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2. Splash Screen

3. Lesson 6-1 Line and Angle Relationships
Lesson 6-2 Triangles and Angles
Lesson 6-3 Special Right Triangles
Lesson 6-4 Classifying Quadrilaterals
Lesson 6-5 Congruent Polygons
Lesson 6-6 Symmetry
Lesson 6-7 Reflections
Lesson 6-8 Translations
Lesson 6-9 Rotations
Contents

4. Example 1 Classify Angles and Angle Pairs
Example 3 Find a Missing Angle Measure
Example 4 Find an Angle Measure
Lesson 1 Contents

5. Classify the angle using all names that apply.
is less than
Answer: So, is an acute angle.
Example 1-1a

6. Classify the angle using all names that apply.
Answer: right
Example 1-1b

7. Classify the angle pair using all names that apply.
are adjacent angles since they have the same vertex, share a common side, and do not overlap. Together they form a straight angle measuring
Answer: are adjacent angles and supplementary angles.
Example 1-2a

8. Classify the angle pair using all names that apply.
Answer: adjacent, complementary
Example 1-2b

9. The two angles below are supplementary. Find the value of x.
Definition of supplementary angles
Subtract 155 from each side.
Simplify.
Answer: 25
Example 1-3a

10. The two angles below are complementary. Find the value of x.
Answer: 35
Example 1-3b

11. Since are alternate interior angles, they are congruent.
BRIDGES The sketch below shows a simple bridge design used in the 19th century. The top beam and floor of the bridge are parallel. If and find and
Since are alternate interior angles, they are congruent.
So,
Example 1-4a

12. Since 1, 2, and 4 form a line, the sum of their measures is 180°.
Therefore, m
Answer:
Example 1-4a

13. BRIDGES The sketch below shows a simple bridge design
BRIDGES The sketch below shows a simple bridge design. The top beam and floor of the bridge are parallel. If and find and
Answer:
Example 1-3b

14. End of Lesson 1

15. Example 1 Find a Missing Angle Measure Example 2 Classify Triangles
Lesson 2 Contents

16. The sum of the measures is 180.
Find the value of x in
The sum of the measures is 180.
Replace m P with 17, m Q with 46, and m R with x.
Simplify.
Subtract 63 from each side.
The value of x is 117.
Answer: 117
Example 2-1a

17. Find the value of x in
Answer: 35
Example 2-1b

18. Classify the triangle by its angles and its sides.
Angles has one right angle.
Sides has two congruent sides.
Answer: is a right isosceles triangle.
Example 2-2a

19. Classify the triangle by its angles and its sides.
Answer: acute equilateral
Example 2-2b

20. Classify the triangle by its angles and its sides.
Angles has one obtuse angle.
Sides has no congruent sides.
Answer: is an obtuse scalene triangle.
Example 2-3a

21. Classify the triangle by its angles and its sides.
Answer: right scalene
Example 2-3b

22. End of Lesson 2

23. Example 1 Find Lengths of a 30–60 Right Triangle
Example 2 Find the Lengths of a 45–45 Right Triangle
Lesson 3 Contents

24. Find each missing length. Round to the nearest tenth if necessary.
Step 1 Find c.
Write the equation.
Replace a with 6.
Multiply each side by 2.
Simplify.
Example 3-1a

25. Replace c with 12 and a with 6. Evaluate Subtract 36 from each side.
Step 2 Find b.
Pythagorean Theorem
Replace c with 12 and a with 6.
Evaluate
Subtract 36 from each side.
Simplify.
Take the square root of each side.
Use a calculator.
Answer: The length of b is about 10.4 inches, and the length of c is 12 inches.
Example 3-1a

26. Find each missing length. Round to the nearest tenth if necessary.
Answer:
Example 3-1b

27. BASEBALL The figure below shows the dimensions of a baseball diamond
BASEBALL The figure below shows the dimensions of a baseball diamond. The distance between home plate and first base is 90 feet. The area between first base, third base, and home plate forms a right triangle. Find the distance from first base to third base and the distance from third base to home plate.
Example 3-2a

28. Sides a and b are the same length. Since b 90 feet, a 90 feet.
Let a equal the distance from home plate to third base. Let b equal the distance from home plate to first base. And let c equal the distance from first base to third base.
Step 1 Find a.
Sides a and b are the same length Since b 90 feet, a 90 feet.
Example 3-2a

29. Replace a with 90 and b with 90.
Step 2 Find c.
Pythagorean Theorem
Replace a with 90 and b with 90.
Evaluate
Simplify.
Take the square root of each side.
Use a calculator.
Answer: The distance from first base to third base is about 127 feet, and the distance from third base to home plate is 90 feet.
Example 3-2a

30. SAILING The sail of a sailboat is in the shape of a. right triangle
SAILING The sail of a sailboat is in the shape of a right triangle. The height of the sail is 12 feet. Find each missing length.
Answer:
Example 3-2b

31. End of Lesson 3

32. Example 1 Find a Missing Angle Measure
Example 2 Classify Quadrilaterals
Example 3 Classify Quadrilaterals
Lesson 4 Contents

33. Find the value of q in quadrilateral PQRS.
Example 4-1a

34. The sum of the measures is 360.
Simplify.
Subtract 280 from each side.
Simplify.
Answer: 80
Example 4-1a

35. Find the value of q in quadrilateral QUAD.
Answer: 150
Example 4-1b

36. Classify the quadrilateral using the name that best describes it.
The quadrilateral has no congruent sides and no special angles.
Answer: It is a quadrilateral.
Example 4-2a

37. Classify the quadrilateral using the name that best describes it.
Answer: rhombus
Example 4-2b

38. Classify the quadrilateral using the name that best describes it.
The quadrilateral has all sides congruent and four right angles.
Answer: It is a square.
Example 4-3a

39. Classify the quadrilateral using the name that best describes it.
Answer: rectangle
Example 4-3b

40. End of Lesson 4

41. Example 1 Identify Congruent Polygons Example 2 Find Missing Measures
Lesson 5 Contents

42. Determine whether the trapezoids shown are congruent
Determine whether the trapezoids shown are congruent. If so, name the corresponding parts and write a congruence statement.
Example 5-1a

43. Angles The arcs indicate that
Sides The side measures indicate that ,
Answer: Since all pairs of corresponding angles and sides are congruent, the two trapezoids are congruent. One congruence statement is trapezoid
Example 5-1a

44. Determine whether the triangles shown are congruent
Determine whether the triangles shown are congruent. If so, name the corresponding parts and write a congruence statement.
Answer: yes;
Example 5-1b

45. According to the congruence statement, are corresponding angles. So,
In the figure, Find
According to the congruence statement, are corresponding angles. So,
Answer: Since
Example 5-2a

46. In the figure, Find
Answer:
Example 5-2b

47. Answer: Since centimeters, centimeters.
In the figure, Find QR.
corresponds to So,
Answer: Since centimeters, centimeters.
Example 5-3a

48. In the figure, Find LN.
Answer: 5 in.
Example 5-3b

49. End of Lesson 5

50. Example 1 Identify Line Symmetry
Example 2 Identify Rotational Symmetry
Example 3 Identify Rotational Symmetry
Lesson 6 Contents

51. Answer: This figure has one vertical line of symmetry.
TRILOBITES The trilobite is an animal that lived millions of year ago. Determine whether the figure has line symmetry. If it does, draw all lines of symmetry. If not, write none.
Answer: This figure has one vertical line of symmetry.
Example 6-1a

52. BOTANY Determine whether the leaf has line symmetry
BOTANY Determine whether the leaf has line symmetry. If it does, draw all lines of symmetry. If not, write none.
Answer:
Example 6-1b

53. FLOWERS Determine whether the flower design has rotational symmetry
FLOWERS Determine whether the flower design has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation.
Example 6-2a

54. Answer: Yes, this figure has rotational symmetry
Answer: Yes, this figure has rotational symmetry. It will match itself after being rotated 90, 180, and 270.
Example 6-2a

55. FLOWERS Determine whether the flower design has rotational symmetry
FLOWERS Determine whether the flower design has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation.
Answer: yes; 180
Example 6-2b

56. FLOWERS Determine whether the flower design has rotational symmetry
FLOWERS Determine whether the flower design has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation.
Example 6-3a

57. Answer: Yes, this figure has rotational symmetry
Answer: Yes, this figure has rotational symmetry. It will match itself after being rotated 60°, 120°, 180°, 240°, and 300°.
Example 6-3a

58. FLOWERS Determine whether the flower design has rotational symmetry
FLOWERS Determine whether the flower design has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation.
Answer: no
Example 6-3b

59. End of Lesson 6

60. Example 1 Draw a Reflection Example 2 Reflect a Figure over the x-axis
Example 3 Reflect a Figure over the y-axis
Example 4 Use a Reflection
Lesson 7 Contents

61. Copy trapezoid STUV below on graph paper
Copy trapezoid STUV below on graph paper. Then draw the image of the figure after a reflection over the given line.
Example 7-1a

62. Step 1
Count the number of units between each vertex and the line of reflection.
Answer:
Step 2
Plot a point for each vertex the same distance away from the line on the other side.
Step 3
Connect the new vertices to form the image of trapezoid STUV, trapezoid S'T'U'V'.
Example 7-1a

63. Copy trapezoid TRAP below on graph paper
Copy trapezoid TRAP below on graph paper. Then draw the image of the figure after a reflection over the given line.
Answer:
Example 7-1b

64. Graph quadrilateral EFGH with vertices E(–4, 4), F(3, 3), G(4, 2) and H(–2, 1). Then graph the image of EFGH after a reflection over the x–axis and write the coordinates of its vertices.
Example 7-2a

65. The coordinates of the vertices of the image are E'(–4, –4), F'(3, –3), G'(4, –2), and H'(–2, –1).
Notice that the y–coordinate of a point reflected over the x–axis is the opposite of the y–coordinate of the original point.
H(–2, 1)
same
opposites
E(–4, 4)
F(3, 3)
G(4, 2)
H'(–2, –1)
G'(4, –2)
F'(3, –3)
E'(–4, –4)
Example 7-2a

66. Answer: E'(–4, –4), F'(3, –3), G'(4, –2), and H'(–2, –1).
Example 7-2a

67. Answer: Q'(2, –4), U'(4, –1), A'(–1, –1), and D'(–3, –3).
Graph quadrilateral QUAD with vertices Q(2, 4), U(4, 1), A(–1, 1), and D(–3, 3). Then graph the image of QUAD after a reflection over the x–axis, and write the coordinates of its vertices.
Answer: Q'(2, –4), U'(4, –1), A'(–1, –1), and D'(–3, –3).
Example 7-2b

68. Graph trapezoid ABCD with vertices A(1, 3), B(4, 0), C(3, –4), and D(1, –2). Then graph the image of ABCD after a reflection over the y–axis, and write the coordinates of its vertices.
Example 7-3a

69. The coordinates of the vertices of the image are A'(–1, 3), B'(–4, 0), C'(–3, –4), and D'(–1, –2).
Notice that the x–coordinate of a point reflected over the y–axis is the opposite of the x–coordinate of the original point.
D(1, –2)
opposites
same
A(1, 3)
B(4, 0)
C(3, –4)
D'(–1, –2)
C'(–3, –4)
B'(–4, 0)
A'(–1, 3)
Example 7-3a

70. Answer: A'(–1, 3), B'(–4, 0), C'(–3, –4), and D'(–1, –2).
Example 7-3a

71. Answer: A'(–2, 2), B'(–5, 0), C'(–4, –2), and D'(–2, –1).
Graph quadrilateral ABCD with vertices A(2, 2), B(5, 0), C(4, –2), and D(2, –1). Then graph the image of ABCD after a reflection over the y–axis, and write the coordinates of its vertices.
Answer: A'(–2, 2), B'(–5, 0), C'(–4, –2), and D'(–2, –1).
Example 7-3b

72. Connect vertices as appropriate.
ARCHITECTURE Copy and complete the office floor plan shown below so that the completed office has a horizontal line of symmetry.
You can reflect the half of the office floor plan shown over the indicated horizontal line.
Find the distance from each vertex on the figure to the line of reflection.
Then plot a point the same distance away on the opposite side of the line.
Connect vertices as appropriate.
Answer:
Example 7-4a

73. GAMES Copy and complete the game board shown below so that the completed game board has a vertical line of symmetry.
Answer:
Example 7-4b

74. End of Lesson 7

75. Example 1 Draw a Translation
Example 2 Translation in the Coordinate Plane
Example 3 Use a Translation
Lesson 8 Contents

76. Move each vertex of the triangle 3 units right and 2 units up.
Copy below on graph paper. Then draw the image of the figure after a translation of 3 units right and 2 units up.
Step 1
Move each vertex of the triangle 3 units right and 2 units up.
Step 2
Connect the new vertices to form the image.
Answer:
Example 8-1a

77. Copy below on graph paper
Copy below on graph paper. Then draw the image of the figure after a translation of 2 units right and 4 units down.
Answer:
Example 8-1b

78. Graph ABC with vertices A(–2, 2), B(3, 4), and C(4, 1)
Graph ABC with vertices A(–2, 2), B(3, 4), and C(4, 1). Then graph the image of ABC after a translation of 2 units left and 5 units down. Write the coordinates of its vertices.
Example 8-2a

79. The coordinates of the vertices of the image are A'(–4, –3), B'(1, –1), and C'(2, –4). Notice that these vertices can also be found by adding –2 to the x–coordinates and –5 to the y–coordinates, or (–2, –5).
Original
Image
Example 8-2a

80. Answer: A'(–4, –3), B'(1, –1), and C'(2, –4)
Example 8-2a

81. Answer: P'(1, 0), Q'(4, 1), and R'(5, –1)
Graph PQR with vertices P(–1, 3), Q(2, 4), and R(3, 2). Then graph the image of PQR after a translation of 2 units left and 3 units down. Write the coordinates of its vertices.
Answer: P'(1, 0), Q'(4, 1), and R'(5, –1)
Example 8-2b

82. MULTIPLE-CHOICE TEST ITEM Point S is moved to a new location, S'
MULTIPLE-CHOICE TEST ITEM Point S is moved to a new location, S'. Which white shape shows where the shaded figure would be if it was translated in the same way?
A A
B B
C C
D D
Example 8-3a

83. Figure A: 3 units left and 1 unit up
Read the Test Item You are asked to determine which figure has been moved according to the same translation as point S.
Solve the Test Item Point S is translated 3 units right and 1 unit down. Identify the figure that is a translation of the shaded figure 3 units right and 1 unit down.
Figure A: 3 units left and 1 unit up
Figure B: 3 units right and 1 unit down
Answer: B
Example 8-3a

84. MULTIPLE-CHOICE TEST ITEM Point W is moved to a new location, W'
MULTIPLE-CHOICE TEST ITEM Point W is moved to a new location, W'. Which white shape shows where the shaded figure would be if it was translated in the same way?
A A
B B
C C
D D
Answer: D
Example 8-3b

85. End of Lesson 8

86. Example 1 Rotations in the Coordinate Plane Example 2 Use a Rotation
Lesson 9 Contents

87. Step 1 Lightly draw a line connecting point Q to the origin.
Graph QRS with vertices Q(1, 1), R(3, 4), and S(4, 1). Then graph the image of QRS after a rotation of counterclockwise about the origin, and write the coordinates of its vertices.
Step 1 Lightly draw a line connecting point Q to the origin.
Example 9-1a

88. Step 2 Lightly draw so that
Example 9-1a

89. Step 3 Repeat steps 1–3 for points R and Q
Step 3 Repeat steps 1–3 for points R and Q. Then erase all lightly drawn lines and connect the vertices to form
Example 9-1a

90. Answer: Triangle Q'R'S' has vertices Q'(–1, –1), R'(–3, –4), and S'(–4, –1).
Example 9-1a

91. Answer: A'(–4, –1), B'(–2, –1), C'(–2, –4)
Graph with vertices A(4, 1), B(2, 1), and C(2, 4). Then graph the image of after a rotation of counterclockwise about the origin, and write the coordinates of its vertices.
Answer: A'(–4, –1), B'(–2, –1), C'(–2, –4)
Example 9-1b

92. QUILTS Copy and complete the quilt piece shown below so that the completed figure has rotational symmetry with 90°, 180°, and 270°, as its angles of rotation.
Example 9-2a

93. Rotate the figure 90, 180, and 270 counterclockwise
Rotate the figure 90, 180, and 270 counterclockwise. Use a 90 rotation clockwise to produce the same rotation as a 270 rotation counterclockwise.
90° counterclockwise
180° counterclockwise
Example 9-2a

94. Answer:
90° clockwise
Example 9-2a

95. QUILTS Copy and complete the quilt piece shown below so that the completed figure has rotational symmetry with 90°, 180°, and 270°, as its angles of rotation.
Answer:
Example 9-2b

96. End of Lesson 9

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