# Chapter 3. Chapter 3 Opener. Section 3.1. Big Ideas Math Blue Worked-Out Solutions. Try It Yourself (p. 101) So, the value of x is PDF Free Download

4.1 Translations – Big Ideas Math Geometry
4.1 Translations – Big Ideas Math Geometry

Chapter 3 Opener Try It Yourself (p. 101) 1. The angles are vertical. x + 8 120 x 112 o, the value of x is 112. 2. The angles are adjacent. ( x ) + 3 + 43 90 x + 46 90 x 44 o, the value of x is 44. 3. The angles are complementary. ( x ) – 8 + 20 90 x + 12 90 x 78 o, the value of x is 78. 4. The angles are supplementary. ( x ) 2 + 4 + 76 180 + 80 180 100 x 50 o, the value of x is 50. ection 3.1 3.1 Activity (pp. 102 103) 1. a. Eight angles are formed. ample answer: 1 2 3 4 5 6 7 8 b. 2, 4, 6, and 8 have equal measures. 1, 3, 5, and 7 have equal measures. ample answer: 2 and 4, 6 and 8, 1 and 3, 5 and 7 are vertical angles. Vertical angles have the same measure. Using a protractor, you can determine that the following angles have equal measures: 1 and 5, 2 and 6, 3 and 7, 4 and 8. You can also determine this by visually placing an angle over the angle that corresponds to it to see that they have equal measures. 3. a. ample answer: b. Answer should include, but is not limited to: tudents should adjust the parallel lines or transversal. The angle measures should increase and decrease as they adjust the lines. 4. ample answer: When two parallel lines are intersected by a transversal, eight angles are formed as shown. In the figure, 1, 3, 5, and 7 are congruent, and 2, 4, 6, and 8 are congruent. 5. G D G D 1 110″ A 70″ The measures of all eight angles are 90. 3.1 On Your Own (pp. 104 106) H 2 3 4 90″ 90″ A 90″ 90″ 1. 1 and the 63 angle are corresponding angles. They are congruent. o, the measure of 1 is 63. 2. 1 and 2 are supplementary. 1 + 2 180 63 + 2 180 2 117 110″ C 70″ 70″ 110″ o, the measure of 2 is 117. 5 C F 6 7 8 90″ 90″ 90″ 90″ H F 70″ 110″ B E B E 2. a. ample answer: Measure the vertical angles and the angles that correspond to them and make sure they are congruent. b. The studs are parallel lines and the diagonal support beam is a transversal. 71

3. The 59 angle is supplementary to both 1 and 3. 1 + 59 180 1 121 o, the measures of 1 and 3 are 121. 2 and the 59 angle are vertical angles. They are congruent. o, the measure of 2 is 59. Using corresponding angles, the measures of 6 are 121, and the measures of 5 and 7 are 59. 4 and 4. Because all of the letters are slanted at a 65 angle, the dashed lines are parallel. The piece of tape is the transversal. Using corresponding angles, the 65 angle is congruent to the angle that is supplementary to 1. o, the measure of 1 is 180-65 115. 5. 3 and 4 are supplementary angles. 3 + 4 180 3 + 84 180 3 96 o, the measure of 3 is 96. 6. 4 and 5 are alternate interior angles. Because the angles are congruent, the measure of 5 is 84. 7. 4 and 5 are alternate interior angles. Because the angles are congruent, the measure of 5 is 84. 5 and 6 are supplementary angles. 5 + 6 180 84 + 6 180 6 96 o, the measure of 6 is 96. 3.1 Exercises (pp. 107 109) Vocabulary and Concept Check 1. ample answer: 4 and 8 are corresponding angles. 2. Because 2 and 6 are corresponding angles, and 6 and 8 are vertical angles, 2, 6, and 8 are congruent. Because 5 and 6 are supplementary angles, the statement The measure of belong with the other three. 4 8 t p q 5 does not Practice and Problem olving 3. Lines m and n are parallel. 4. Line t is the transversal. 5. Eight angles are formed by the transversal. 6. Using vertical angles and corresponding angles, 1, 3, 5, and 7 are congruent, and 2, 4, 6, and 8 are congruent. 7. 1 and the 107 angle are corresponding angles. They are congruent. o, the measure of 1 is 107. 1 and 2 are supplementary angles. 1 + 2 180 107 + 2 180 2 73 o, the measure of 2 is 73. 8. 3 and the 95 angle are corresponding angles. They are congruent. o, the measure of 3 is 95. 3 and 4 are supplementary angles. 3 + 4 180 95 + 4 180 4 85 o, the measure of 4 is 85. 9. 5 and the 49 angle are corresponding angles. They are congruent. o, the measure of 5 is 49. 5 and 6 are supplementary angles. 5 + 6 180 49 + 6 180 6 131 o, the measure of 6 is 131. 10. The lines are not parallel, so the corresponding angles 5 and 6 are not congruent. 11. Because 1 and 2 are corresponding angles, the measure of 2 is 60. 12. ample answer: The yard lines on a football field are parallel. The lampposts on a road are parallel. 13. ample answer: Rotate the figure 180 and translate down. 14. The least number of angle measures you need to know is one angle measure. Three other angles will be congruent to the known angle. The rest of the angle measures are supplementary to the known angle. 72

15. 1 and the 61 angle are corresponding angles. They are congruent. o, the measure of 1 is 61. 1 is supplementary to both 2 and 4. 1 + 2 180 61 + 2 180 2 119 o, the measures of 2 and 4 are 119. 1 and 3 are vertical angles. They are congruent. o, the measure of 3 is 61. Using corresponding angles, the measures of 5 and 7 are 119, and the measure of 6 is 61. 16. The 99 angle is supplementary to both 1and 3. 1 + 99 180 1 81 o, the measures of 1and 3 are 81. 2 and the 99 angle are vertical angles. They are congruent. o, the measure of 2 is 99. Using corresponding angles, the measures of 4 and 6 are 99, and the measures of 5 and 7 are 81. 17. The right angle is supplementary to both 1 and 3. 90 + 1 180 1 90 o, the measures of 1 and 3 are 90. 2 and the right angle are vertical angles. They are congruent. o, the measure of 2 is 90. Using corresponding angles, the measures of 4, 5, 6, and 7 are 90. 18. Using corresponding angles, 1is congruent to 8, which is supplementary to 4. 1 + 4 180 124 + 4 180 4 56 o, if the measure of 1 124, then the measure of 4 56. 19. Using corresponding angles, 2 is congruent to 7, which is supplementary to 3. 2 + 3 180 48 + 3 180 3 132 o, if the measure of 2 48, then the measure of 3 132. 20. Because 4 and 2 are alternate interior angles, 4 is congruent to 2. o, if the measure of 4 55, then the measure of 2 55. 21. Because 6 and 8 are alternate exterior angles, 6 is congruent to 8. o, if the measure of 6 120, then the measure of 8 120. 22. Using alternate exterior angles, 7 is congruent to 5, which is supplementary to 6. 7 + 6 180 50.5 + 6 180 6 129.5 o, if the measure of 7 50.5, then the measure of 6 129.5. 23. Using alternate interior angles, 3 is congruent to 1, which is supplementary to 2. 3 + 2 180 118.7 + 2 180 2 61.3 o, if the measure of 3 118.7, then the measure of 2 61.3. 24. Because the two rays of sunlight are parallel, 1 and 2 are alternate interior angles. Because the angles are congruent, the measure of 1 is 40. 25. Because the transversal is perpendicular to two parallel lines, they intersect at right angles. o, all the angles formed are 90. 26. ample answer: Using vertical angles, 1 is congruent to 3. 3 and 7 are congruent because they are corresponding exterior angles. o, 1 is congruent to 7. Using corresponding angles, 1 is congruent to 5, and 5 and 7 are congruent because they are vertical angles. o, 1 is congruent to 7. 27. The 50 angle is congruent to the alternate interior angle formed by the intersection of line a and line c. This angle is congruent to the corresponding angle formed by the intersection of line a and line d. This angle is supplementary to the x angle. o, the 50 angle is supplementary to the x angle. 50 + x 180 x 130 o, the value of x is 130. 73

5. Answer should include, but is not limited to: tudents should draw three triangles and repeat parts (b) (d) from Activity 1 for each triangle. Yes, students should get the same results. 6. The sum of the measures of the interior angles of a triangle is 180. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 3.2 On Your Own (pp. 112 113) 1. x + 81 + 25 180 x + 106 180 2. x ( x ) x 74 + – 35 + 43 180 3. 40 + 30 y 70 y + 8 180 172 x 86 o, the measure of the exterior angle is 70. n – n + + n 4. 4 20 ( 2 20) 4n – 20 3n + 20 4n 3n + 40 n 40 o, the measure of the exterior angle is 4( 40) – 20 140. 5. x ( x ) + + 7.5 + 63.9 180 + 71.4 180 The value of x is 54.3. 108.6 x 54.3 3.2 Exercises (pp. 114 115) Vocabulary and Concept Check 1. ubtract the sum of the given angle measures from 180. 2. 2; When the sides forming a vertex are extended, 2 angles are formed that are adjacent to the interior angle. 3. 60 + 55 115 ; 65 + 55 120 ; 60 + 65 125 Practice and Problem olving 4. 30 + 90 + x 180 120 + x 180 x 60 o, the measures of the interior angles are 30, 60, and 90. 5. 65 + 40 + x 180 105 + x 180 x 75 o, the measures of the interior angles are 40, 65, and 75. 6. 35 + 45 + x 180 80 + x 180 x 100 o, the measures of the interior angles are 35, 45, and 100. 7. ( x ) + 65 + 25 + x 180 90 + 180 90 x 45 o, the measures of the interior angles are 25, 45, and ( 45 + 65) 110. 48 + – 44 + x 180 8. ( x ) 4 + 180 176 x 88 o, the measures of the interior angles are 88-44 44, 48, and 88. 73 + – 11 + x 180 9. ( x ) 62 + 180 118 x 59 o, the measures of the interior angles are 59-11 48, 59, and 73. 10. 60 + x + x 180 60 + 180 120 x 60 The value of x in the billiard rack is 60. 11. + 45 + x 180 45 + 3x 180 3x 135 x 45 The value of x is 45. 12. 38 + 90 x 128 x o, the measure of the exterior angle is 128. 75

13. 64 + 76 k 140 k o, the measure of the exterior angle is 140. 14. ( a ) + 10 + 44 2a a + 54 2a 54 a o, the measure of the exterior angle is 2( 54) 108. 15. The measure of the exterior angle is equal to the sum of the measures of the two nonadjacent interior angles. – 12 x + 30 x + 42 x 42 The exterior angle is ( ) 16. + 3x + 5x 180 10x 180 x 18 2 42-12 72. The interior angle measures are 2( 18) 36, 3( 18) 54, 5( 18) 90. 17. 90 + 3x 180 – ( 5x – 6) 90 + 3x – 5x + 186 3x – 5x + 96 8x 96 x 12 ( ) 180-5 12-6 126 The measure of the exterior angle is 126. 18. no; The two nonadjacent interior angles could be any two angle measures that sum to 120. 19. sometimes; The sum of the angle measures must equal 180. 20. always; Because the sum of the interior angle measures must equal 180 and one of the interior angles is 90, the other two interior angles must sum to 90. 21. never; If a triangle had more than one vertex with an acute exterior angle, then it would have to have more than one obtuse interior angle which is impossible. 22. You know that x + y + w 180 and w + z 180. ubstitute w + z for 180 in the first equation to get x + y + w w + z. ubtract w from each side of the equation to get x + y z. Fair Game Review 23. – 4x + 3 19 Check: – 4x + 3 19-3 – 3? – 4( – 4) + 3 19-4x 16? – 4x 16 16 + 3 19-4 – 4 19 19 x – 4 2-1 + 6y – 10 24. ( y ) ( y) 2-2 1 + 6y – 10 2y – 2 + 6y – 10 8y – 2-10 + 2 + 2 8y – 8 8y – 8 8 8 y – 1 Check: ( y ) 2-1 + 6y – 10 2-1 – 1 + 6-1 – 10 25. ( n) 2-2 + – 6-10 – 4 + – 6-10??? – 10-10 5 + 0.5 6 + 14 3 5 + 0.5 6 + 0.5 14 3 5 + 3n + 7 3 3 3n + 12 3-12 – 12 3n – 9 3n – 9 3 3 n – 3 Check: 5 + 0.5 6 + 14 3 5 + 0.5 6-3 + 14 3 5 + 0.5-18 + 14 3 5 + 0.5-4 3???? 5-2 3 3 3 26. A 76

tudy Help Available at BigIdeasMath.com. Quiz 3.1 3.2 1. Because the 82 angle and 2 are alternate exterior angles, the angles are congruent. o, the measure of 2 is 82. 2. Because the 82 angle and 6 are vertical angles, the angles are congruent. o, the measure of 6 is 82. 3. Because the 82 angle and 4 are corresponding angles, the angles are congruent. o, the measure of 4 is 82. 4. Using corresponding angles, the 82 angle is congruent to 4, which is supplementary to 1. o, the measure of 1 is 180-82 98. 5. Using alternate exterior angles, 1 is congruent to 7. o, if the measure of 1 123, then the measure of 7 123. 6. Using corresponding angles, 2 is congruent to 6, which is supplementary to 5. 2 + 5 180 58 + 5 180 5 122 o, if the measure of 2 58, then the measure of 5 122. 7. Because 5 and 3 are alternate interior angles, 5 is congruent to 3. o, if the measure of 5 119, then the measure of 3 119. 8. Because 4 and 6 are alternate exterior angles, 4 is congruent to 6. o, if the measure of 4 60, then the measure of 6 60. 9. x + 60 + 60 180 x + 120 180 x 60 o, the measures of the interior angles are 60, 60, and 60. 10. x + 25 + 40 180 x + 65 180 x 115 o, the measures of the interior angles are 25, 40, and 115. 11. x + x + 90 180 + 90 180 90 x 45 o, the measures of the interior angles are 45, 45, and 90. 12. 55 + 50 b 105 b o, the measure of the exterior angle is 105. z + 10 + 4z z + 50 13. 5z + 10 z + 50 4z + 10 50 4z 40 z 10 o, the measure of the exterior angle is 10 + 50 60. 14. Using corresponding angles, the 72 angle is congruent to the angle that is supplementary to both 1 and 2. 1 + 72 180 1 108 o, the measures of 1 and 2 are 108. 15. x + 5x + 90 180 6x + 90 180 6x 90 x 15 Exterior angle with wall: 180-15 165 Exterior angle with ground: 180-5( 15) 105 ection 3.3 3.3 Activity (pp. 118 119) 1. a. Quadrilateral: n 4 Yes, there is more than one way to divide the figure into two triangles. ample answer: Draw one line that divides the quadrilateral into two triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the angle 360. measures of the quadrilateral is 77

b. Pentagon: n 5 ample answer: f. Number of ides, n Number of Triangles 3 4 5 6 7 8 1 2 3 4 5 6 Draw two lines that divide the pentagon into three triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the pentagon is 3 180 540. c. Hexagon: n 6 ample answer: 2. a. Angle um, The sum of the interior angle measures of a polygon 10 180 1800. with 12 sides is C B 180 360 540 720 900 1080 A E 78 Draw three lines that divide the hexagon into four triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the hexagon is 4 180 720. d. Heptagon: n 7 ample answer: Draw four lines that divide the heptagon into five triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the heptagon is 5 180 900. e. Octagon: n 8 ample answer: Draw five lines that divide the octagon into six triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the octagon is 6 180 1080. b. tudents arrange the angles to be adjacent and joined at the vertices. The sum of the angles is 360. c. Answer should include, but is not limited to: tudents repeat the procedure in parts (a) and (b) for the quadrilateral and the hexagon. The sum of the measures of the exterior angles of a convex polygon is 360. The sum does not depend on the number of sides of the polygon. 3. ( n – 2) 180 4. To find the sum of the interior angle measures of a n -. polygon with n sides, use the expression The sum of the exterior angle measures of a convex polygon is 360. 3.3 On Your Own (pp. 120 122) 1. The spider web is in the shape of a heptagon. It has 7 sides. – 7-5 180 900 The sum of the interior angle measures is 900. 2. The honeycomb is in the shape of a hexagon. It has 6 sides. – 6-4 180 720 D The sum of the interior angle measures is 720.

3. The polygon has 6 sides. – 6-4 180 720 The sum of the interior angle measures is 720. 125 + 120 + 125 + 110 + 135 + x 720 The value of x is 105. 4. The polygon has 4 sides. 4-360 615 + x 720 x 105 The sum of the interior angle measures is 360. 115 + 80 + 90 + x 360 285 + x 360 The value of x is 75. x 75 5. The polygon has 5 sides. – 5-3 180 540 The sum of the interior angle measures is 540. + 145 + 145 + + 110 540 The value of x is 35. 6. An octagon has 8 sides. – 8-6 180 1080 4x + 400 540 4x 140 x 35 The sum of the interior angle measures is 1080. 1080 8 135 The measure of each interior angle is 135. 7. A decagon has 10 sides. – 10-8 180 1440 The sum of the interior angle measures is 1440. 1440 10 144 The measure of each interior angle is 144. 8. An 18-gon has 18 sides. – 18-16 180 2880 The sum of the interior angle measures is 2880. 2880 18 160 The measure of each interior angle is 160. 9. 90 + 90 + x + 90 + x 360 270 + 360 90 x 45 o, the measures of the exterior angles are 90, 90, 45, 90, and 45. 3.3 Exercises (pp. 123 125) Vocabulary and Concept Check 1. A three-sided polygon is a triangle. Because it is a regular polygon, all sides are congruent. o, the polygon is an equilateral triangle. 2. Because the second figure is not made up entirely of line segments and the other three are, the second figure does not belong. 79

3. Because the question What is the measure of an interior angle of a regular pentagon? asks for one interior angle measure and the other three ask for the sum of the interior angle measures, it is different. A regular pentagon has 5 sides. – 5-3 180 540 540 5 108 The measure of an interior angle of a regular pentagon is 108. The sum of the interior angle measures of a regular, convex, or concave pentagon is 540. Practice and Problem olving 4. 5. 6. Draw one line that divides the quadrilateral into two triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the quadrilateral is 2( 180 ) 360. Draw six lines that divide the 9-gon into seven triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the 9-gon is 7( 180 ) 1260. Draw four lines that divide the heptagon into five triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the heptagon is 5( 180 ) 900. 7. The polygon has 4 sides. – 4-360 The sum of the interior angle measures is 360. 8. The polygon has 8 sides. – 8-6 180 1080 The sum of the interior angle measures is 1080. 9. The polygon has 9 sides. – 9-7 180 1260 The sum of the interior angle measures is 1260. 10. The formula is incorrect. The correct formula is the product of two less than the number of sides and 180. – 13-11 180 1980 11. no; ( n – ) ( – ) 5 3 180 540 Because the sum of the given interior angle measures is 120 + 105 + 65 + 150 + 95 535 and not 540, a pentagon cannot have the given interior angle measures. 12. The polygon has 4 sides. – 4-360 155 + 25 + 137 + x 360 317 + x 360 x 43 The measures of the interior angles are 155, 25, 137, and 43. 80

13. The polygon has 6 sides. 2 180 6 2 180 4 180 720 90 + 90 + x + x + x + x 720 180 + 4x 720 4x 540 x 135 The measures of the interior angles are 90, 90, 135, 135, 135, and 135. 14. The polygon has 6 sides. 2 180 6 2 180 4 180 720 3x + 45 + 135 + x + 135 + 45 720 4x + 360 720 4x 360 x 90 The measures of the interior angles are 3( 90) 270, 45, 135, 90, 135, and 45. 15. Find the number of sides. 2 180 1260 2 180 7 n 2 9 n Because the regular polygon has 9 sides, the measure of each interior angle is 1260 9 140. 16. The sum of the interior angle measures of a triangle is 180. 180 3 60 The measure of each interior angle is 60. 17. The polygon has 9 sides. 2 180 9 2 180 7 180 1260 1260 9 140 The measure of each interior angle is 140. 18. The polygon has 12 sides. 2 180 12 2 180 10 180 1800 1800 12 150 The measure of each interior angle is 150. 19. The sum should have been divided by the number of interior angles, which is 20, not 18. 3240 20 162 The measure of each interior angle is 162. 20. a. The bolt has 5 sides. 2 180 5 2 180 3 180 540 540 5 108 The measure of each interior angle is 108. b. ample answer: Because the standard shape of a bolt is a hexagon, most people have tools to remove a hexagonal bolt and not a pentagonal bolt. o, a special tool is needed to remove the bolt from a fire hydrant. 21. The sum of the interior angles of the regular polygon is n 165, where n is the number of sides. 2 180 n 165 2 180 165n 180n 360 15n 360 n 24 The polygon has 24 sides. 22. 140 + 110 + x 360 250 + x 360 x 110 o, the measures of the exterior angles are 140, 110, and 110. 23. 107 + 85 + 93 + w 360 285 + w 360 w 75 o, the measures of the exterior angles are 107, 85, 93, and 75. 81

24. ( z ) + 45 + z + 74 + 78 + 55 360 252 + 2z 360 2z 108 z 54 o, the measures of the exterior angles are 54 + 45 99, 54, 74, 78, and 55. 25. 60 ; The sum of the interior angle measures of a hexagon is (6 2) 180 720. Because it is regular, each angle has the same measure. o, the measure of each interior angle is 720 6 120, and the measure of each exterior angle is 180 120 60. 26. n + n + 90 + n + n + 90 360 180 + 4n 360 4n 180 n 45 o, the measures of the exterior angles are 45, 45, 90, 45, 45, and 90. 27. Because the interior angles of the triangle have the same measure, the measure of each interior angle is 180 3 60. Therefore, the measure of each exterior angle is 180 60 120. o, the measures of the exterior angles are 120, 120, and 120. 28. The exterior angle for 55 is 180 55 125. The exterior angle for 125 is 180 125 55. Two sides of the quadrilateral are parallel and the other sides act as transversals. Using alternate interior angles, the other exterior angles are 55 and 125. o, the measures of the exterior angles are 125, 55, 125, and 55. 29. 2 180 8 2 180 6 180 1080 The measure of each interior angle of the polygon is 1080 8 135. 8x 360 x 45 The measure of each of exterior angle is 45. 31. Find the sum of the interior angle measures of the heptagon. 2 180 7 2 180 5 180 900 Let x represent the value of each of the three remaining interior angle measures, in degrees. 4 135 + 3 x 900 540 + 3x 900 3x 360 x 120 The measure of each remaining interior angle is 120. 32. a. The polygon has 11 sides. b. 2 180 11 2 180 9 180 1620 1620 11 147 The measure of each interior angle is about 147. 33. a. ample answer: b. ample answer: quares: Regular hexagons: c. ample answer: The tessellation is formed using equilateral triangles and squares. 30. 45 45 270 82

d. Answer should include, but is not limited to: a discussion of the interior and exterior angles of the polygons in the tessellation and how they add to 360 where the vertices meet. Fair Game Review 34. 36. x 3 12 4 x 4 12 3 4x 36 x 9 14 x 35. 21 3 14 3 21 x 42 21x 2 x The value of x is 9. The value of x is 2. 9 6 x 2 9 2 x 6 18 6x 3 x 10 15 37. 4 x 10 x 4 15 10x 60 x 6 The value of x is 3. The value of x is 6. 38. D; Because the ratio of tulips to daisies is 3 : 5, the total number is a multiple of 3 + 5 8. The multiples of 8 are 8, 16, 24, 32, R. o, 16 is the only choice that could be the total number of tulips and daisies. ection 3.4 2. a. A 3 B 3 C 3 A 2 B 2 C 2 A 50 30 B C A 1 B 1 C 1 A 4 3.4 Activity (pp. 1268127) 1. a. ample answer: Draw a line segment that is 8 centimeters long. Then use the line segment and a protractor to draw a triangle that has a 60 and a 40 angle. Label the triangle JKL. b. ample answer: Draw a line segment that is 2 centimeters long. Then use the line segment and a protractor to draw a triangle that has a 60 and a 40 angle. Label the triangle PQR. c. yes; ample answer: The corresponding angles of the triangles are congruent and the corresponding side lengths are proportional. B 4 C 4 b. The third angle measure is 100 in each triangle. c. Because two angles in the first triangle are congruent to two angles in the second triangle, the sum of the measures of the two angles in each triangle is the same. Therefore, when you subtract this sum from 180, you get the measure of the third angle in each triangle. o, the triangles are similar. 3. a. Because the uns rays are parallel, B and E are corresponding angles and are therefore congruent: B E. Because Aand D are both right angles, A D. Because two angles of ABC are congruent to two angles of DEF, C F. Therefore, ABC and DEF are similar triangles. b. Because ABC and DEF are similar triangles, the ratios of the corresponding side lengths are equal. o, write and solve a proportion to find the height of the flagpole. Height of flagpole Height of boy x 36 5 3 3x 180 x 60 The height of the flagpole is 60 feet. Length of flagpoles shadow Length of boys shadow 83

4. When two angles in one triangle are congruent to two angles in another triangle, you can conclude that the third angles are also congruent. o, the triangles are similar. 5. a. Because the flagpole is not being measured directly, the process is called VindirectW measurement. bbc. Answer should include, but is not limited to: The student will use indirect measurement to measure the height of something outside. The student will include a diagram of the process used with all measurements and calculated lengths labeled. 6. ample answer: In the figure, you know that the streetlight forms a right angle with the ground and the person forms a right angle with the ground. Each triangle shares the same angle formed by the ground and the top of the streetlight. Because two angles in one triangle are congruent to two angles in another triangle, the third angles are also congruent. o, the triangles are similar. 3.4 On Your Own (p. 129) 1. no; x + 28 + 80 180 x + 108 180 x 72 The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 2. yes; 66 + 90 + x 180 3. 156 + x 180 x 24 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. x 40 55 50 x 40 55 55 55 50 x 44 The distance across the river is 44 feet. 3.4 Exercises (pp. 1308131) Vocabulary and Concept Check 1. Because the ratios of the corresponding side lengths in similar triangles are equal, a proportion can be used to find a missing measurement. 2. The angle measures of ABC, DEF, and GHI are 35, 82, and 63, so they are similar triangles. The angle measures of JKL are 32, 85, and 63. o, JKL does not belong with the other three. Practice and Problem olving 4B 5. Answer should include, but is not limited to: tudents should draw a triangle with the same angle measures as the triangle in the textbook. The ratios of the corresponding side lengths should be equal. (Ratios may differ slightly due to rounding.) 6. yes; The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 7. no; x + 36 + 72 180 x + 108 180 x 72 The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 8. no; x + 64 + 85 180 x + 149 180 x 31 The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 9. yes; x + 48 + 81 180 x + 129 180 x 51 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 10. x + x + 90 180 + 90 180 90 x 45 The leftmost and rightmost rulers have two pairs of congruent angles. o, the third angles are congruent, and the rulers are similar in shape. 11. yes; The measure of the exterior angle is 51 + 90 141. 141 102 39 180 102 39 39 51 102 39 39 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 3. ample answer: The two angles in each triangle have the same sum. When you subtract this sum from 180, you get the same third angle. 84

12. Using vertical angles, the triangle on the left has an angle measure of 29. Using supplementary angles, the triangle on the right has an angle measure of 180 91 89. The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 13. Using vertical angles, the triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. Find the missing dimension using indirect measurement. x 80 300 240 240x 24,000 x 100 You take 100 steps from the pyramids to the treasure. 14. no; Consider the two similar triangles below. A 1 bh 2 1 xy 2 1 A bh 2 1 1.5 1.5 2 1 ( 2.25) xy 2 ( x)( y) o, the area of the larger triangle is 2.25 times the area of the smaller triangle, which is a 125% increase. 15. Find the missing dimension using indirect measurement. Height of person Length of persons shadow 6 h 3 15 90 3h 30 h The height of the pine tree is 30 feet. 16. ample answer: 10 ft Assume that you are 5 feet tall. 5 ft y x 1.5x 3 ft 6 ft x 1.5y Height of tree Length of trees shadow 17. maybe; They are similar when both have measures of 30, 60, 90 or both have measures of 45, 45, 90. They are not similar when one has measures of 30, 60, 90 and the other has measures of 45, 45, 90. ~ ; ~ ; ~ 18. ABG ACF ABG ADE ACF ADE; Because AB, BC, and CD are all equal, and the length of segment BD is 6.32 feet, the length of segment AB is 6.32 2 3.16 feet and the length of segment AD is 6.32 + 3.16 9.48 feet. Let x BG. AB BG AD DE 3.16 x 9.48 6 18.96 9.48x 2 x o, the length of segment BG is 2 feet. The length of segment AC is 3.16 2 6.32 feet. Let y CF. CF DE AC AD y 6 6.32 9.48 9.48y 37.92 y 4 o, the length of segment CF is 4 feet. Fair Game Review 19. y 5x 3 y 5x + 5x 3 + 5x y 3 + 5x 20. 4x + 6y 12 21. 4x 4x + 6y 12 4x 6y 12 4x 6y 12 4x 6 6 2 y 2 x 3 1 y 1 4 1 y 1 4 1 y 1 4 1 4 y 4 1 2 4 y 4 + 8x ( x) 85

40 + + 16 + x 180 22. B; ( x ) Quiz 3.3B3.4 + 56 180 124 x 62 1. The polygon has 10 sides. 2 180 10 2 180 8 180 1440 The sum of the interior angle measures is 1440. 2. The polygon has 13 sides. 2 180 13 2 180 11 180 1980 The sum of the interior angle measures is 1980. 3. The polygon has 4 sides. 2 180 4 2 180 2 180 360 x + 122 + 134 + 46 360 x + 302 360 x 58 The measures of the interior angles are 58, 122, 134, and 46. 4. The polygon has 7 sides. 2 180 7 2 180 5 180 900 x + 130 + 140 + 120 + 115 + 154 + 115 900 x + 774 900 x 126 The measures of the interior angles are 126, 130, 140, 120, 115, 154, and 115. 5. The polygon has 5 sides. 2 180 5 2 180 3 180 540 x + 40 + 4x + 40 + 110 540 5x + 190 540 5x 350 x 70 The measures of the interior angles are 70, 40, 4 70 280, 40, and 110. 115 + 2 + 100 + + 25 360 6. r ( r ) 3r + 240 360 3r 120 r 40 The measures of the exterior angles are 115, 2 40 80, 40 + 25 65. 100, and 7. 90 + 90 + x + 360 3x + 180 360 3x 180 x 60 The measures of the exterior angles are 90, 2 60 120, 90, and 60. 8. x + 46 + 95 180 x + 141 180 x 39 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 9. x + 40 + 51 180 x + 91 180 x 89 The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 10. 2 180 4140 2 180 23 n 2 25 n The polygon has 25 sides. 86

11. a. Angles W and Z are right angles, so they are congruent. Angles WXV and ZXY are vertical angles, so they are congruent. Because two angles in VWX are congruent to two angles in YXZ, the third angles are also congruent and the triangles are similar. b. 60 100 30 60 3000 50 The distance across the patch of swamp is 50 feet. Chapter 3 Review 1. The 140 angle and 8 are alternate exterior angles. They are congruent. o, the measure of 8 is 140. 2. The 140 angle and 5 are corresponding angles. They are congruent. o, the measure of 5 is 140. 3. The 140 angle and 3 are supplementary. o, the measure of 3 is 180 140 40. 3 and 7 corresponding angles. They are congruent. o, the measure of 7 is 40. 4. The 140 angle and 2 are supplementary. o, the measure of 2 is 180 140 40. 5. x + 49 + 90 180 x + 139 180 x 41 are The measures of the interior angles are 49, 41, and 90. 6. x + 35 + 110 180 x + 145 180 x 35 The measures of the interior angles are 35, 110, and 35. 7. 50 + 75 s 125 s The measure of the exterior angle is 125. + 10 + + + 20 180 8. ( t ) t ( t ) 3t + 30 180 3t 150 t 50 The measure of the exterior angle is 180 50 + 20 110. 9. The polygon has 4 sides. 2 180 4 2 180 2 180 360 60 + 128 + 95 + x 360 283 + x 360 x 77 o, the measures of the interior angles are 60, 128, 95, and 77. 10. The polygon has 7 sides. 2 180 7 2 180 5 180 900 135 + 125 + 135 + 105 + 150 + 140 + x 900 790 + x 900 x 110 o, the measures of the interior angles are 135, 125, 135, 105, 150, 140, and 110. 11. The polygon has 6 sides. 2 180 6 2 180 4 180 720 100 + 120 + 60 + + 65 + x 720 3x + 345 720 3x 375 x 125 o, the measures of the interior angles are 100, 120, 2 125 250, 65, and 125. 60, 12. 135 + 100 + y 360 235 + y 360 y 125 o, the measures of the exterior angles are 135, 100, and 125. 13. 6z 360 z 60 o, the measures of the 6 exterior angles are 60 each. 87

14. yes; x + 68 + 90 180 x + 158 180 x 22 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 15. yes; x + 30 + 100 180 x + 130 180 x 50 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. Chapter 3 Test 1. 1 and the 47 angle are supplementary. o, the measure of 1 is 180 47 133. 2. 1 and the 47 angle are supplementary. o, the measure of 1 is 180 47 133. 1 and 8 are alternate exterior angles. They are congruent. o, the measure of 8 is 133. 3. 4 and the 47 angle are supplementary. o, the measure of 4 is 180 47 133. 4. 1 and the 47 angle are supplementary. o, the measure of 1 is 180 47 133. 1 and 5 corresponding angles. They are congruent. o, the measure of 5 is 133. 5. x + 23 + 129 180 x + 152 180 are x 28 o, the measures of the interior angles are 23, 28, and 129. 6. x + 44 + 68 180 x + 112 180 x 68 o, the measures of the interior angles are 68, 68, and 44. 7. x + x + x 180 3x 180 x 60 o, the measures of the interior angles are 60, 60, and 60. 8. j 90 + 40 j 130 9. ( 2 p + 15) p + ( p + 15) 2 p + 15 2 p + 15 The exterior angle can have any measure greater than 15 and less than 180. 10. The polygon has 5 sides. 2 180 5 2 180 3 180 540 125 + 90 + 125 + + 540 4x + 340 540 4x 200 x 50 o, the measures of the interior angles are 125, 90, 2 50 100, 2 50 100. 125, and + 111 + + 17 + 90 360 11. y ( y ) 2y + 218 360 2y 142 y 71 o, the measures of the exterior angles are 71, 111, 90, 71 + 17 88. and 12. no; x + 61 + 70 180 x + 131 180 x 49 The triangles do not have two pairs of congruent angles. o, the triangles are not similar. 13. yes; x + 35 + 90 180 x + 125 180 x 55 The triangles have two pairs of congruent angles. o, the third angles are congruent, and the triangles are similar. 14. ample answer: 1) The 65 angle and 3 are supplementary, so 3 is 180 65 115 ; 3 and 5 are alternate interior angles. They are congruent. o, the measure of 5 is 115. 2) The 65 angle and 8 are alternate exterior angles. They are congruent. o, the measures of 8 is 65. 5 and 8 are supplementary, so 5 is 180 65 115. o, the measure of the exterior angle is 130. 88

15. The triangles are similar, so the ratios of the corresponding side lengths are equal. d 80 105 140 140d 8400 d 60 The distance across the pond is 60 meters. Chapter 3 tandards Assessment 1. 147; Find the sum of the interior angle measures. 2 180 11 2 180 1620 Divide the sum by the number of interior angles, 11. 1620 11 147 The measure of each interior angle is about 147. 2. B; C 11 + 1.6t C 11 1.6t C 11 t 1.6 5 4 3 x 3. I; ( x ) ( x) 5 5 4 3 x 5x 20 3 x + 20 + 20 5x 3x + 20 3 x 3 x 20 20 2 2 x 10 12 cm x cm 4. C; 16 cm 18 cm 16x 216 5. 55; x 13.5 Oak Trail 125 1 2 10 15 30 5 5 2 5 ( 10x 15) 30 2 5 2 10x 15 75 6. F; ( x ) 7. B; 10x 90 x 9 Reflection in the y-axis: ( x, y) ( x, y) X Y Z ( 6, 1) X ( 6, 1) ( 3, 5) Y ( 3, 5) ( 2, 3) Z ( 2, 3) 8. Part A: 2 180 Part B: A quadrilateral has 4 sides. 2 180 4 2 180 2 180 360 x + 100 + 90 + 90 360 x + 280 360 x 80 The measure of the fourth angle is 80. Part C: ample answer: Divide the pentagon into 3 triangles. Because the sum of the interior angle measures of each triangle is 180, the sum of the interior angle measures of the pentagon is 3 180 540. Pine Trail 2 x Ash Trail The 125 and 1 are supplementary. o, the measure of 1 is 180 125 55. 1 and the x angle are corresponding angles. They are congruent. o, the value of x is 55. 89

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