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Name
Geometric IdeasLines, line segments, and rays are basic geometric ideas. Theyare sometimes described by the relationship they have to otherlines, line segments, and rays.
Use the diagram at the right. Name the following.
1. two parallel lines
2. a line segment
3. two perpendicular lines
4. two lines
5. a ray
6. two perpendicular rays
7. a point
R 6-1
!""#
!""# !""# !""#
!""#
!""#!""# !""#
Draw Write Say Description
!""# !""#EF !GH !""# !""#
Line NP intersects
line GO.
NP intersects GO
Line EF is parallel to
line GH.
SU !VT Line SU is
perpendicular
to line VT.
S
U
TV
F
EH
G
N
P
O
G
EF and GH are parallel
lines. They are the same
distance apart and will not
cross each other.
!""# !""#NP and GO are intersecting
lines. They pass through the
same point.
!""# !""#SU and VT are perpendicular
lines. They intersect and form
square corners.
H
I
C
B
K
VL A
D G
E F
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Use with Lesson 6-1. 73
Name
Geometric IdeasUse the diagram at the right. Name the following.
1. three points
2. a ray
3. two intersecting lines but not perpendicular
4. two parallel lines
5. a line segment
6. two perpendicular lines
7. Reasoning Can a line segment have two midpoints? Explain.
Test Prep
8. Which type of lines are shown by the figure?
A. Intersecting B. Parallel
C. Perpendicular D. Curved
9. Writing in Math Draw and label two perpendicular linesegments KL!!! and MN!!!.
P 6-1
FIR
P J
Q
E
N A
A
B
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74 Use with Lesson 6-2.
Name
Measuring and Classifying AnglesThe chart below can help you to describe and identify an angle.
Classify each angle as acute, right, obtuse, or straight. Thenmeasure each angle.
1. 2. 3.
Draw an angle with each measure.
4. 100° 5. 170° 6. 90°
7. Reasoning !ABC measures less than 180° but more than90°. Is !ABC a right, an acute, an obtuse, or a straight angle?
R 6-2
180
90
0
!""
!""
Measure !EFG.
!EFG measures 45°.
Draw an angle of 65°.
Remember to place
the 0° mark on one
side of the angle.
Place the center of the
protractor on S. Line up SW
with the 0° mark. Place a point
at 65°. Label it J. Draw SJ.
Classifying Angles
Acute — measures between 0° and 90°
Right — measures exactly 90°
Obtuse — measures between 90° and 180°
Straight — measures exactly 180°
F
E
G
S W
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Name
Measuring and Classifying AnglesClassify each angle as acute, right, obtuse, or straight. Thenmeasure each angle. (Hint: Draw longer sides if necessary.)
1. 2.
Draw an angle with each measure.
3. 120° 4. 180°
5. Draw an acute angle. Label it with the lettersA, B, and C. What is the measure of the angle?
Test Prep
6. Which kind of angle is shown in the figure below?
A. Acute B. Obtuse
C. Right D. Straight
7. Writing in Math Explain how to use a protractor to measure an angle.
P 6-2
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Name
Segments and Angles Related to Circles
Use the terms at the top of this page to identify each figure in circle F.
1. point F
2. !GFS
3. R!S!
4. U!V!
5. G!F!
6. Reasoning If a circle has a radius of 12 in., what is the circle’s diameter?
R 6-3
All the points on a circle are the same
distance from the center. The center of
circle C is at point C.
A radius goes from the center to any point
on the circle. RS is a radius of circle R.
(plural: radii)
A chord has both endpoints on a circle. HD
is a chord of circle O.
A diameter passes through the center
of a circle. DR is a diameter of circle M.
A diameter of a circle is also the longest
chord of that circle. A circle’s diameter is
twice as long as its radius.
A central angle is an angle whose vertex is
the center. !ACB is a central angle.
S
R
AD
B
C
R
D
M
DH
O
C
F
G
S
V
U
R
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Name
Segments and Angles Related to CirclesIdentify each figure in circle D.
1. !NDI
2. LE!!!
3. DI!!
4. AN!!!
5. point D
6. DA!!!
7. Reasoning Is the radius of a circle half of the diameter?Explain.
8. Construct a circle that has a radius with an equal length to the given line segment.
Test Prep
9. A line segment that connects two points on a circle passing through the center is called a
A. Radius B. Diameter C. Chord D. Central angle
10. Writing in Math Brenda thinks that a chord can be many differentlengths on a given circle. Is she right? Explain your thinking.
P 6-3
L
I
D
E
N
A
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76 Use with Lesson 6-4.
Name
Polygons
Name each polygon. Then tell if it appears to be a regular polygon.
1. 2.
3. 4.
5. Reasoning Shakira sorted shapes into two different groups. Use geometric terms to describe how she sorted the shapes.
R 6-4
A regular polygon has sides of equal length and angles of equal measure.
Each side is 3 in. long. Each angle is 60°.
A polygon is a closed plane figure made up of line segments. Common polygons have names that tell the number of sides the polygon has.
60°60°
60°
3 in.
Open Figure
3 in. 3 in.Octagon
8 sides
Triangle
3 sides
Pentagon
5 sides
Hexagon
6 sides
Quadrilateral
4 sides
Group A Group B
© P
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Name
PolygonsName each polygon. Then tell if it appears to be a regular polygon.
1. 2.
3. Name the polygon. Name the vertices.
Test Prep
4. Which polygon has eight sides?
A. Quadrilateral B. Pentagon C. Hexagon D. Octagon
5. Writing in Math Draw two regular polygons and two thatare irregular. Use geometric terms to describe onecharacteristic of each type.
P 6-4
E
H
T
B
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Name
Classifying TrianglesYou can classify triangles by the lengths of their sides and thesizes of their angles.
Remember that the sum of the measures of the angles of atriangle is 180°.
Classify each triangle by its sides and then by its angles.
1. 2. 3.
The measures of two angles of a triangle are given. Find themeasure of the third angle.
4. 40°, 100°, 5. 14°, 98°, 6. 38°, 38°,
R 6-5
acuteall angles less than 90°
equilateralall sides the same length
This triangle is both equilateral and acute.
Not all acute triangles are equilateral.
rightone right angle
isoscelestwo sides the same length
This triangle is both isosceles and right.
Not all right triangles are isosceles.
obtuseone obtuse angle
scaleneno sides the same length
This triangle is both scalene and obtuse.
Not all obtuse triangles are scalene.
60°60°
60°
8 cm
8 cm 8 cm
24 in.
5 ft
4 ft
7.5 ft
16 in. 16 in.
120°
8 cm 8 cm
14 cm
128°
26° 26°60°60°
60°
3 ft
3 ft 3 ft
4 in.
3 in.5 in.
© P
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Name
Classifying TrianglesClassify each triangle by its sides and then by its angles.
1. 2.
The measures of two angles of a triangle are given. Find themeasure of the third angle.
3. 47°, 62°, 4. 29°, 90°,
5. 75°, 75°, 6. 54°, 36°,
7. Judy bought a new tent for a camping trip. Look at the side of the tent with the opening to classify the triangle by its sides and its angles.
Test Prep
8. Which describes a scalene triangle?
A. 4 equal sides B. 3 equal sides C. 2 equal sides D. 0 equal sides
9. Writing in Math The lengths of two sides of a triangle are 15 in. each. The third side measures 10 in. What type of triangle is this? Explain your answer usinggeometric terms.
P 6-5
8 ft
12 ft9.5 ft
4 m
4 m4 m
60°
60° 60°
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78 Use with Lesson 6-6.
Name
Classifying Quadrilaterals
Classify each quadrilateral. Be as specific as possible.
1. 2. 3.
The measures of three angles of a quadrilateral are given. Find the measure of the fourth angle.
4. 65°, 150°, 89°, 5. 100°, 80°, 100°,
6. 82°, 78°, 90°,
R 6-6
Quadrilateral Definition Example
A quadrilateral with both pairs of opposite sides parallel and equal in length
A parallelogram with four right angles
A parallelogram with all sides the same length
A rectangle with all sides the same length
A quadrilateral with only one pair of parallel sides
5 ft
5 ft
2 ft 2 ft
2 in.
6 in.
2 in. 3 in.
1 ft
1 ft
1 ft 1 ft
5 in.
5 in.
2 in. 2 in.
4 in.
4 in.
4 in. 4 in.
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Remember that the sum of the measures of the angles of a quadrilateral is 360°.
6 ft
6 ft
3 ft 3 ft
4 in.
4 in.
4 in. 4 in.
6 ft
7 ft
3 ft
9 ft
© P
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78 Use with Lesson 6-6.
Name
Classifying QuadrilateralsClassify each quadrilateral. Be as specific as possible.
1. 2.
3. 4.
The measures of three angles of a quadrilateral are given. Findthe measure of the fourth angle.
5. 90°, 145°, 78°, 6. 110°, 54°, 100°,
7. Name the vertices of the square.
Test Prep
8. Three of the angles of a quadrilateral measure 80°, 100°,and 55°. Which is the measure of the fourth angle?
A. 115° B. 120° C. 125° D. 130°
9. Writing in Math Can a trapezoid have four obtuse angles? Explain.
P 6-6
8 cm 8 cm
15 cm
15 cm
8 in.
8 in.
4 in.4 in.
17 cm
17 cm
17 cm17 cm
GF
AL
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Name PROBLEM-SOLVING STRATEGY
Solve a Simpler ProblemGive Away During a grand opening, a mall provides prizes forthe first 70 customers. The 1st customer receives a couponbook, the 2nd customer, a travel mug, and the 3rd customer, aT-shirt. If the prizes continue to be given away in the samepattern, which prize would the 70th customer receive?
Is your work correct?
Yes, 69 can be divided by 3 evenly to find the 3rd prize. The 70th customer wouldbe next for the 1st prize.
Solve the simpler problem. Use the solution to help you solvethe original problem.
1. Anna can choose from white or yellow paper. She canchoose from thin or wide paintbrushes. She can choosefrom green, purple, or blue paint. How many differentcombinations of art supplies are possible?
Look Back and Check
Plan and Solve
Read and Understand
R 6-7
Step 1: What do you know?
There are 3 different prizes given away in a pattern.
Step 2: What are you trying to find?
Which gift the 70th customer will receive
Step 3: What strategy will youuse?
Strategy: Solve a simpler problem
I can divide 69 by 3. I know the69th customer will receive the3rd prize. The 70th customerwill receive the coupon book.
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Name PROBLEM-SOLVING STRATEGY
Solve a Simpler ProblemSolve the simpler problems. Use the solutions to help you solve the original problem.
1. Reggie is designing a triangular magazine rack with 5 shelves. The top shelf will hold 1 magazine. The second shelf will hold 3 magazines, and the third shelf will hold 5 magazines. This pattern continues to the bottom shelf. How many magazines will themagazine rack hold altogether?
Simpler Problem What is the pattern?
How many magazines will the fourth shelf hold?
How many magazines will the bottom shelf hold?
Solution:
2. At the deli, you receive 1 free sub after you buy 8 subs.How many free subs will you receive from the deli if youbuy 24 subs?
3. The chef has 5 different kinds of pasta and 3 different flavorsof sauce. How many different meals is she able to make?
P 6-7
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Name PROBLEM-SOLVING SKILL
Writing to DescribeUse geometric terms to describe one characteristic of the lines in each group.
You can brainstorm to help you describe the characteristics.
Group 1 Group 2
no right angles all right anglesnot parallel not parallel all intersecting all intersecting no perpendicular lines all perpendicular lines
When you write your description, use mathematical termscorrectly and be brief.
All of the lines in Group 1 are intersecting and not perpendicular.
All of the lines in Group 2 are intersecting and are perpendicular.
1. Abdul sorted the polygons to the right into two groups. Use geometric terms to describe a characteristic in each group.
R 6-8
Group 1 Group 2
K
J
VU
T
U
YX
C
D
RQ
E
F
B
A
Group A Group B
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Name PROBLEM-SOLVING SKILL
Writing to DescribeSolve each problem. You may use a brainstorming table to helpplan your description.
1. Use geometric terms to describe three properties of a square.
2. Use mathematical terms to describe the numbers in each set.
3. Mariam created two patterns. Use mathematical terms todescribe the numbers in each pattern.
Pattern 1: 12, 16, 20, 24, 28, 32
Pattern 2: 1, 2, 4, 8, 16, 32, 64
4. Use geometric terms to describe three properties of this trapezoid.
P 6-8
Set C
83
3
1
27
35
9
28
72
42
86
6
8
Set D
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Name
Congruence and Similarity
Do the figures in each pair appear to be similar? If so, are they also congruent?
1. 2.
3. 4.
R 6-9
Similar polygons have the same angle measures, but the
lengths are different. Think: same shape, different size
Congruent polygons have the same angle measures and the
lengths of the sides are the same. Think: same shape, same size
37°
53°
6 ft
7.5 ft4.5 ft
37°
53°
9 ft
14 ft12 ft
10 m
10 m
5 m 5 m
10 m
10 m
5 m 5 m
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Name
Congruence and Similarity Do the figures in each pair appear to be similar? If so, are they also congruent?
1. 2.
Is each situation an example of congruent or similar figures?Explain your reasoning.
3. When baking bread, the baker uses two sizes of loaf pans.One loaf pan fits inside the other.
4. A friend gave you and five friends each a quarter.
Test Prep
5. If Figure A is similar to Figure B, which is the measure of !XYZ in Figure A?
A. 90° B. 58°
C. 45° D. 32°
6. Writing in Math If two circles are congruent, are thediameters the same length? Explain.
P 6-9
Use with Lesson 6-9. 81
Figure A Figure B
58°
32°
X
Y
Z A
B
C
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Name
TransformationsThe size and shape of a figure do not change when it is slid,flipped, or turned.
Turns can be measured in degrees. Here are some common turns.
Tell whether the figures in each pair are related by a slide, a flip,or a turn. If a turn, tell what type of turn.
1. 2.
3. 4.
R 6-10
1. A slide, or translation, moves a figure in astraight direction.
2. A flip, or reflection, ofa figure gives itsmirror image.
3. A turn, or rotation, moves a figure about a point.
14
90° or turn12
180° or turn34
270° or turn 360° or full turn
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Name
TransformationsTell whether the figures in each pair are related by a flip, slide,or turn. If a turn, describe it.
1. 2.
3. On a compass, if you are standing at 0° and you turn to your right to make a 90° turn, what fraction of a turn is that?
4. If a figure is flipped over the dashed line and then rotated a !14! turn counterclockwise, which of the figures below shows the result?
Test Prep
5. Which term describes the mirror image of a figure?
A. Slide B. Flip C. Turn D. Pattern
6. Writing in Math Mark says that the figure above wasflipped. Faith says that it made a !34! turn. Steven says that itmade a 90° turn. Who is correct?
P 6-10
A. C.B. D.
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Name
SymmetryIf a figure is symmetric, it can be split in half so that one side reflectsonto the other side. The fold line is called the line of symmetry.
How many lines of symmetry does each figure have? You maytrace the figure to check.
1. 2. 3.
Part of a symmetric trademark is shown. Complete the drawing.
4.
R 6-11
This figure has two lines of symmetry. If you folded along one of the lines, the two sides would fit on top of each other.
This figure has one line of symmetry. There is only one way to fold it exactly in half.
This figure has no lines of symmetry. It cannot be folded in half so that one side reflects onto the other side.
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SymmetryHow many lines of symmetry does each figure have? You mayfold your paper to check.
1. 2.
3. Reasoning How many lines of symmetry does anequilateral triangle have? Explain.
Part of a symmetric trademark is shown. Complete each drawing.
4. 5.
Test Prep
6. Which figure has more than two lines of symmetry?
7. Writing in Math Draw a shape orfigure that has more than 2 lines ofsymmetry. Put the lines of symmetryinto your drawing.
P 6-11
A. B. C. D.
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Name PROBLEM-SOLVING APPLICATIONS
The Geometry of LifeJuanita and her friend Kim decided to meet at the park. Usegeometric terms to describe the location of their houses andthe park in relation to the line segment.
Juanita’s house is located at point J on !JK. Kim’s house is located atpoint K on !JK. The park is located at P, the midpoint of !JK.
Use the diagram above for question 1.
1. What is the shape of the chimney at Juanita’s house?
2. Kenneth’s mother ordered a pizza for his graduation party. The pizza has a diameter of 12 in. What is the measurement of the radius of the pizza?
3. If it is 3:00 P.M., what is the measure of the angle that can be formed between the hour hand and the minute hand?
4. Angelina saw the sign at the right. What is the name of this polygon?
Juanita'shouse
Kim'shouse
Park
J KP
R 6-12
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Name PROBLEM-SOLVING APPLICATION
The Shape We Are In1. Tarah hung an American flag from the front
porch of her house. She placed it in a flag stand on one of the posts of the porch. Measure the angle of the flagpole from the post.
2. Construct a circle. Use the measure of the radius below.
3. Karen and Willie took a walk on the streets near their home. The path of their walk is shown at the right. Classify the triangle formed by the path by its angles and its sides.
4. Part of a symmetric figure is shown. Complete the drawing.
P 6-12
A
B
