geometry workbook answer

Prentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.

97

A N S W E R S

For each isometric drawing, make an orthographic drawing. Assume there are no hidden cubes.

9. 10. 11.

12. Visualization Look at the orthographic drawing at the right. Make an isometric drawing of the structure.

13. Choose the nets that will fold to make a cube. A. B. C.

14. Writing To make a net from a container, you start by cutting one of the seams along an edge where two sides meet. If you wanted to make a diff erent net for the container, what would you do diff erently?

15. Multiple Representations Draw two diff erent nets for the solid shown at the right.

FrontRigh

tFront Righ

t Front Right

Top Front Right

1-1 Practice (continued) Form GNets and Drawings for Visualizing Geometry

Top Front Right Top Front Right Top Front Right

Front Right

Sample: I would not separate the same edges when making a second net. Also, I would make sure that the result cannot be rotated or ipped so that it is the same as the rst.

A and B

Answers may vary. Samples:

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1-1 Practice Form GNets and Drawings for Visualizing GeometryMatch each three-dimensional fi gure with its net.

1. 2. 3.

A. B. C.

Make an isometric drawing of each cube structure on isometric dot paper.

4. 5. 6.

7. Error Analysis Two students draw nets for the solid shown below. Who is correct, Student A or Student B? Explain.

Student A:

Student B:

8. You want to make a one-piece cardboard cutout for a child to fold and tape together to make a dollhouse. It includes a fl oor, a complete roof, and four walls. Draw a net for the dollhouse.

A BC

Front Right Front Rig

htFront Right

Student B; the rst net has two faces that overlap when folded into a 3-D shape.

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1-1 Think About a PlanNets and Drawings for Visualizing GeometryMultiple Representations Th ere are eight diff erent nets for the solid shown at the right. Draw as many of them as you can. (Hint: Two nets are the same if you can rotate or fl ip one to match the other.)

Understanding the Problem

1. What is the net of a solid?

2. What is a result of fl ipping the net below? Of rotating it?

Planning the Solution

3. Visualize unfolding the solid so that the base shares an edge with all four triangles. Th en visualize unfolding the solid so that the base shares an edge with three triangles. Are the nets that result the same? Explain.

4. In Step 3, you saw that a net can have three or four triangles that share an edge with the square base. Are there other possibilities? If so, what are they? Are these the only possibilities?

Finding the Answer

5. Are there other nets that have three or four triangles that share an edge with the square base? Explain.

6. Th ere are four nets that have two triangles that share an edge with the base. For each of these, the triangles may either be on opposite or adjacent sides of the base. Draw each net.

7. How many nets have only one triangle touching the base? Draw as many of them as you can.

A net is a two-dimensional pattern that you can fold to form a three-dimensional gure.

Answers may vary. Sample:

The nets are different because you cannot superimpose them with a ip or a turn.

The base can share one, two, three, or four edges with the triangles. These are the

only possibilities.

No; any other such net can be superimposed on one of the nets from Step 3 by a ip or

a rotation.

There are two such nets.

Flip Rotate

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1-1 ELL SupportNets and Drawings for Visualizing GeometryAdd the phrases to the web. Place them where you think they should go.

corner view folds into a solid isometric drawing

right-side view top view

orthographic drawing

two-dimensional diagram of a three-dimensional fi gure

front view

net

folds into a solid

isometric drawing

corner view right-side view top view

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98

A N S W E R S

1-1 EnrichmentNets and Drawings for Visualizing GeometryOrthographic ProjectionsAn orthographic projection is a way of drawing a three-dimensional object on a two-dimensional surface. Th e projections you have seen so far have had solid lines and dashed lines to indicate visible edges and hidden edges, respectively.

In this exercise you will make isometric drawings from the top view, front view, and right side view of a structure of square blocks. However, the locations of the edges on the outline of the fi gure are not given. Draw two isometric drawings for each fi gure, one that uses the maximum number of blocks, and one that uses the minimum number of blocks.

1. Make two isometric drawings.

Minimum number of blocks: u Maximum number of blocks: u

2. Make two isometric drawings.

Minimum number of blocks: u Maximum number of blocks: u

Top

Front

Right

Top

Front

Right

Front

Sample:

FrontRight Rig

ht

Front

Sample:

FrontRig

ht Right

17

10

20

23

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1-1 Standardized Test PrepNets and Drawings for Visualizing GeometryMultiple Choice

For Exercises 13, choose the correct letter.

1. Which three-dimensional fi gure matches the net shown at the right?

2. Which cube structure matches the isometric drawing shown at the right?

3. Which top view of an orthographic drawing matchesthe isometric drawing shown at the right?

Short Response

4. You are building a small shed. You want to describe the area of the ground the shed will cover. Which type of drawing, isometric or orthographic, would best represent the area? Why?

FrontRigh

t

Front Right

[2] orthographic; sample: It includes a birds-eye or top-down view, which gives the buildings footprint on the ground. [1] correct answer with incomplete explanation [0] incorrect answer with no explanation given

A

B

I

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1-1 Practice (continued) Form KNets and Drawings for Visualizing Geometry 8. Visualization If the net shown at the right is

folded so that side A is the front of the cube, what letters will be on the top, bottom, right, left, and back?

9. Multiple Representations How many diff erent nets can you make for a cube? Draw at least fi ve nets.

10. Reasoning Are there more, fewer, or the same number of nets possible for a rectangular prism than for a cube? Explain.

11. Open-Ended Make an isometric drawing of a structure you can build using six cubes.

12. Error Analysis A classmate drew the net of a triangular prism shown at the right. Explain the error in your classmates drawing. Draw the net correctly.

Match the package with its net.

13. 14. 15.

A. B. C.

B A CE

DF

C A B

E; F; C; B; D

Eleven; check students work.

Answers may vary. Sample: Same number; a cube is a rectangular prism.

Check students work.

The bases need to be on opposite sides of the net.

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1-1 Practice Form KNets and Drawings for Visualizing GeometryMatch each three-dimensional fi gure with its net.

1. 2. 3.

A. B. C.

Draw a net for each fi gure. Label the net with its dimensions. To start, visualize opening the end fl aps of the prism.

4. 5.

Make an isometric drawing of each cube structure on isometric dot paper. To start, draw the front edge.

6. 7.

10 in.

7 in.

22 in.

14 cm

8 cm

12 cm

12 cm

10 in.

7 in.

7 in.

10 in.

22 in.

12 cm

14 cm12 cm

8 cm

C B A

6.

7.

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99

A N S W E R S

1-2 Think About a Plan Points, Lines, and PlanesEstimation You can represent the hands on a clock at 6:00 as opposite rays. Estimate the other 11 times on a clock that you can represent as opposite rays.

Know

1. Opposite rays are .

2. Th e hands on the clock represent rays. At 6:00, these rays form opposite rays.

Th is means they form a .

Need

3. To solve the problem I need to fi nd the 11 other times that

.

Plan

4. When the hour hand is between 1 and 2 oclock, what will the minute hand be between?

5. On the two clock faces at the right, draw the hands of a clock at 1:35 and at 1:38.

6. At which time, 1:35 or 1:38, do you think opposite rays form? Explain.

7. Complete the table to show all of the times when the hands on a clock represent opposite rays.

+

12

3

4

567

8

9

1011 12

+

12

3

4

567

8

9

1011 12

Hour

Time when oppositerays form

6

6:00

7 8 9 10 11

straight line

two rays that share the same endpoint and form a line

the hands of a clock form a straight line

somewhere between the 7 and 8

Answers may vary. Sample: Opposite rays form at 1:38. Even though 1 and 7 are

opposite each other on the clock face, the minute hand is not at 7 when the hour hand

is at 1, so the hands are not opposite rays at 1:35.

Answers may vary. Note that 6:00 is the only exact time. Reasonable estimates might be within 2 min of these times.

12 1 2 3 4

4:543:492:431:3812:3311:2710:229:168:117:05

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1-2 ELL SupportPoints, Lines, and PlanesComplete the vocabulary chart by fi lling in the missing information.

Word or Word Phrase

De nition Picture or Example

intersection An intersection is the set of points two or more fi gures have in common.

Point E is the intersection of the lines.

E

ray A ray is part of a line that consists of one endpoint and all the points of the line on one side of the endpoint.

1.

opposite rays 2. Point Q is the endpoint shared by these two rays.

segment A segment is part of a line that consists of two endpoints and all points between them.

3.

collinear Points that lie on the same line are collinear.

4.

postulate 5. Example: Th rough any two points there is exactly one line.

coplanar 6. Points G, H, and Z are coplanar.

Q

GZ

H

AB

X

Y

Opposite rays are two rays that share the same endpoint and form a line.

A postulate is an accepted statement of fact.

Points that lie on the same plane are coplanar.

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1-1 Reteaching (continued)Nets and Drawings for Visualizing GeometryAn isometric drawing is a corner-view drawing of a three-dimensional fi gure. It shows the top, front, and side views.

To make an isometric drawing, you can start by visualizing picking up a block structure and turning it so that you are looking directly at one face. Draw the edges of that face. Th en visualize and draw the two other faces.

Follow the steps below to make an isometric drawing of the block structure at the right.

Step 1 Step 2 Step 3

Draw the edges that Start at a vertex of the front Draw the back and topsurround the front face. face and draw an edge for the edges to connect the right side view. Edges only fi gure. Draw all occur at bends and folds. parallel edges.

Problem

What is an isometric drawing of the block structure at the right?

Isometric drawing:

Exercises 2. Draw a possible net for this fi gure. 3. Make the isometric drawing for

this structure.

FrontRig

ht

FrontRig

ht

Front Right

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A net is a two-dimensional fl at diagram that represents a three-dimensional fi gure. It shows all of the shapes that make up the faces of a solid.

Stepping through the process of building a three-dimensional fi gure from a net will help you improve your ability to visualize the process. Here are the steps you would take to build a square pyramid.

Step 1 Step 2 Step 3

Start with the net. Fold up on the Tape the adjacent dotted lines. triangle sides together.

Here are other examples of nets that also fold up into a square pyramid.

Problem

How can you be sure that none of the nets shown above are the same?

Make sure you cannot rotate or fl ip any net and place it on top of any other net.

Exercise

1. What is a possible net for the fi gure shown at the right?

A. B. C.

1-1 ReteachingNets and Drawings for Visualizing Geometry

A

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100

A N S W E R S

1-2 Practice (continued) Form KPoints, Lines, and PlanesUse the fi gure at the right for Exercises 1321.

Name the intersection of each pair of planes. To start, identify the points that both planes contain.

13. planes DCG and EFG

14. planes EFG and ADH

15. planes BCG and ABF

Name two planes that intersect in the given line. To start, identify the planes that contain the given line.

16. *CD) 17.

*DH) 18.

*EF)

Copy the fi gure. Shade the plane that contains the given points.

19. A, B, C 20. C, D, H 21. E, H, B

Postulate 1-4 states that any three noncollinear points lie in one plane. Find the plane that contains the fi rst three points listed. Th en determine whether the fourth point is in that plane. Write coplanar or noncoplanar to describe the points.

22. P, T, R, N 23. P, O, S, N

24. T, R, N, U 25. P, O, R, S

Use the diagram at the right. How many planes contain each line and point?

26. *KL) and G 27.

*HM) and F

28. *JI) and G 29.

*NM) and M

BA

F

D

E HG

C

OP

NU

R

S

T

N M

HIJ

L

F

K

O

G

ABC and DCG DHG and DHE EFG and EFB

coplanar

*GH)

*EH)

*BF)

coplanar

noncoplanar

1

1

1

in nitely many

noncoplanar

19. BA

F

D

E HG

C

20. BA

F

D

E HG

C

21. BA

F

D

E HG

C

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1-2 Practice Form KPoints, Lines, and PlanesUse the fi gure at the right for Exercises 14. Note that line r pierces the plane at X. It is not coplanar with V.

1. What are two other ways to name *QX)?

To start, remember you can name a line by any 9 point(s) on the line or by 9 lowercase letter(s).

Two other ways to name *QX) are line 9 and 9.

2. What are two other ways to name plane V?

3. Name three collinear points.

4. Name four coplanar points.

Use the fi gure at the right for Exercises 57.

5. Name six segments in the fi gure. To start, remember that a segment is part of a line that consists of 9 endpoints.

Six segments are AB, BC, 9, 9, 9, and 9.

6. Name the rays in the fi gure.

7. a. Name the pairs of opposite rays with endpoint C.

b. Name another pair of opposite rays.

For Exercises 812, determine whether each statement is always, sometimes, or never true.

8. Plane ABC and plane DEF are the same plane.

9. *DE) and *DF) are the same line.

10. Plane XYZ does not contain point Z.

11. All the points of a line are coplanar.

12. Two rays that share an endpoint form a line.

R V

XQZ

rs

AB

CD

two; one

never

always

sometimes

sometimes

sometimes

two

CD; AC; AD; BD

s; Answers may vary. Sample: *XQ)

Answers may vary. Samples: AB) or AC

)

or AD); BC) or BD

); CD); DC) or DB

) or DA

); CB) or CA

); BA)

Answers may vary. Samples: BA) and BD

) or BC)

Answers may vary. Samples: CD) and CA

) or CB)

Answers may vary. Sample: XQZ; RZQ

R, X, and Q

Z, R, X, and Q

0015_hsm11gmtr_0102.indd 15 3/23/09 6:48:10 AM

1-2 Practice (continued) Form GPoints, Lines, and Planes 17. Draw a fi gure to fi t each description. a. Th rough any two points there is exactly one line.

b. Two distinct lines can intersect in only one point.

18. Reasoning Point F lies on EG) and point M lies on EN

). If F, E, and M are

collinear, what must be true of these rays?

19. Writing What other terms or phrases mean the same as postulate?

20. How many segments can be named from the fi gure at the right?

Use the fi gure at the right for Exercises 2129. Name the intersection of each pair of planes or lines.

21. planes ABP and BCD

22. *RQ) and *RO)

23. planes ADR and DCQ

24. planes BCD and BCQ

25. *OP) and *QP)

Name two planes that intersect in the given line.

26. *RO) 27.

*CQ) 28.

*DA) 29.

*BP)

Coordinate Geometry Graph the points and state whether they are collinear.

30. (0, 0), (4, 2), (6, 3) 31. (0, 0), (6, 0), (9, 0)

32. (21, 1), (2, 22), (4, 23) 33. (1, 2), (2, 3), (4, 5)

34. (22, 0), (0, 4), (2, 0) 35. (24, 21), (21, 22), (2, 23)

D EF

G

C

A B

O

R Q

PD

6

point R

ODR and OPR CQR and CQP OAD and BAD CBP and APB

point P

Sample answer:

Sample answer:

Answers may vary. Samples given:

They are either opposite rays or identical rays.

Answers may vary. Samples: axiom, statement of fact, assumption, statement of truth

*BC)

Check students graphs.

yes

no

no

yes

yes

yes

*AB)

*DR)

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1-2 Practice Form GPoints, Lines, and PlanesUse the fi gure below for Exercises 18. Note that

*RN) pierces the plane at N. It is

not coplanar with V.

AR

NM

X

CV

1. Name two segments shown in the fi gure.

2. What is the intersection of *CM) and *RN)?

3. Name three collinear points.

4. What are two other ways to name plane V?

5. Are points R, N, M, and X coplanar?

6. Name two rays shown in the fi gure.

7. Name the pair of opposite rays with endpoint N.

8. How many lines are shown in the drawing?

For Exercises 914, determine whether each statement is always, sometimes, or never true.

9. GH) and HG

) are the same ray.

10. JI) and JL

) are opposite rays.

11. A plane contains only three points.

12. Th ree noncollinear points are contained in only one plane.

13. If *EG) lies in plane X, point G lies in plane X.

14. If three points are coplanar, they are collinear.

15. Reasoning Is it possible for one ray to be shorter in length than another? Explain.

16. Open-Ended Draw a fi gure of two planes that intersect in *ST).

Answers may vary. Sample: NM and NX

point N

Answers may vary. Sample: points C, N, and M

Answers may vary. Sample: plane CNX and AXM

no

Answers may vary. Sample: NM) and CN

)

Answers may vary. Sample: NM)

and NC)

3

never

sometimes

sometimes

never

always

always

It is not possible. Each ray has an endpoint, but each continues on in one direction without end. They are the same length because they are both in nitely long.

Sample:S T

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101

A N S W E R S

1-2 Reteaching (continued) Points, Lines, and PlanesA postulate is a statement that is accepted as true.

Postulate 14 states that through any three noncollinear points, there is only one plane. Noncollinear points are points that do not all lie on the same line.

In the fi gure at the right, points D, E, and F are noncollinear. Th ese points all lie in one plane.

Th ree noncollinear points lie in only one plane. Th ree points that are collinear can be contained by more than one plane. In the fi gure at the right, points P, Q, and R are collinear, and lie in both plane O and plane N.

Exercises

Identify the plane containing the given points as front, back, left side, right side, top, or bottom.

F G

I H

Z

W

Y

X

Top

Right

Front

7. F, G, and X 8. F, G, and H

9. H, I, and Z 10. F, W, and X

11. I, W, and Z 12. Z, X, and Y

13. H, G, and X 14. W, Y, and Z

Use the fi gure at the right to determine how many planes contain the given group of points. Note that

*GF) pierces the plane at R,

*GF) is not

coplanar with X, and *GF) does not intersect

*CE).

15. C, D, and E 16. D, E, and F

17. C, G, E, and F 18. C and F

D F E

QR

P

N

O

C D

F

E

G

R X

back

back

left side

right side

in nite number of planes

in nite number of planes

bottom

bottom

1 plane

0 planes

front

top

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1-2 Reteaching Points, Lines, and PlanesReview these important geometric terms.

Remember:

1. When you name a ray, an arrowhead is not drawn over the beginning point.

2. When you name a plane with three points, choose no more than two collinear points.

3. An arrow indicates the direction of a path that extends without end.

4. A plane is represented by a parallelogram. However, the plane actually has no edges. It is fl at and extends forever in all directions.

Exercises

Identify each fi gure as a point, segment, ray, line, or plane, and name each.

1. 2. 3.

4. 5. 6.

Term Examples of Labels Diagram

Point Italicized capital letter: D

Line

Line Segment

Ray

Plane

Three capital letters: ABF, AFB, BAF, BFA,

FAB, or FBA

One italicized capital letter: W

D

FA

BW

B

A

BmA

AB

Two capital letters (called endpoints)

with a segment drawn over them: AB or BA

Two capital letters with a line drawn

over them: AB or BA

One italicized lowercase letter: m

Two capital letters with a ray symbol

drawn over them: AB

LL N

M CD

P

O

E F G T S

point; point L line; *CD)

ray; ST)segment; PO

plane; Answers may vary. Sample: plane LMN

line; answers may vary. Sample: *EG)

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1-2 Enrichment Points, Lines, and PlanesAssembly Required

Th e bookshelf shown at the right comes in separate pieces, with assembly instructions. Th e pieces are listed below. T is the top piece, B is the base, and S1 and S2 are shelves. R is the right side, L is the left side, and X is the back piece.

Use the diagrams to answer each question.

1. After the bookshelf is assembled, will lines j and o intersect?

2. Will lines k and j intersect?

At how many points will the following lines intersect?

3. j and q 4. m and p 5. h and o

Name the intersection of each pair of planes. If they do not intersect, write none.

6. B and S1 7. B and L 8. B and R

Name two planes that intersect in the given line.

9. k 10. h 11. m

12. Draw a diff erent bookshelf. Th en draw each separate component of the bookshelf. How many planes are in your drawing? Which of these planes intersect?

q

r

n

T

LS1

XR

S2

B

o

jk

h

m

p

l

i

S1

S2

T

B

L R X

no

1 0 1

yes

none

Check students drawings.

line n

T and L X and R S2 and X

line o

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1-2 Standardized Test Prep Points, Lines, and PlanesMultiple Choice


1. Look at the fi gure at the right. Where do planes ACE and BCD intersect?

*AD)

*CB)

*CD)

*BF)

2. Which of the following are opposite rays?

TS) and XS

) TS

) and TZ

)

TX) and TZ

) TS

) and TX

)

3. What is the smallest number of distinct points that can defi ne a plane?

2 3 4 infi nite

4. At how many points can two distinct lines intersect?

1 2 3 4

5. In the fi gure at the right, which line is

the same as *ED)?

*ML)

*NL)

*DM)

*MN)

6. If two lines are coplanar, which of the following must be true?

Th e lines intersect.

Th e lines never intersect.

All points on the lines are coplanar.

Th e lines share at least one point.

7. What is the intersection of two distinct, non-parallel planes?

a point a line a line segment a ray

Short Response

8. Point C does not lie on *XY). Can point C lie in the same plane as

*XY)? Explain.

A

BC

D

E

F

TS

ZX

L

NM D

E

B

B

F

B

B

H

I

[2] Yes; X, Y, and C are noncollinear, and any three noncollinear points are coplanar.[1] answer correct, but explanation missing, incomplete, or incorrect [0] no explanation given

0012_hsm11gmtr_0102.indd 17 3/23/09 7:19:24 AM

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A N S W E R S

1-3 Practice (continued) Form GMeasuring SegmentsOn a number line, the coordinates of P, Q, R, and S are 212, 25, 0, and 7, respectively.

22. Draw a sketch of this number line. Use this sketch to answer Exercises 2326.

23. Which line segment is the shortest?

24. Which line segment is the longest?

25. Which line segments are congruent?

26. What is the coordinate of the midpoint of PR?

27. You plan to drive north from city A to town B and then continue north to city C. Th e distance between city A and town B is 39 mi, and the distance between town B and city C is 99 mi.

a. Assuming you follow a straight driving path, after how many miles of driving will you reach the midpoint between city A and city C?

b. If you drive an average of 46 mi/h, how long will it take you to drive from city A to city C?

28. Algebra Point O lies between points M and P on a line. OM 5 34z and OP 5 36z 2 7. If point N is the midpoint of MP, what algebraic equation can you use to fi nd MN?

Algebra Use the diagram at the right for Exercises 2932.

29. If AD 5 20 and AC 5 3x 1 4, fi nd the value of x. Th en fi nd AC and DC.

30. If ED 5 5y 1 6 and DB 5 y 1 30, fi nd the value of y. Th en fi nd ED, DB, and EB.

31. If DC 5 6x and DA 5 4x 1 18, fi nd the value of x. Th en fi nd AD, DC, and AC.

32. If EB 5 4y 2 12 and ED 5 y 1 17, fi nd the value of y. Th en fi nd ED, DB, and EB.

33. Writing Is it possible that PQ 1 QR , PR? Explain.

A B

CED

QR

PS

PQ and RS; also PR and QS

69 mi

3 h

MN 5 12[34z 1 (36z 2 7)]

12; 40; 20

6; 36; 36; 72

9; 54; 54; 108

23; 40; 40; 80

Answers may vary. Sample: No; if Q lies on the segment connecting P and R, then PQ 1 QR 5 PR, but this is the least the sum PQ 1 QR can be, so the sum is always greater than or equal to PR.

26

P Q R S

12 10 8 6 4 2 0 2 4 6 8

0022_hsm11gmtr_0103.indd 24 3/23/09 7:18:55 AM

1-3 Practice Form GMeasuring SegmentsIn Exercises 16, use the fi gure below. Find the length of each segment.

A

65432101234

B C D

1. AB 2. BC 3. AC

4. AD 5. BD 6. CD

For Exercises 711, use the fi gure at the right.

7. If PQ 5 7 and QR 5 10, then PR 5u. 8. If PQ 5 20 and QR 5 22, then PR 5u. 9. If PR 5 25 and PQ 5 12, then QR 5u. 10. If PR 5 19 and QR 5 12, then PQ 5u. 11. If PR 5 10 and PQ 5 4, then QR 5u.Use the number line below for Exercises 1216. Tell whether the segments are congruent.

G H I J K

20246810

12. GH and HI 13. GH and IK 14. HJ and IK

15. IJ and JK 16. HJ and GI 17. HK and GI

18. Reasoning Points A, Q, and O are collinear. AO 5 10, AQ 5 15, and OQ 5 5. What must be true about their positions on the line?

Algebra Use the fi gure at the right for Exercises 19 and 20.

19. Given: ST 5 3x 1 3 and TU 5 2x 1 9. a. What is the value of ST? b. What is the value of TU?

20. Given: ST 5 x 1 3 and TU 5 4x 2 6. a. What is the value of ST? b. What is the value of SU?

21. Algebra On a number line, suppose point E has a coordinate of 3, EG 5 6, and EX 5 12. Is point G the midpoint of EX ? What are possible coordinates for G and X?

P Q R

T US

3 5 8

6 1

17

42

13

7

6

yes yes

no

Point O is between points A and Q.

no

yes yes

9

21

126

21

No; sample: Point G is at 9 or 23; point X is at 29 or 15.

0022_hsm11gmtr_0103.indd 23 3/23/09 7:18:52 AM

1-3 Think About a Plan Measuring SegmentsIf AD 5 12 and AC 5 4y 2 36, fi nd the value of y. Th en fi nd AC and DC.


1. What are the two congruence relationships that the diagram shows?

2. What is the value of DC? u

3. Write an equation that describes the relationship between AC, DC, and AD.


4. How can you use the equation in Exercise 3 above to fi nd the value of y?

Getting an Answer

5. Write an equation for y using the method described in Exercise 4 above.

6. Solve for y.

7. AC 5 4y 2 36. Substitute the value of y to fi nd AC.

8. Check your answer. Does it make the equation that you wrote in Step 5 true?

A B

CD

E

AD 5 DC, BD 5 DE

12

AC 5 AD 1 DC

Substitute the given values for AC and AD, and the value of DC, into the

equation above.

4y 2 36 5 12 1 12 or 4y 2 36 5 24

y 5 15

24

yes

0022_hsm11gmtr_0103.indd 22 3/23/09 7:18:51 AM

1-3 ELL SupportMeasuring SegmentsTh e column on the left shows the steps used to fi nd the lengths of line segments. Use the column on the left to answer each question in the column on the right.

Problem Using the Midpoint

T is the midpoint of SU . What are ST, TU, and SU?

1. What is the midpoint of a line segment?

Write an equation to relate ST to TU.

ST > TU , so ST 5 TU .

2. What does ST > TU mean? Why is this true?

Write an equation to nd x by substituting.

5x 2 2 5 3x 1 4

3. What does it mean to substitute here?

Subtract 3x from each side. Add 2 to each side.

2x 2 2 5 4

2x 5 6

4. Why do you add 2 to each side?

Divide each side by 2.

x 5 3

Use x to nd ST and TU. Substitute 3 for x.

ST 5 5(3) 2 2 5 13

TU 5 3(3) 1 4 5 13

5. What does it mean to substitute 3 for x?

Check that ST 5 TU.

ST 5 13, TU 5 13, ST 5 TU

6. What would you do if ST were not equal to TU when you checked the problem?

Find SU by adding ST and TU.

ST 1 TU 5 13 1 13 5 26

7. What is another way to fi nd SU?

S T U

5x 2 3x 4

a point that divides the line segment

into two congruent segments

It means the segments are congruent.

The midpoint divides SU into two

congruent segments.

It means to replace ST and TU with the

given algebraic expressions.

To get the term with x alone on one

side of the equation.

It means to replace x in the algebraic

expression with the number 3.

Solve for x again.

Multiply ST or TU by 2.

0021_hsm11gmtr_0103.indd 21 3/23/09 7:03:14 AM

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103

A N S W E R S

1-3 Enrichment Measuring SegmentsSegment Addition

1. Use the following segments to form a path between points A and B. Th e path must reach B exactly, without going past it. You may place the segments end-to-end, or with their endpoints perpendicular, but you must place the vertical segments vertically and the horizontal segments horizontally. You do not have to use all of the segments, but you must place each segment so that the path goes in the direction of the arrow. See how many solutions you can fi nd.

2. Use the following segments to form a path between points A and B. Remember that the path must go in the direction of the arrow. How many solutions can you fi nd?

3

4

5

6

9

23

46

7

1

A

B

1

2

6

7

8

10

23

56

7

A

B



5 6

6

4 2

5

7

7

8

6

610

2

0022_hsm11gmtr_0103.indd 28 3/23/09 7:19:01 AM

1-3 Standardized Test Prep Measuring SegmentsGridded Response

Solve each exercise and enter your answer on the grid provided.

1. What is the length of BD?

A B C D

10 8 6 4 2 0 2

2. Points G, H, and I are collinear and H is between G and I. If GH 5 12 and GI 5 23, what is HI?

3. Look at the diagram below. If XY 5 7 and XZ 5 30, what is the value of t?

X Y Z

2t 17

For Exercises 4 and 5, use the fi gure at the right.

4. M is the midpoint of LN . What is LM?

5. What is LN?

L M N

2x 23x

Answers

1. 2. 3. 4. 5.

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

a

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

a

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

a

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

a

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

a

0022_hsm11gmtr_0103.indd 27 3/23/09 7:18:57 AM

Suppose the coordinate of P is 2, PQ 5 8, and PR 5 12. What are the possible coordinates of the midpoint of the given segment?

17. PQ 18. PR 19. QR

Visualization Without using your ruler, sketch a segment with the given length. Use your ruler to see how well your sketch approximates the length provided.

20. 5 cm 21. 8 in. 22. 8 cm

23. 12 cm 24. 85 mm 25. 5 in.

26. Suppose point J has a coordinate of 22 and JK 5 4. What are the possible coordinates of point K?

27. Suppose point X has a coordinate of 5 and XY 5 10. What are the possible coordinates of point Y?

Algebra Use the diagram at the right for Exercises 2832.

28. If NO 5 17 and NP 5 5x 2 6, fi nd the value of x. Th en fi nd NP and OP.

29. If RO 5 6 1 x and OQ 5 2x 1 1, fi nd the value of x. Th en fi nd RO, OQ, and RQ.

30. If NO 5 3x 1 4 and NP 5 10x 2 10, fi nd the value of x. Th en fi nd NO, NP, and OP.

31. If RO 5 5x and RQ 5 12x 2 20, fi nd the value of x. Th en fi nd RO, OQ, and RQ.

32. Vocabulary What term describes the relationship between NP and RQ?

33. Reasoning If KL 5 5 and KJ 5 10, is it possible that LJ 5 5? Explain.

N

OPR

Q

1-3 Practice (continued) Form KMeasuring Segments

6 or 22


26 and 2

15 or 25

8; NP 5 34; OP 5 17

5; RO 5 11; OQ 5 11; and RQ 5 22

4.5; NO 5 17.5; OP 5 17.5; NP 5 35

8 or 24 12, 4, 0, or 28

10; RO 5 50; OQ 5 50; and RQ 5 100

They are segment bisectors of each other because they divide each other in half.

Yes; if L is between K and J, and L is on the segment KJ, then LJ 5 5.

0025_hsm11gmtr_0103.indd 26 3/23/09 6:48:44 AM

1-3 Practice Form KMeasuring SegmentsFind the length of each segment. To start, fi nd the coordinate of each endpoint.

1. PR 2. QT 3. QS

Use the number line at the right for Exercises 46.

4. If GH 5 31 and HI 5 11, then GI 5u.

5. If GH 5 45 and GI 5 61, then HI 5u.

6. Algebra GH 5 7y 1 3, HI 5 3y 2 5, and GI 5 9y 1 7.

a. What is the value of y?

b. Find GH, HI, and GI.

Use the number line below for Exercises 79. Tell whether the segments are congruent. To start, use the defi nition of distance. Use the coordinates of the points to write an equation for each distance.

7. CE and FD 8. CD and FG 9. GE and BD

For Exercises 1012, use the fi gure below. Find the value of KL.

10. KL 5 3x 1 2 and LM 5 5x 2 10

11. KL 5 8x 2 5 and LM 5 6x 1 3

12. KL 5 4x 1 7 and LM 5 5x 2 4

On a number line, the coordinates of D, E, F, G, and H are 29, 22, 0, 3, and 5, respectively. Find the lengths of the two segments. Th en tell whether they are congruent.

13. DG and DH 14. DE and EH

15. EG and GH 16. EG and FH

5 2 4 5 8

P Q SR T

G H I

15 0510 5 10

GFEDCB

K L M

9

9

66; 22; 88

no

20

27

51

12 and 14; no

5 and 2; no

7 and 7; yes

5 and 5; yes

yes yes

10 7

42

16

0025_hsm11gmtr_0103.indd 25 3/23/09 6:48:41 AM

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104

A N S W E R S

1-4 Think About a Plan Measuring AnglesUse the diagram at the right. Solve for x. Find the angle measures to check your work.

m/AOB 5 4x 2 2, m/BOC 5 5x 1 10, m/COD 5 2x 1 14


1. Th e diagram shows that /AOB and are congruent.

2. So, m/AOB 5 .


3. How can you use the information in Step 2 to write an equation for x?

4. Write an equation for x.

Getting an Answer

5. Solve for x.

6. Find the measures of the angles by substituting for x.

m/AOB 5u m/BOC 5u m/COD 5u 7. Measure the angles using a protractor to check your answers. Are

they reasonable?

A B

C

DO

lCOD

mlCOD

Substitute the given values of mlAOB and mlCOD into the equation in Exercise 2.

4x 2 2 5 2x 1 14

x 5 8

30

30

50

yes

0032_hsm11gmtr_0104.indd 32 3/23/09 7:18:22 AM

1-4 ELL SupportMeasuring AnglesTamsin wants to fi nd the measures of lSOP and lPOT , when m/SOT 5 100.

She wrote the following steps on note cards, but they got mixed up.

Substitute the expressions for the measures of the angles.

(2n 1 4) 1 (3n 2 19) 5 100

Combine like terms and solve for n.

5n 2 15 5 100

5n 5 115

n 5 23

Substitute the value of n to find mlPOT and mlSOP.

mlPOT 5 3(23) 2 19 5 50;

mlSOP 5 2(23) 1 4 5 50

Write the equation

mlSOP 1 mlPOT 5 mlSOT.

Use the note cards to complete the steps below. Follow the instructions on each note card to solve the problem.

1. First, ______________________________________________________

___________________________________________________________

2. Second, ____________________________________________________

___________________________________________________________

3. Next, ______________________________________________________

___________________________________________________________

4. Finally, _____________________________________________________

___________________________________________________________

T

(3n 19)

O

S

P

(2n 4)

write the equation mlSOP 1 mlPOT 5 mlSOT .

substitute the expressions for the measures of the angles.

(2n 1 4) 1 (3n 2 19) 5 100

combine like terms and solve for n.

5n 2 15 5 100, 5n 5 115, n 5 23

substitute the value of n to nd mlPOT and mlSOP.

mlPOT 5 3(23) 2 19 5 50; mlSOP 5 2(23) 1 4 5 50

0031_hsm11gmtr_0104.indd 31 3/23/09 7:04:51 AM

1-3 Reteaching (continued) Measuring SegmentsTh e midpoint of a line segment divides the segment into two segments that are equal in length. If you know the distance between the midpoint and an endpoint of a segment, you can fi nd the length of the segment. If you know the length of a segment, you can fi nd the distance between its endpoint and midpoint.

W X Y

X is the midpoint of WY . XW 5 WY , so XW and WY are congruent.

Problem

C is the midpoint of BE. If BC 5 t 1 1, and CE 5 15 2 t , what is BE?

B C E

BC 5 CE De nition of midpoint

t 1 1 5 15 2 t Substitute.

t 1 t 1 1 5 15 2 t 1 t Add t to each side.

2t 1 1 5 15 Simplify.

2t 1 1 2 1 5 15 2 1 Subtract 1 from each side.

2t 5 14 Simplify.

t 5 7 Divide each side by 2.

BC 5 t 1 1 Given.

BC 5 7 1 1 Substitute.

BC 5 8 Simplify.

BE 5 2(BC) De nition of midpoint.

BE 5 2(8) Substitute.

BE 5 16 Simplify.

Exercises

12. W is the midpoint of UV . If UW 5 x 1 23, and WV 5 2x 1 8, what is x?

13. W is the midpoint of UV . If UW 5 x 1 23, and WV 5 2x 1 8, what is WU?

14. W is the midpoint of UV . If UW 5 x 1 23, and WV 5 2x 1 8, what is UV?

15. Z is the midpoint of YA. If YZ 5 x 1 12, and ZA 5 6x 2 13, what is YA?

15

38

76

34

0022_hsm11gmtr_0103.indd 30 3/23/09 7:19:07 AM

Th e Segment Addition Postulate allows you to use known segment lengths to fi nd unknown segment lengths. If three points, A, B, and C, are on the same line, and point B is between points A and C, then the distance AC is the sum of the distances AB and BC.

A B C

AC 5 AB 1 BC

Problem

If QS 5 7 and QR 5 3, what is RS?

Q R S

QS 5 QR 1 RS Segment Addition Postulate

QS 2 QR 5 RS Subtract QR from each side.

7 2 3 5 RS Substitute.

4 5 RS Simplify.

Exercises

For Exercises 15, use the fi gure at the right.

1. If PN 5 29 cm and MN 5 13 cm, then PM 5 z z. 2. If PN 5 34 cm and MN 5 19 cm, then PM 5 z z. 3. If PM 5 19 and MN 5 23, then PN 5 z z. 4. If MN 5 82 and PN 5 105, then PM 5 z z. 5. If PM 5 100 and MN 5 100, then PN 5 z z.For Exercises 68, use the fi gure at the right.

6. If UW 5 13 cm and UX 5 46 cm, then WX 5 z z. 7. UW 5 2 and UX 5 y. Write an expression for WX.

8. UW 5 m and WX 5 y 1 14. Write an expression for UX.

On a number line, the coordinates of A, B, C, and D are 26, 22, 3, and 7, respectively. Find the lengths of the two segments. Th en tell whether they are congruent.

9. AB and CD 10. AC and BD 11. BC and AD

P M N

U W X

1-3 Reteaching Measuring Segments

16 cm

42

23

200

15 cm

33 cm

4; 4; yes 9; 9; yes 5; 13; no

m 1 y 1 14

y 2 2

0022_hsm11gmtr_0103.indd 29 3/23/09 7:19:04 AM

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105

A N S W E R S

15. If m/FHI 5 142, what are m/FHG and m/GHI ?

16. /JKL is a right angle. What are m/JKM and m/MKL?

Use a protractor. Measure and classify each angle.

17. 18.

19. 20.

Algebra Use the diagram at the right for Exercises 2123. Solve for x. Find the angle measures to check your work.

21. m/CGD 5 4x 1 2, m/DGE 5 3x 2 5, m/EGF 5 2x 1 10

22. m/CGD 5 2x 2 2, m/EGF 5 37, m/CGF 5 7x 1 2

23. If m/DGF 5 72, what equation can you use to fi nd m/EGF ?

24. Th e fl ag of the United Kingdom is shown at the right. Copy the fl ag on a separate piece of paper. Label at least two of each type of angle:

a. acute b. obtuse

c. right d. straight

FG

H

(9x 8)(3x 6)I K

JM

(6x 3)(12x 3)

L

G

C

D E

F

1-4 Practice (continued) Form KMeasuring Angles

42; 100

162; obtuse

100; obtuse

58; acute

90; right

15; 62; 40; 40

14; 26; 37; 37; 100

2x 5 72


63; 27

0035_hsm11gmtr_0104.indd 36 3/23/09 6:49:16 AM

1-4 Practice Form KMeasuring AnglesName each shaded angle in three diff erent ways. To start, identify the rays that form each angle.

1. 2. 3.

Use the diagram below. Find the measure of each angle. Th en classify the angle as acute, right, obtuse, or straight.

4. /AFB

To start, identify /AFB. Th en use the defi nition of the measure of an angle to fi nd m/AFB.

m/AFB 5 uu 2u u 5uTh is angle is a(n) 9 angle.

5. /AFD

6. /CFD

7. /BFD

8. /AFE

9. /BFE

10. /AFC

Use the diagram at the right. Complete each statement.

11. /MIG > z z

12. /PMJ > z z

13. If m/KJL 5 30, then m/z z 5 30.

14. If m/LMP 5 100, then m/QHG 5 z z.

C

B

A

H

F

G

JM

LK

1

A

B

C

F

D

E

9090

80

100

10080

11070

12060 13050 14040 15030 16020

17010

1800

70

11060

12050

130

40140

30150

20160

10 170

0 180

1800

P

M

G

Q

H

JL

K

I

lABC, lCBA, lB

acute

140; obtuse

65; acute

90; right

180; straight

130; obtuse

75; acute

lFGH, lHGF, lG

lMKL, lLKM, l1

180 50130

MJI

100

lJIH

lGHQ

0035_hsm11gmtr_0104.indd 35 3/23/09 6:49:13 AM

1-4 Practice (continued) Form GMeasuring Angles 19. /JKL and /CDE are congruent. If m/JKL 5 137, what is m/CDE?

Use the fi gure at the right for Exercises 2023. mlFXH 5 130 and mlFXG 5 49.

20. /FXG >u 21. m/GXH 5u 22. Name a straight angle in the fi gure.

23. /IXJ >u 24. Algebra If m/RZT 5 110, m/RZS 5 3s,

and m/TZS 5 8s, what are m/RZS and m/TZS?

25. Algebra m/OZP 5 4r 1 2, m/PZQ 5 5r 2 12, and m/OZQ 5 125. What are m/OZP and m/PZQ?

26. Reasoning Elsa draws an angle that measures 56. Tristan draws a congruent angle. Tristan says his angle is obtuse. Is he correct? Why or why not?

27. Lisa makes a cherry pie and an apple pie. She cuts the cherry pie into six equal wedges and she cuts the apple pie into eight equal wedges. How many degrees greater is the measure of a cherry pie wedge than the measure of an apple pie wedge?

28. Reasoning /JNR and /RNX are congruent. If the sum of the measures of the two angles is 180, what type of angle are they?

29. A new pizza place in town cuts their circular pizzas into 12 equal slices. What is the measure of the angle of each slice?

GH

I

JF

X

R

Z

S

T

O

P

QZ

137

30 and 80

lIXJ or lJXI

right

30

62 and 63

No; if his angle is congruent to Elsas, it must have the same measure, 56. Because this is between 0 and 90, the angle is acute.

81

15

lFXI or lIXF or lGXJ or lJXG

lGXF or lFXG

0032_hsm11gmtr_0104.indd 34 3/23/09 7:18:26 AM

1-4 Practice Form GMeasuring AnglesUse the diagram below for Exercises 111. Find the measure of each angle.

1. /MLN

2. /NLP

3. /NLQ

4. /OLP

5. /MLQ

Classify each angle as acute, right, obtuse, or straight.

6. /MLN 7. /NLO 8. /MLP

9. /OLP 10. /OLQ 11. /MLQ

Use the fi gure at the right for Exercises 12 and 13.

12. What is another name for /XYW ?

13. What is another name for /WYZ?

Use a protractor. Measure and classify each angle.

14. 15. 16.

17. 18.

M L

NO

P

Q

9090

80

100

10080

11070

12060 13050 14040 15030 16020

17010

1800

70

11060

12050

130

40140

30150

20160

10 170

0 180

1 2Y

W

ZX

60

70

120

40

180

acute

acute

33; acute

90; right97; obtuse

125; obtuse 172; obtuse

acute

right straight

l1 or lWYX

l2 or lZYW

obtuse

0032_hsm11gmtr_0104.indd 33 3/23/09 7:18:23 AM

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106

A N S W E R S

1-4 Reteaching (continued) Measuring AnglesTh e Angle Addition Postulate allows you to use a known angle measure to fi nd an unknown angle measure. If point B is in the interior of /AXC, the sum of m/AXB and m/BXC is equal to m/AXC.

m/AXB 1 m/BXC 5 m/AXC

Problem

If m/LYN 5 125, what are m/LYM and m/MYN ?

Step 1 Solve for p.

m/LYN 5 m/LYM 1 m/MYN Angle Addition Postulate

125 5 (4p 1 7) 1 (2p 2 2) Substitute.

125 5 6p 1 5 Simplify.

120 5 6p Subtract 5 from each side.

20 5 p Divide each side by 6.

Step 2 Use the value of p to fi nd the measures of the angles.

m/LYM 5 4p 1 7 Given

m/LYM 5 4(20) 1 7 Substitute.

m/LYM 5 87 Simplify.

m/MYN 5 2p 2 2 Given

m/MYN 5 2(20) 2 2 Substitute.

m/MYN 5 38 Simplify.

Exercises

19. X is in the interior of /LIN . m/LIN 5 100, m/LIX 5 14t , and m/XIN 5 t 1 10.

a. What is the value of x? b. What are m/LIX and m/XIN ?

20. Z is in the interior of /GHI . m/GHI 5 170, m/GHZ 5 3s 2 5, and m/ZHI 5 2s 1 25.

a. What is the value of s? b. What are m/GHZ and m/ZHI ?

A

B C

X

L Y

M

(4p 7)

(2p 2)N

6 84, 16

30 85, 85

0032_hsm11gmtr_0104.indd 40 3/23/09 7:18:39 AM

1-4 Reteaching Measuring AnglesTh e vertex of an angle is the common endpoint of the rays that form the angle. An angle may be named by its vertex. It may also be named by a number or by a point on each ray and the vertex (in the middle).

Th is is /Z, /XZY , /YZX , or /1.

It is not /ZYX , /XYZ, /YXZ, or /ZXY .

Angles are measured in degrees, and the measure of an angle is used to classify it.

Exercises

Use the fi gure at the right for Exercises 1 and 2.

1. What are three other names for /S?

2. What type of angle is /S?

3. Name the vertex of each angle. a. /LGH b. /MBX

Classify the following angles as acute, right, obtuse, or straight.

4. m/LGH 5 14 5. m/SRT 5 114

6. m/SLI 5 90 7. m/1 5 139

8. m/L 5 179 9. m/P 5 73

Use the diagram below for Exercises 1018. Find the measure of each angle.

10. /ADB 11. /FDE

12. /BDC 13. /CDE

14. /ADC 15. /FDC

16. /BDE 17. /ADE

18. /BDF

Y

X

1Z

The measure ofan acute angleis between 0 and 90.

The measure of aright angle is 90.

The measure of anobtuse angle is between90 and 180.

The measure of astraight angle is 180.

3

R

S T

A

B

C

D

E

F

9090

80

100

10080

11070

12060 13050 14040 15030 16020

17010

1800

70

11060

12050

130

40140

30150

20160

10 170

0 180

1800

right

right

point G

25

55

80

120

155

35

65

100

145

acute

acute

obtuse

obtuse

obtuse

point B

lRST , lTSR, l3

0032_hsm11gmtr_0104.indd 39 3/23/09 7:18:36 AM

1-4 Enrichment Measuring AnglesSpreading SeedsA student follows the path that three seeds take on their way to the ground. She draws a diagram to show how each seed drops from Point A. Th e seeds land on the ground (the number line) at points E, H, and I. Complete the diagram using this information.

Maple Seed Moves from point A to B, right to C, down to D, and left to E.

m/ABC 5 48, m/BCD 5 90, m/CDE 5 130, BC 5 112 in., and CD 558 in.

Ash Seed Moves from point A to F, down to G, and right to H.

m/AFG 5 83, FG 5 114 in., and GH 5 112 in.

Hemlock Seed Moves left from point A to I. m/AIH 5 59.

Try this exercise with your classmates.

1. Make your own directions for two falling seeds.

2. Give your directions to three classmates.

3. Did everyone draw the path the same way?

4. If not, can you clarify your instructions so that your classmates will all make the same drawing? Explain.

2 1 0 1 2 3

A

B

F

C

D

G

HI E


0032_hsm11gmtr_0104.indd 38 3/23/09 7:18:35 AM

1-4 Standardized Test Prep Measuring AnglesMultiple Choice


1. What is m/BAC?

25

50

130

155

2. What is another name for /4?

/VWS /SVW

/SWV /WVT

3. m/KLM 5 129 and m/MNO 5 129. What is true about these two angles?

Th ey are both acute angles. Th ey are both right angles.

Th ey are congruent. Th ey are both straight angles.

4. m/MRT 5 133. What is m/MRN ?

24 46

48 87

5. /LJB and /IJM are congruent. If the sum of the measures of the angles is 90, what type of angle are they?

acute obtuse right straight

Short Response

6. m/RNY 1 m/GNC 5 128 and /RNY > /GNC. What is true about these two angles?

B

A

C

9090

80

100

10080

11070

12060 13050 14040 15030 16020

17010

1800

70

11060

12050

130

40140

30150

20160

10 170

0 180

1800

W S

VT

45

R

M

N

(4g 9)(2g 2)

T

H

B

H

A

[2] They are both acute angles and they each measure 64. [1] only one of two characteristics listed [0] no characteristics listed

A

0032_hsm11gmtr_0104.indd 37 3/23/09 7:18:28 AM

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107

A N S W E R S

QS) bisects lPQR. Solve for x and fi nd mlPQR.

17. m/PQS 5 3x; m/SQR 5 5x 2 20

18. m/PQS 5 2x 1 1; m/RQS 5 4x 2 15

19. m/PQR 5 3x 2 12; m/PQS 5 30

20. m/PQS 5 2x 1 10; m/SQR 5 5x 2 17

For Exercises 2124, can you make each conclusion from the information in the diagram below? Explain.

A B C

D

21. /DAB and /CDB are congruent.

22. /ADB and /CDB are complementary.

23. /ADB and /CDB are congruent.

24. /ADB and /BCD are congruent.

25. Algebra /MLN and /JLK are complementary, m/MLN 5 7x 2 1, and m/JLK 5 4x 1 3.

a. Solve for x. b. Find m/MLN and m/JKL. c. Show how you can check your answer.

26. Reasoning Describe all the situations in which the following statements are true.

a. Two vertical angles are also complementary. b. A linear pair is also supplementary. c. Two supplementary angles are also a linear pair. d. Two vertical angles are also a linear pair.

27. Open-Ended Write and solve an equation using an angle bisector to fi nd the measure of an angle.

1-5 Practice (continued) Form GExploring Angle Pairs

10; 60

8; 34

24; 60

55; 35 55 1 35 5 90

9; 56

8

Yes; the angles are marked as congruent.

Yes; their complements are congruent.

The measure of each angle must be 45.This is always true.

The angles are also adjacent.

Answers may vary. Sample: BC) bisects lABD so that mlDBC 5 5x and

mlABC 5 2x 1 30. Solve for x and nd mlABD. Answer: x 5 10; mlABD 5 100.

This is never true.

No; it is impossible to tell if they are congruent unless a measure of 45 is given.

Yes; lDAB and lADB are complementary and lDAB is congruent to lCDB.

0042_hsm11gmtr_0105.indd 44 3/23/09 7:17:45 AM

1-5 Practice Form GExploring Angle PairsUse the diagram at the right. Is each statement true? Explain.

1. /2 and /5 are adjacent angles.

2. /1 and /4 are vertical angles.

3. /4 and /5 are complementary.

Name an angle or angles in the diagram described by each of the following.

4. complementary to /BOC

5. supplementary to /DOB

6. adjacent and supplementary to /AOC

Use the diagram below for Exercises 7 and 8. Solve for x. Find the angle measures.

7. m/AOB 5 4x 2 1; m/BOC 5 2x 1 15; m/AOC 5 8x 1 8

8. m/COD 5 8x 1 13; m/BOC 5 3x 2 10; m/BOD 5 12x 2 6

9. /ABC and /EBF are a pair of vertical angles; m/ABC 5 3x 1 8 and m/EBF 5 2x 1 48. What are m/ABC and m/EBF ?

10. /JKL and /MNP are complementary; m/JKL 5 2x 2 3 and m/MNP 5 5x 1 2. What are m/JKL and m/MNP?

For Exercises 1114, can you make each conclusion from the information in the diagram? Explain.

11. /3 > /4 12. /2 > /4

13. m/1 1 m/5 5 m/3 14. m/3 5 90

15. KM) bisects /JKL. If m/JKM 5 86, what is m/JKL?

16. SV) bisects /RST . If m/RST 5 62, what is m/RSV ?

12

34

5

B

C

D

O

A

E

CB

AO

D

12

34

5

False; the angles are not next to each other.

False; the measures sum to 180, not 90.

lBOA

lDOC

lBOA and lDOE

True; they are on opposite sides of a vertex formed by two lines.

3; mlAOB 5 11; mlBOC 5 21; mlAOC 5 32

9; mlCOD 5 85; mlBOC 5 17; mlBOD 5 102

mlJKL 5 23; mlMNP 5 67

mlABC 5 mlEBF 5 128

No; adjacent angles are not always congruent.

Yes; they are vertical angles.

Yes; they are vertical angles. No; a right angle mark is needed.172

31

0042_hsm11gmtr_0105.indd 43 3/23/09 7:17:42 AM

1-5 Think About a PlanExploring Angle PairsReasoning When BX

) bisects /ABC, /ABX > /CBX . One student claims there

is always a related equation m/ABX 5 12 m/ABC. Another student claims the related equation is 2m/ABX 5 m/ABC. Who is correct? Explain.


1. What does it mean for BX) to bisect /ABC?

2. How is m/ABC related to m/ABX and m/CBX ?

3. How are m/ABX and m/CBX related?


4. Based on your answers, write an equation relating m/ABC and m/ABX .

5. Based on your answers, write an equation relating m/ABC and m/CBX .

6. Based on your answers, write an equation relating m/ABX and m/CBX .

Getting an Answer

7. Do any of your equations match an equation given in the exercise?

8. Can you show using algebra that one of your equations is equivalent to another equation in the exercise? Explain.

9. Which student is correct? Explain.

AX

B C

Answers may vary. Sample: BX) cuts

the angle in half.

Answers may vary. Sample: mlABC 5 mlABX 1 mlCBX

They are equal.

yes

Yes; answers may vary. Sample: Divide both sides of the equation by 2 to get the

equation mlABX 5 12mlABC.

Answers may vary. Sample: 2mlABX 5 mlABC

Answers may vary. Sample: 2mlCBX 5 mlABC

Answers may vary. Sample: mlABX 5 mlCBX

Both students are correct. Answers may vary. Sample: The two equations are equivalent.

0042_hsm11gmtr_0105.indd 42 3/23/09 7:17:41 AM

1-5 ELL SupportExploring Angle Pairs adjacent angles angle bisector complementary angles

supplementary angle vertical angles

Choose the word from the list above that is defi ned by each statement.

1. two angles whose measures have a sum of 180

2. two angles whose measures have a sum of 90

3. two angles whose sides are opposite rays

4. two angles that share a side but do not overlap

5. a ray that divides one angle into two angles with the same measure

Use a word from the list to complete each sentence.

6. If the measure of one angle in a pair of is n, the measure of the other angle is 180 2 n.

7. never share a side.

8. always share a side.

Draw the angle bisector for each angle.

9. 10.

Multiple Choice

11. Which of the following are a pair of vertical angles?

/GBA and /CBJ /ABM and /MBC

/GBA and /GBM /ABJ and /MBC

12. Which of the following are supplementary angles?

/GBA and /CBJ /ABM and /MBC

/GBA and /GBM /ABJ and /MBC

AB

C

J

MG

supplementary angles

complementary angles

vertical angles

adjacent angles

angle bisector

supplementary angles

Vertical angles

Adjacent angles

A

H

0041_hsm11gmtr_0105.indd 41 3/23/09 7:04:33 AM

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108

A N S W E R S

1-5 EnrichmentExploring Angle PairsAngling for More TimeExamine the analog clock shown below. Let the hour hand defi ne OA

), the minute

hand defi ne OB), and the second hand defi ne OC

).

O

A

B

C

12

9

10

1

2

4

5

11

7

8

3

6

Make a sketch of a clock that matches the following descriptions. Estimate the time shown on each sketch. Show all three hands on each sketch.

1. /AOB and /BOC are supplementary.

2. /AOB and /BOC are complementary.

3. OC) bisects /AOB and m/AOB 5 60.

4. OB) bisects /AOC and m/AOB 5 60.

5. OA) bisects /BOC, and /BOC is a straight angle.

6. /AOB, /BOC, and /AOC are congruent.

After you have answered Exercises 16, exchange answers with a partner. Check your partners answers.

7. Explain why the exercises can have more than one answer.

8. Use only terms found in Chapter 1. Try to describe a clock face such that only one possible clock face matches the description. Th en give your description to another student and see if that student can determine the correct time.

Answers may vary. Sample: 4:00:50






The rays can always be rotated so that the hands point to a different time and still maintain their relationship.


0042_hsm11gmtr_0105.indd 48 3/23/09 7:17:55 AM

1-5 Standardized Test PrepExploring Angle PairsMultiple Choice


1. /CDE and /FDE are supplementary, m/CDE 5 3x 1 10, and m/FDE 5 6x 1 8. What is m/FDE?

18 64 108 116

2. SV) bisects /RST . If m/RSV 5 64, what is m/RST ?

32 64 116 128

Use the diagram at the right for Exercises 3 and 4.

3. Which of the following pairs are vertical angles?

/1 and /2 /2 and /5

/2 and /3 /4 and /5

4. Which of the following pairs are supplementary?

/1 and /2 /2 and /3

/2 and /5 /4 and /5


5. Which of the following conclusions can you make from the information in the diagram?

/MNL > /LMN LM > MN

m/MNL 5 2m/LMN LM 5 2MN

6. Which of the following conclusions cannot be made from the information in the diagram?

MN > LN /NLM is supplementary to /NML.

/NLM > /NML /NLM is complementary to /NML.

Short Response

7. /ABC and /DBE are vertical angles, m/ABC 5 3x 1 20, and m/DBE 5 4x 2 10. Write and solve an equation to fi nd m/ABC and m/DBE.

1

23

45

M

L N

D

C

F

B

H

[2] 3x 1 20 5 4x 2 10; mlABC 5 mlDBE 5 110 [1] correct equation and wrong solution or correct answer with no equation [0] both equation and solution wrong

I

0042_hsm11gmtr_0105.indd 47 3/23/09 7:17:47 AM

14. Algebra In the diagram, XY) bisects /WXZ.

a. Solve for x and fi nd m/WXY.

b. Find m/YXZ.

c. Find m/WXZ.

Algebra QR) bisects lPQS. Solve for x and fi nd mlPQS.

15. m/PQR 5 3x, m/RQS 5 4x 2 9

16. m/PQS 5 4x 2 6, m/PQR 5 x 1 11

17. m/PQR 5 5x 2 4, m/SQR 5 3x 1 10

18. m/PQR 5 8x 1 1, m/SQR 5 6x 1 7

Algebra Find the measure of each angle in the angle pair described.

19. Th e measure of one angle is 5 times the measure of its complement.

20. Th e measure of an angle is 30 less than twice its supplement.

21. Draw a Diagram Make a diagram that matches the following description.

/1 is adjacent to /2.

/2 and /3 are a linear pair.

/2 and /4 are vertical angles.

/4 and /5 are complementary.

In the diagram at the right, mlHKI 5 48. Find each of the following.

22. m/HKJ 23. m/IKJ

24. m/FKG 25. m/FKH

26. m/FKJ 27. m/GKI

X

ZY

W

(7x 7)

(5x 3)

48

IH

G

F

KJ

1-5 Practice (continued) Form KExploring Angle Pairs

5; 28

56

28

9; 54

14; 50

75 and 15

70 and 110


7; 62

3; 50

90

42 132

138

42

138

1

2 34

5

0045_hsm11gmtr_0105.indd 46 3/23/09 6:49:56 AM

1-5 Practice Form KExploring Angle PairsUse the diagram at the right. Is each statement true? Explain.

1. /5 and /4 are supplementary angles.

2. /6 and /5 are adjacent angles.

3. /1 and /2 are a linear pair.

Name an angle or angles in the diagram described by each of the following.

4. a pair of vertical angles

5. supplementary to /RPS

To start, remember that supplementary angles are two angles whose measures

have a sum of u.

6. a pair of complementary angles

To start, remember that complementary angles are two angles whose

measures have a sum of u.

7. adjacent to /TPU

For Exercises 811, can you make each conclusion from the information in the diagram? Explain.

8. /CEG > /FED 9. DE > EF

10. /BCE > /BAD 11. /ADB and /FDE are vertical angles.


12. Name two pairs of angles that form a linear pair.

13. Name two pairs of angles that are complementary.

12

3456

40

R

Q

ST

UP

F

GEC

BD

A

N

S R

Q

P

O

No; they are complementary.

Yes; they have a common side and vertex.

lRPQ and lTPU

lQPR and lRPS or lRPS and lUPT

lSPU

180

90

Yes; they are adjacent angles whose noncommon sides are opposite rays.

lSPT, lRPT, and lQPT

lPSQ and lQSR, lPSQ and lNSO

Answers may vary. Sample: lNSO and lQSO; lNSP and lQSP; lOSP and lRSP; lOSQ and lRSQ

Yes; they are vert. '.

no; not enough information Yes; their sides are opposite rays.

Yes; they are marked as O.

0045_hsm11gmtr_0105.indd 45 3/23/09 6:49:53 AM

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109

A N S W E R S

1-6 Think About a Plan Basic Constructions a. Draw a large triangle with three acute angles. Construct the bisector of the

three angles. What appears to be true about the three angle bisectors?

b. Repeat the constructions with a triangle that has one obtuse angle.

c. Make a Conjecture What appears to be true about the three angle bisectors of any triangle?

1. In this problem, you will draw three rays for each triangle. Each ray starts at a vertex and passes through the interior of the triangle.

a. What is the maximum number of points of intersection between the rays? u b. What is the minimum number? u 2. On a separate piece of paper, draw nABC such that all three angles are acute

angles and the triangle takes up most of the paper.

3. In your own words, what are the steps for drawing the angle bisector of /A?

4. Draw the angle bisector of /A. Th en draw the angle bisectors for /B and /C.

5. On a separate piece of paper, draw nDEF such that one of the three angles is an obtuse angle and the triangle takes up most of the paper. Construct the angle bisectors for /D, /E, and /F .

6. How many points of intersection are there for the three angle bisectors of nABC? u

7. How many points of intersection are there for the three angle bisectors of nDEF? u 8. Suppose a friend draws nDHI . Without looking at the triangle, what do you

think will be true of the three angle bisectors of its angles?

3

1

Check students drawings of an acute kABC.

Answers may vary. Check students drawings of angle bisectors drawn at lA, lB, and lC.

Answers may vary. Check students drawings of obtuse kDEF with angle bisectors at lD, lE, and lF.

Answers may vary. Sample: Draw an arc centered at A that crosses its two rays at X

and Y. Draw arcs with the same radius from X and Y that intersect at Z. Draw a ray

from A through Z.

1

1

They will intersect at a single point.

0052_hsm11gmtr_0106.indd 52 3/23/09 7:17:06 AM

1-6 ELL SupportBasic ConstructionsConcept List

angle bisector arc congruent segments

is greater than is perpendicular to midpoint

parallel lines perpendicular bisector perpendicular lines

Choose the concept from the list above that best represents the item in each box.

1. 2. ' 3. AB 5 DE

4. 5. 6. Point P

7. . 8. 9.

A B12

DE

12

S TM

N M P R

UV

XT

perpendicular lines is perpendicular to congruent segments

perpendicular bisector

is greater than angle bisector arc

parallel lines midpoint

0051_hsm11gmtr_0106.indd 51 3/23/09 7:04:17 AM

1-5 Reteaching (continued)Exploring Angle PairsSupplementary Angles and Complementary AnglesTwo angles that form a line are supplementary angles. Another term for these angles is a linear pair. However, any two angles with measures that sum to 180 are also considered supplementary angles. In both fi gures below, m/1 5 120 and m/2 5 60, so /1 and /2 are supplementary.

Two angles that form a right angle are complementary angles. However, any two angles with measures that sum to 90 are also considered complementary angles. In both fi gures below, m/1 5 60 and m/2 5 30, so /1 and /2 are complementary.

Exercises

3. Copy the diagram at the right.

a. Label /ABD as /1.

b. Label an angle that is supplementary to /ABD as /2.

c. Label as /3 an angle that is adjacent and complementary to /ABD.

d. Label as /4 a second angle that is complementary to /ABD.

e. Name an angle that is supplementary to /ABE.

f. Name an angle that is complementary to /EBF .

1 2

1

2

1 2 1 2

A B

DE

F

C60

60

Label lABD as l1.

Label lDBC as l2.

lCBE

lCBF

Label lDBE as l3.

Label lFBC as l4.

0042_hsm11gmtr_0105.indd 50 3/23/09 7:17:59 AM

1-5 ReteachingExploring Angle PairsAdjacent Angles and Vertical AnglesAdjacent means next to. Angles are adjacent if they lie next to each other. In other words, the angles have the same vertex and they share a side without overlapping.

Adjacent Angles Overlapping Angles

Vertical means related to the vertex. So, angles are vertical if they share a vertex, but not just any vertex. Th ey share a vertex formed by the intersection of two straight lines. Vertical angles are always congruent.

Vertical Angles Non-vertical Angles

Exercises

1. Use the diagram at the right.

a. Name an angle that is adjacent to /ABE.

b. Name an angle that overlaps /ABE.

2. Use the diagram at the right.

a. Mark /DOE and its vertical angle as congruent angles.

b. Mark /AOE and its vertical angle as congruent angles.

21 2

1

2121

A

D E

F

B C

B

C

D

O

A

E

lEBF or lEBC

lAOB

lDOB

Answers may vary. Sample: lDBF or lDBC

0042_hsm11gmtr_0105.indd 49 3/23/09 7:17:56 AM

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110

A N S W E R S

Tell whether each statement is true or false. If false, rewrite the statement correctly.

11. To construct a perpendicular bisector, the fi rst step is to place the compass point on the center of the segment. Th en draw arcs on either end of the segment.

12. To construct a congruent segment, fi rst draw a ray. Th en open the compass to the length of the original segment.

13. To construct an angle bisector, fi rst place your compass point on one side of the angle and draw an arc in the center of the angle. Th en from a point on the other side, draw an arc using the same compass setting.

14. Th e steps for constructing /B > /E are listed below in the wrong order. Rewrite the steps in the correct order.

A. Open the compass to the length of DF. Using this setting, put the compass point on C. Draw an arc to locate point A.

B. Draw BA.

C. Put the compass point on E and draw an arc that intersects the sides of /E. Label the points of intersection D and F.

D. Draw a ray with endpoint B.

E. Keep the compass on the same setting. Th en put the point of the compass on B and draw an arc. Label the point of intersection with the ray as C.

For Exercises 15 and 16, copy l1 and l2. Construct each angle described.

15. /R; m/R 5 m/1 1 m/2

16. /S; m/S 5 m/2 2 m/1

DA

BC

EF

1 2

1-6 Practice (continued) Form K Basic Constructions

False; the rst step is to put the compass point on both endpoints, and, using the same compass setting, draw arcs.

False; rst place your compass point on the vertex and draw an arc that intersects both sides of the angle. Then from each point of intersection draw two intersecting arcs using the same compass setting.

true

D, C, E, A, B

S

16. Sample:

R

15. Sample:

0055_hsm11gmtr_0106.indd 56 4/2/09 6:17:32 PM

1-6 Practice Form KBasic ConstructionsFor Exercises 19, draw a diagram similar to the given one. Th en do the construction. Check your work with a ruler or protractor.

1. Construct QR congruent to BC.

To start, draw a ray with endpoint Q.

2. Construct LM so that LM 5 3BC.

3. Construct XY so that XY 5 RS 1 TU.

To start, construct a segment congruent to RS.

4. Construct AB so that AB 5 RS 2 TU.

5. Construct /J so that /J > /D.

6. Construct /R so that m/R 5 2m/D.

7. Construct the perpendicular bisector of BC.

8. Construct the perpendicular bisector of RS.

9. Draw acute /FGH. Th en construct its bisector.

To start, draw acute angle FGH.

Sketch the fi gure described. Explain how to construct it. Th en do the construction.

10. OM) bisects obtuse angle /NOP.

B C

R S T U

D

Answers may vary. Sample: Draw an obtuse angle. Put the compass on the vertex and draw an arc that intersects both sides of the angle. Put the compass on one of the points where the arc intersects the side and draw an arc. Do this on the other point, using the same compass setting. Draw a line through the intersection of the two arcs and the vertex. PO

N

M

Q R

L M

X Y


F

GH

A B

J

5.

R

6.

B C

7.

R S

8.

0055_hsm11gmtr_0106.indd 55 3/23/09 6:50:28 AM

1-6 Practice (continued) Form GBasic Constructions 14. Draw a segment AB. Construct a segment whose length is 14AB. a. How is the line segment you need to construct related to the perpendicular

bisector of AB?

b. How can you use the previous constructions to help you?

15. Answer the questions about an angle in a plane. Explain each answer. a. How many angle bisectors does an angle have in the plane? b. How many rays in the plane bisect the angle? c. How many rays in space bisect the angle?

16. Writing Explain how to do each construction with a compass and straightedge. a. Draw an /A. Construct the angle bisector of /A.

b. Divide /A into four congruent angles.

17. Draw a segment ST . a. Construct a right triangle with two sides

that have the measure 12 ST . b. Reasoning Describe how to construct a

458 angle. Th en describe how to construct an isosceles right triangle.

18. Draw a segment VW . a. Construct a square ABCD whose sides have length VW. b. Describe how to construct the perpendicular bisectors

of sides AB and BC.

Answers may vary. Sample: The bisector divides line segment AB into two segments to measure 12AB.

Answers may vary. Sample: Constructing the midpoint of either half of line segment AB will make a segment with the measure 14 AB.

11

1

Answers may vary. Sample: 1) Put the compass on vertex A. Draw an arc that intersects both sides of lA. Label the points of intersection B and C. 2) Put the compass point on C. Draw an arc. With the same setting, put the compass point on B and draw an arc. Label the point of intersection D. 3) Draw AD

). This is the angle bisector of lA.

Answers may vary. Sample: 1) Construct AD), the

angle bisector of lA or lBAC. 2) Construct AE), the

angle bisector of lDAC. 3) Construct AF), the angle

bisector of lDAB. The resulting angles lBAF, lFAD, lDAE, and lEAC are congruent.

Answers may vary. Sample: First, construct the perpendicular bisector UP of ST to make a 908 angle. Label the point of intersection U. Then construct the angle bisector of lSUP or lPUT . Mark a spot along UP that is TU units from U. Label that point V. Then draw TV.

Sample:

P

US T

V

TS

D

AX

C

Y

B

Sample: V W

Answers may vary. Sample: Use a compass to draw arcs with the same radius from points A and B that intersect in two points. Draw the line through those two points. Repeat for B and C.

0052_hsm11gmtr_0106.indd 54 3/23/09 7:17:14 AM

1-6 Practice Form GBasic ConstructionsFor Exercises 113, do the construction using the fi gures below. Check your work with a ruler or a protractor.

1. Construct AB congruent to XY .

2. Construct the perpendicular bisector of XY .

3. Construct a triangle whose sides are all the same length as XY .

4. Construct AB so that AB 5 MN 1 OP.

5. Construct KL so that KL 5 OP 2 MN .

6. Construct the perpendicular bisector of MN .

7. Construct the perpendicular bisector of OP.

8. Construct /A so that m/A 5 m/1 1 m/2.

9. Construct /B so that m/B 5 m/1 2 m/2.

10. Construct /C so that m/C 5 2m/2.

11. Construct /D so that m/D 5 2m/1.

12. Construct /Y so that m/Y 5 12 m/2.

13. Construct /Z so that m/Z 5 12 m/X .

X Y

M N

O P

1 2

X

3.

A B1.

A

8.

B

9.

C

10.

K L5.

4. A B

7.

6.

2.

11.

D

12.

Y

13.

Z

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111

A N S W E R S

1-6 Reteaching (continued) Basic Constructions 2. Analyze the construction of a congruent angle. First draw/X . a. THINK about what it means to construct a congruent angle. What is

your goal?

b. SKETCH /Y congruent to /X . Your goal is to construct an angle the same size as /Y .

c. EXPLAIN the fi rst step (drawing a ray). What is the purpose of the fi rst step?

d. EXPLAIN the second step (drawing an arc). What is the purpose of the second step?

e. EXPLAIN the third step (drawing an arc). What is the purpose of the third step?

f. EXPLAIN the fourth step (drawing an arc). What is the purpose of the fourth step?

g. EXPLAIN the fi fth step (drawing a segment). What is the purpose of the fi fth step?

3. Analyze the construction of an angle bisector. First draw /W . a. THINK about what it means to construct an angle bisector. What is

your goal?

b. SKETCH a ray that makes /V congruent to 12 /W , where /V shares a ray with /W and has its other ray inside /W . Your goal is to construct a ray that bisects an angle.

c. EXPLAIN the fi rst step (drawing an arc). What is the purpose of the fi rst step?

d. EXPLAIN the second step (drawing two arcs). What is the purpose of the second step?

e. EXPLAIN the third step (drawing a ray). What is the purpose of the third step?

Answers may vary. Sample: to draw an angle of the same size as the given angle

Answers may vary. Sample: to divide a given angle into two equal angles

Answers may vary. Sample: to divide the given angle into two equal smaller angles


Answers may vary. Sample: to measure the opening of the given angle

Answers may vary. Sample: to mark the size of the given angle on the ray

Answers may vary. Sample: to draw the congruent angle

Answers may vary. Sample: to make a ray to serve as one side of the new congruent angle

Answers may vary. Sample: to locate two points of intersection to use to draw the other endpoint of the angle bisector

Answers may vary. Sample: to locate the point of intersection that will be another point on the angle bisector

Answers may vary. Sample: to nd the point of intersection so you can draw the other side of the angle


0052_hsm11gmtr_0106.indd 60 3/23/09 7:17:24 AM

1-6 Reteaching Basic ConstructionsBefore you start the construction, THINK about the goal. Th en SKETCH the segment or angle you are trying to construct. Next EXPLAIN the purpose of each step in the construction as you complete it.

Problem

Construct AB so that AB is congruent to XY .

X

Y

THINK: Can you describe what the problem is asking for in your own words? You want to draw a line segment that has the same length as one you are given.

SKETCH: Sketch a segment and label it as AB. Why do you start with a sketch? A sketch helps you to see what you need to construct.

EXPLAIN: First, draw a ray. What is the purpose of drawing a ray? Th e ray is a part of a line on which you can mark off the correct

length of AB.

Second, measure the length of XY , using the compass. What is the purpose of measuring the length of XY ? You need this measure to mark the same length on the ray.

Finally, mark the length of XY on the ray, using the compass. Why do you mark the length of XY on your ray?

Th is completes your construction of AB, which is congruent to XY .

Exercises

Analyze the construction of a congruent angle and bisectors.

1. Analyze the construction of a perpendicular bisector. First draw XY . a. THINK about what it means to construct a perpendicular bisector. What is

your goal?

b. SKETCH a perpendicular bisector to XY .

c. EXPLAIN the fi rst two steps. What is the purpose of drawing an arc from each endpoint?

d. EXPLAIN the last step. What is the purpose of drawing the segment between the intersections of the arcs?

Answers may vary. Sample: to draw a segment that is perpendicular to XY and that divides it into two equal pieces


Answers may vary. Sample: to form two intersections to become two points of the perpendicular bisector

Answers may vary. Sample: to draw the perpendicular bisector

0052_hsm11gmtr_0106.indd 59 3/23/09 7:17:23 AM

1-6 Enrichment Basic ConstructionsAn optical illusion is an image that often is misleading to the human eye. Th e following images are optical illusions. How do you view each image?

Do you see the cube from How many balls are on each Are the inside vertical lines above or below? shelf? Which shelf are they on? straight or curved?

Th e optical illusion at the right can be viewed in two ways. It can be viewed as a cube with another cube on its corner or as a cube with a corner cut out.

Use the following steps to make this optical illusion. Use a pencil for steps 16 so that your pencil marks can be erased when you have fi nished.

1. Use a compass to draw a circle and keep the compass settings.

2. Place a point on the circumference of the circle. Label this point A.

3. Place the tip of the compass at point A, and mark an arc on the circle counterclockwise from A. Label the point where the arc intersects the circle as point B.

4. Place the tip of the compass at point B, and mark a similar arc on the circle next to B. Label the point where the arc intersects the circle as point C. Repeat this step to mark D, E, and F.

5. Use a straightedge to draw AD, BE, and CF . Label the intersection of these three line segments as point I.

6. Use a compass to fi nd the midpoint of AI . Label this point Ar. Repeat this step to fi nd midpoints of BI , CI , DI , EI , and FI .

7. Use a straightedge to draw hexagon ABCDEF.

8. Use a straightedge to draw hexagon ArBrCrDrErFr.

9. Name the two polygons you shade to make the optical illusion.

10. Name the two polygons you shade to cause the illusion of a room w

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