

.pro

trac

tor

2.U

se t

he f

igur

e to

nam

e ea

ch o

f th

e fo

llow

ing.

a.a

righ

t an

gle

!A

BE

or !

EB

Gb.

an o

btus

e an

gle

!A

BF

or !

AB

Cc.

an a

cute

ang

le!

EB

F,!

FBC

,!C

BG

,!E

BC

,or

!FB

Gd.

a po

int

in t

he in

teri

or o

f "E

BC

Fe.

a po

int

in t

he e

xter

ior

of "

EB

AF,

C,o

r G

f.th

e an

gle

bise

ctor

of "

EB

CB

F""#

g.a

poin

t on

"C

BE

C,B

,or

Eh

.th

e si

des

of "

AB

FB

A""#

and

BF""#

i.a

pair

of

oppo

site

ray

sB

A""#

and

BG

""#

j.th

e co

mm

on v

erte

x of

all

angl

es s

how

n in

the

fig

ure

Bk.

a pa

ir o

f co

ngru

ent

angl

es!

EB

Fan

d !

FBC

,or

!A

BE

and

!E

BG

l.th

e an

gle

wit

h th

e gr

eate

st m

easu

re!

AB

G

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r re

late

d ge

omet

ric

idea

s is

to

com

pare

the

m a

nd s

ee h

ow t

hey

are

alik

e an

d ho

w t

hey

are

diff

eren

t.G

ive

som

e si

mila

riti

es a

nd d

iffe

renc

es b

etw

een

cong

ruen

t se

gmen

tsan

d co

ngru

ent

angl

es.

Sam

ple

answ

er:C

ongr

uent

seg

men

ts a

nd c

ongr

uent

ang

les

are

alik

ebe

caus

e th

ey b

oth

invo

lve

a pa

ir o

f fig

ures

with

the

sam

e m

easu

re.T

hey

are

diff

eren

t be

caus

e co

ngru

ent

segm

ents

hav

e th

e sa

me

leng

th,w

hich

can

be m

easu

red

in u

nits

suc

h as

inch

es o

r ce

ntim

eter

s,w

hile

con

grue

ntan

gles

hav

e th

e sa

me

degr

ee m

easu

re.

BG

28% 28

%

A

EF

CD

©G

lenc

oe/M

cGra

w-H

ill24

Gle

ncoe

Geo

met

ry

Ang

le R

elat

ions

hips

Ang

les

are

mea

sure

d in

deg

rees

(').

Eac

h de

gree

of

an a

ngle

is d

ivid

-ed

into

60

min

utes

((),

and

each

min

ute

of a

n an

gle

is

divi

ded

into

60

seco

nds

()).

60(

$1'

60)

$1(

67"1 2" '

$67

'30(

70.4

'$

70°2

4(90

'$

89°6

0(

Tw

o an

gles

are

com

ple

men

tary

if

the

sum

of

thei

r m

easu

res

is 9

0%.

Fin

d t

he

com

ple

men

t of

eac

h o

f th

e fo

llow

ing

angl

es.

1.35

'15(

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o an

gles

are

su

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lem

enta

ry i

f th

e su

m o

f th

eir

mea

sure

s is

180

%.F

ind

th

e su

pp

lem

ent

of e

ach

of

the

foll

owin

g an

gles

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rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-4

1-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Ang

le R

elat

ions

hips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill25

Gle

ncoe

Geo

met

ry

Lesson 1-5

Pair

s o

f A

ng

les

Ad

jace

nt

angl

esar

e an

gles

in t

he s

ame

plan

e th

at h

ave

a co

mm

onve

rtex

and

a c

omm

on s

ide,

but

no c

omm

on in

teri

or p

oint

s.V

erti

cal

angl

esar

e tw

ono

nadj

acen

t an

gles

for

med

by

two

inte

rsec

ting

line

s.A

pai

r of

adj

acen

t an

gles

who

seno

ncom

mon

sid

es a

re o

ppos

ite

rays

is c

alle

d a

lin

ear

pai

r.

Iden

tify

eac

h p

air

of a

ngl

es a

s a

dja

cen

t a

ngl

es,v

erti

cal

an

gles

,an

d/o

r as

a l

inea

r pa

ir.

Exam

ple

Exam

ple

a.

"S

RT

and

"T

RU

have

a c

omm

onve

rtex

and

a c

omm

on s

ide,

but

noco

mm

on in

teri

or p

oint

s.T

hey

are

adja

cent

ang

les.

c.

"6

and

"5

are

adja

cent

ang

les

who

seno

ncom

mon

sid

es a

re o

ppos

ite

rays

.T

he a

ngle

s fo

rm a

line

ar p

air.

D

CB

A

56

RUT

Sb.

"1

and

"3

are

nona

djac

ent

angl

es f

orm

edby

tw

o in

ters

ecti

ng li

nes.

The

y ar

e ve

rtic

alan

gles

."2

and

"4

are

also

ver

tica

l ang

les.

d.

"A

and

"B

are

two

angl

es w

hose

mea

sure

sha

ve a

sum

of

90.T

hey

are

com

plem

enta

ry.

"F

and

"G

are

two

angl

es w

hose

mea

sure

sha

ve a

sum

of

180.

The

y ar

e su

pple

men

tary

.

AB

FG

30%

60%

60%

120%

N

RP

SM

14

32

Exercis

esExercis

es

Iden

tify

eac

h p

air

of a

ngl

es a

s a

dja

cen

t,ve

rtic

al,

and

/or

as a

lin

ear

pair

.

1."

1 an

d "

22.

"1

and

"6

adja

cent

linea

r pa

ir;a

djac

ent

3."

1 an

d "

54.

"3

and

"2

vert

ical

adja

cent

For

Exe

rcis

es 5

–7,r

efer

to

the

figu

re a

t th

e ri

ght.

5.Id

enti

fy t

wo

obtu

se v

erti

cal a

ngle

s.!

RN

Tan

d !

SN

U

6.Id

enti

fy t

wo

acut

e ad

jace

nt a

ngle

s.!

RN

V a

nd!

VN

Tor

!

VN

Tan

d !

TNU

7.Id

enti

fy a

n an

gle

supp

lem

enta

ry t

o "

TN

U.

!U

NS

or !

TNR

8.F

ind

the

mea

sure

s of

tw

o co

mpl

emen

tary

ang

les

if th

e di

ffer

ence

in t

heir

mea

sure

s is

18.

36 a

nd 5

4

RS

NU

TV

RS

TU

V

P

Q5

43

21

6

©G

lenc

oe/M

cGra

w-H

ill26

Gle

ncoe

Geo

met

ry

Perp

end

icu

lar

Lin

esL

ines

,ray

s,an

d se

gmen

ts t

hat

form

fou

r ri

ght

angl

es a

re p

erp

end

icu

lar.

The

rig

ht a

ngle

sym

bol i

ndic

ates

tha

t th

e lin

es

are

perp

endi

cula

r.In

the

fig

ure

at t

he r

ight

,AC

! "#

is p

erpe

ndic

ular

to

BD

!"# ,

or A

C! "

#!

BD

!"# .

Fin

d x

so t

hat

D !Z!

!P!

Z!.

If D !

Z!!

P!Z!,t

hen

m"

DZ

P$

90.

m"

DZ

Q#

m"

QZ

P$

m"

DZ

PS

um o

f par

ts $

who

le

(9x

#5)

#(3

x#

1)$

90S

ubst

itutio

n

12x

#6

$90

Sim

plify

.12

x$

84S

ubtr

act 6

from

eac

h si

de.

x$

7D

ivid

e ea

ch s

ide

by 1

2.

1.F

ind

xan

d y

so t

hat

NR

!"#

!M

Q!"

# .x

$15

,y$

8

2.F

ind

m"

MS

N.9

0

3.m

"E

BF

$3x

#10

,m"

DB

E$

x,an

d B

D""

#!

BF

""# .

Fin

d x.

x$

20

4.If

m"

EB

F$

7y!

3 an

d m

"F

BC

$3y

#3,

find

yso

th

at E

B" "

# !B

C""

# .9

5.F

ind

x,m

"P

QS,

and

m"

SQ

R.

x$

8,m

!P

QS

$24

,m!

SQ

R$

66

6.F

ind

y,m

"R

PT,

and

m"

TP

W.

y$

15,m

!R

PT

$55

,m!

TPW

$35

P

S

V

R

W

T(4

y #

5)%

(2y

" 5

)%QR

PS

3x% (8

x "

2)%

BC

A

DE

F

M

N RSQP

x%5x

%

(9y

" 1

8)%

Z

D

P

Q(9

x "

5)% (3

x "

1)%

B CDA

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Ang

le R

elat

ions

hips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-5

1-5

Exam

ple

Exam

ple

Exercis

esExercis

es



An

swer

s

Skil

ls P

ract

ice

Ang

le R

elat

ions

hips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill27

Gle

ncoe

Geo

met

ry

Lesson 1-5

For

Exe

rcis

es 1

–6,u

se t

he

figu

re a

t th

e ri

ght

and

a

pro

trac

tor.

1.N

ame

two

acut

e ve

rtic

al a

ngle

s.!

EK

H,!

FKG

2.N

ame

two

obtu

se v

erti

cal a

ngle

s.!

EK

F,!

HK

G

3.N

ame

a lin

ear

pair

.S

ampl

e an

swer

:!E

KH

,!E

KF

4.N

ame

two

acut

e ad

jace

nt a

ngle

s.!

FKG

,!G

KJ

5.N

ame

an a

ngle

com

plem

enta

ry t

o "

EK

H.

!G

KJ

6.N

ame

an a

ngle

sup

plem

enta

ry t

o "

FK

G.

!E

KF

or !

GK

H

7.F

ind

the

mea

sure

s of

an

angl

e an

d it

s co

mpl

emen

t if

one

ang

le m

easu

res

18 d

egre

esm

ore

than

the

oth

er.

36,5

4

8.T

he m

easu

re o

f th

e su

pple

men

t of

an

angl

e is

36

less

tha

n th

e m

easu

re o

f th

e an

gle.

Fin

d th

e m

easu

res

of t

he a

ngle

s.72

,108

ALG

EBR

AF

or E

xerc

ises

9–1

0,u

se t

he

figu

re a

t th

e ri

ght.

9.If

m"

RT

S$

8x#

18,f

ind

xso

tha

t T

R""

#!

TS

""# .

9

10.I

f m

"P

TQ

$3y

!10

and

m"

QT

R$

y,fi

nd y

so t

hat

"P

TR

is a

rig

ht a

ngle

.25

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

can

be

assu

med

fr

om t

he

figu

re.E

xpla

in.

11."

WZ

Uis

a r

ight

ang

le.

Yes;

it is

mar

ked

with

a r

ight

ang

le s

ymbo

l.

12."

YZ

Uan

d "

UZ

V a

re s

uppl

emen

tary

.

Yes;

the

sum

of

thei

r m

easu

res

is 1

80 s

ince

the

an

gles

form

a li

near

pai

r.

13."

VZ

Uis

adj

acen

t to

"Y

ZX

.

No;

the

angl

es d

o no

t sh

are

a co

mm

on s

ide.

ZX

W

V

YU

TP

QR

S

K

G

J

FE H

©G

lenc

oe/M

cGra

w-H

ill28

Gle

ncoe

Geo

met

ry

For

Exe

rcis

es 1

–4,u

se t

he

figu

re a

t th

e ri

ght

and

a

pro

trac

tor.

1.N

ame

two

obtu

se v

erti

cal a

ngle

s.

Sam

ple

answ

er:!

GFH

,!C

FE2.

Nam

e a

linea

r pa

ir w

hose

ver

tex

is B

.!

GB

C,!

CB

A

3.N

ame

an a

ngle

not

adj

acen

t to

but

com

plem

enta

ry t

o "

FG

C.

!FE

D

4.N

ame

an a

ngle

adj

acen

t an

d su

pple

men

tary

to

"D

CB

.!

BC

Gor

!D

CH

5.T

wo

angl

es a

re c

ompl

emen

tary

.The

mea

sure

of

one

angl

e is

21

mor

e th

an t

wic

e th

em

easu

re o

f th

e ot

her

angl

e.F

ind

the

mea

sure

s of

the

ang

les.

23,6

7

6.If

a s

uppl

emen

t of

an

angl

e ha

s a

mea

sure

78

less

tha

n th

e m

easu

re o

f th

e an

gle,

wha

tar

e th

e m

easu

res

of t

he a

ngle

s?12

9,51

AL

GE

BR

AF

or E

xerc

ises

7–8

,use

th

e fi

gure

at

the

righ

t.

7.If

m"

FG

E$

5x#

10,f

ind

xso

tha

t F

C! "

#!

AE

!"# .

16

8.If

m"

BG

C$

16x

!4

and

m"

CG

D$

2x#

13,

find

xso

tha

t "

BG

Dis

a r

ight

ang

le.

4.5

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

can

be

assu

med

fro

m t

he

figu

re.E

xpla

in.

9."

NQ

Oan

d "

OQ

Par

e co

mpl

emen

tary

.

No;

m!

NQ

Pis

not

kno

wn

to b

e 90

.

10."

SR

Qan

d "

QR

Pis

a li

near

pai

r.Ye

s;th

ey a

re a

djac

ent

angl

es w

hose

non

com

mon

sid

es a

re o

ppos

ite r

ays.

11."

MQ

Nan

d "

MQ

Rar

e ve

rtic

al a

ngle

s.

No;

the

angl

es a

re a

djac

ent.

12.S

TREE

T M

APS

Dar

ren

sket

ched

a m

ap o

f th

e cr

oss

stre

ets

near

est

to h

is h

ome

for

his

frie

nd M

igue

l.D

escr

ibe

two

diff

eren

t an

gle

rela

tion

ship

s be

twee

n th

e st

reet

s.

Sam

ple

answ

er:B

eaco

n !

Mai

n;O

live

divi

des

two

of t

hean

gles

form

ed b

y B

acon

and

Mai

n in

to p

airs

of

com

plem

enta

ry a

ngle

s.

Oliv

e

Beacon

Main

Q

O

P

RS

M

N

GF

AB

E

D

C

DA

E

HG

BC

F

Pra

ctic

e (A

vera

ge)

Ang

le R

elat

ions

hips

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-5

1-5



Rea

din

g t

o L

earn

Math

emati

csA

ngle

Rel

atio

nshi

ps

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill29

Gle

ncoe

Geo

met

ry

Lesson 1-5

Pre-

Act

ivit

yW

hat

kin

ds

of a

ngl

es a

re f

orm

ed w

hen

str

eets

in

ters

ect?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

5 at

the

top

of

page

37

in y

our

text

book

.•

How

man

y se

para

te a

ngle

s ar

e fo

rmed

if t

hree

line

s in

ters

ect

at a

com

mon

poin

t? (D

o no

t us

e an

ang

le w

hose

inte

rior

incl

udes

par

t of

ano

ther

ang

le.)

6•

How

man

y se

para

te a

ngle

s ar

e fo

rmed

if n

lines

inte

rsec

t at

a c

omm

onpo

int?

(Do

not

coun

t an

ang

le w

hose

inte

rior

incl

udes

par

t of

ano

ther

ang

le.)

2n

Rea

din

g t

he

Less

on

1.N

ame

each

of

the

follo

win

g in

the

fig

ure

at t

he r

ight

.a.

two

pair

s of

con

grue

nt a

ngle

s!

1 an

d !

3,!

2 an

d !

4b.

a pa

ir o

f ac

ute

vert

ical

ang

les

!2

and

!4

c.a

pair

of

obtu

se v

erti

cal a

ngle

s!

1 an

d !

3d.

four

pai

rs o

f ad

jace

nt a

ngle

s!

1 an

d !

2,!

2 an

d !

3,!

3 an

d !

4,!

4 an

d !

1e.

two

pair

s of

ver

tica

l ang

les

!1

and

!3,

!2

and

!4

f.fo

ur li

near

pai

rs!

1 an

d !

2,!

2 an

d !

3,!

3 an

d !

4,!

4 an

d !

1g.

four

pai

rs o

f su

pple

men

tary

ang

les

!1

and

!2,

!2

and

!3,

!3

and

!4,

!4

and

!1

2.Te

ll w

heth

er e

ach

stat

emen

t is

alw

ays,

som

etim

es,o

r ne

ver

true

.a.

If t

wo

angl

es a

re a

djac

ent

angl

es,t

hey

form

a li

near

pai

r.so

met

imes

b.If

tw

o an

gles

for

m a

line

ar p

air,

they

are

com

plem

enta

ry.

neve

rc.

If t

wo

angl

es a

re s

uppl

emen

tary

,the

y ar

e co

ngru

ent.

som

etim

esd.

If t

wo

angl

es a

re c

ompl

emen

tary

,the

y ar

e ad

jace

nt.

som

etim

ese.

Whe

n tw

o pe

rpen

dicu

lar

lines

inte

rsec

t,fo

ur c

ongr

uent

ang

les

are

form

ed.

alw

ays

f.V

erti

cal a

ngle

s ar

e su

pple

men

tary

.so

met

imes

g.V

erti

cal a

ngle

s ar

e co

mpl

emen

tary

.so

met

imes

h.

The

tw

o an

gles

in a

line

ar p

air

are

both

acu

te.

neve

ri.

If t

wo

angl

es f

orm

a li

near

pai

r,on

e is

acu

te a

nd t

he o

ther

is o

btus

e.so

met

imes

3.C

ompl

ete

each

sen

tenc

e.a.

If t

wo

angl

es a

re s

uppl

emen

tary

and

xis

the

mea

sure

of

one

of t

he a

ngle

s,th

en t

hem

easu

re o

f th

e ot

her

angl

e is

.

b.If

tw

o an

gles

are

com

plem

enta

ry a

nd x

is t

he m

easu

re o

f on

e of

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desi

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1-5

1-5



An

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n be

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e ea

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y it

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um

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en c

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l its

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d so

it is

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1-6

1-6

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1-6

1-6



An

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Rea

din

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Geo

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Geo

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En

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1-6

1-6

Peri

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Documents

Chapter 1 Resource Masters - Wikispaces1.pdf · Chapter 1 Resource Masters ... list of the key vocabulary terms you will learn in Chapter 1.As you study the chapter, ... McGraw-Hill