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Chapter 5 b. Chapter 5 Opener Try It Yourself (p. 183) 2. 30° 60° 90° 120° 150° y 150° 120° 90° 60° 30° 5 7 = x 14 70 = 7 x When x = 60°, y = 180 − 60 = 120°. 10 = x When x = 90°, y = 180 − 90 = 90°. So, x is 10 inches. When x = 120°, y = 180 − 120 = 60°. When x = 30°, y = 180 − 30 = 150°. When x = 150°, y = 180 − 150 = 30°. 2 25 = x 1 2 x = 25 Angle measure (degrees) 1. x x = 12.5 So, x is 12.5 millimeters. Section 5.1 5.1 Activity (pp. 184 –185) 1. a. y 160 120 80 40 0 x 15° 30° 45° 60° 75° y 75° 60° 45° 30° 15° When x = 15°, y = 90 − 15 = 75°. When x = 30°, y = 90 − 30 = 60°. When x = 45°, y = 90 − 45 = 45°. When x = 60°, y = 90 − 60 = 30°. Angle measure (degrees) When x = 75°, y = 90 − 75 = 15°. 0 40 80 120 160 x Angle measure (degrees) Because the graph is a straight line, the function is linear. Because x + y = 180°, the equation of the function is y = 180 − x. The input values of the function are between 0 and 180, so the domain is 0 < x < 180. 2. a. If x and y are complementary angles, then both x and y are always acute. b. If x and y are supplementary angles, then x is y sometimes acute. 80 c. If x is a right angle, then x is never acute. 60 3. a. ∠ BDE and ∠ DBE , ∠ BDE and ∠ BDC , ∠ DBE 40 and ∠ CBD, and ∠ CBD and ∠ BDC are all 20 complementary angles. 0 0 20 40 60 80 x Angle measure (degrees) Because the graph is a straight line, the function is linear. Because x + y = 90°, the equation of the function is y = 90 − x. The input values of the function are between 0 and 90, so the domain is 0 < x < 90. b. ∠ A and ∠ ABE , ∠ A and ∠ BEF , ∠ A and ∠ F , ∠ A and ∠ CBE , ∠ A and ∠ C , ∠ A and ∠ CDE , ∠ A and ∠ BED, ∠ ABE and ∠ BEF , ∠ ABE and ∠ F , ∠ ABE and ∠ CBE , ∠ ABE and ∠ C , ∠ ABE and ∠ CDE , ∠ ABE and ∠ BED, ∠ BEF and ∠ F , ∠ BEF and ∠ CBE , ∠ BEF and ∠ C , ∠ BEF and ∠ CDE , ∠ BEF and ∠ BED, ∠ F and ∠ CBE , ∠ F and ∠ C , ∠ F and ∠ CDE , ∠ F and ∠ BED, ∠ CBE and ∠ C , ∠ CBE and ∠ CDE , ∠ CBE and ∠ BED, ∠ C and ∠ CDE , ∠ C and ∠ BED, ∠ CDE and ∠ BED are all supplementary angles. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 137 Chapter 5 4. Two angles are complementary if the sum of their measures is 90°. Two angles are supplementary if the sum of their measures is 180°. Sample answer: 32° and 58° are complementary angles and 25° and 155° are supplementary angles. 5. Sample answer should include, but is not limited to: a real life example of complementary and supplementary angles with a drawing of each object and approximate angle measures. 5.1 On Your Own (pp. 186 –187) 1. Because 26° + 64° = 90°, the angles are complementary. 2. Because 44° + 136° = 180°, the angles are supplementary. 3. Because 70° + 19° = 89°, the angles are neither complementary nor supplementary. 4. The angles are supplementary. So, the sum of their measures is 180°. x + 85 = 180 x = 95 So, x is 95. 5. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. So, x is 90. 6. The angles are complementary. So, the sum of their measures is 90°. x + 69 = 90 x = 21 So, x is 21. 5.1 Exercises (pp. 188–189) Vocabulary and Concept Check 1. Two angles are complementary if the sum of their measures is 90°. Two angles are supplementary if the sum of their measures is 180°. 2. When two lines intersect, four angles are formed. Because there are two pairs of opposite angles formed, two pairs of vertical angles are formed. Practice and Problem Solving 3. Because the sum of the angle measures of x and y is 180°, x can be acute, obtuse or 90°. So, the statement is sometimes true. 4. Because the sum of the angle measures of x and y is 90° + 90° = 180°, the angles are supplementary. So, the statement is always true. 138 Big Ideas Math Blue Worked-Out Solutions 5. Because the sum of the angle measures of x and y is 90°, the measure of angle y is less than 90°. So, the statement is never true. 6. Because 122° + 68° = 190°, the angles are neither complementary nor supplementary. 7. Because 42° + 48° = 90°, the angles are complementary. 8. Because 59° + 31° = 90°, the angles are complementary. 9. Because 115° + 65° = 180°, the angles are supplementary. 10. Because 24° + 156° = 180°, the angles are supplementary. 11. Because 45° + 55° = 100°, the angles are neither complementary nor supplementary. 12. The angles are complementary. So, the sum of their measures is 90°. x + 35 = 90 x = 55 So, x is 55. 13. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. So, x is 128. 14. The angles are supplementary. So, the sum of their measures is 180°. x + 117 = 180 x = 63 So, x is 63. 15. Vertical angles are congruent, not complementary. So, the angles have the same measure, and the value of x is 35. 16. The angles are supplementary. So, the sum of their measures is 180°. x + 127 = 180 x = 53 So, x is 53. 17. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. 2 x + 1 = 75 2 x = 74 x = 37 So, x is 37. Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 18. The angles are complementary. So, the sum of their measures is 90°. 4 x + 2 x = 90 6 x = 90 x = 15 So, x is 15. 19. The angles are supplementary. So, the sum of their measures is 180°. 7 x + x + 20 = 180 8 x + 20 = 180 8 x = 160 x = 20 So, x is 20. 20. Sample answer: 100° is the measure of an angle that can be supplementary but not complementary. Because complementary angles are all less than 90°, any angle that is greater than 90° and less than 180° cannot be complementary. So, an obtuse angle can be a supplementary angle but cannot be a complementary angle. 21. a. Because ∠ CBE is a right angle, ∠ CBD and ∠ DBE are complementary angles. Because ∠ ABE is a right angle, ∠ ABF and ∠ FBE are complementary angles. b. Because AC is a straight line, the following pairs of angles are supplementary angles: ∠ ABF and ∠ FBC , ∠ ABE and ∠ EBC , and ∠ ABD and ∠ DBC. 22. Because the 50° angle and ∠1 are supplementary, the sum of their measures is 180°. x + 50 = 180 x = 130 So, the measure of ∠1 is 130°. Because the 50° angle and ∠ 2 are vertical angles, the angles have the same measure. So, the measure of ∠ 2 is 50°. Because ∠1 and ∠ 3 are vertical angles, the angles have the same measure. So, the measure of ∠ 3 is 130°. 23. Let 3x represent the measure of the larger angle and 2x represent the measure of the smaller angle. Because the angles are complementary, the sum of their measures is 90°. 3x + 2 x = 90 5 x = 90 x = 18 So, the measure of the larger angle is 3x = 3(18) = 54°. 24. Because vertical angles are congruent, the angles have the same measure. So, let x represent the measure of one complementary angle. x + x = 90 2 x = 90 x = 45 So, the measures of two vertical angles that are complementary angles are 45°. Let y represent the measure of one supplementary angle. y + y = 180 2 y = 180 y = 90 So, the measures of two vertical angles that are supplementary angles are 90°. 25. Because the angles with measures 7x° and y° are supplementary to the right angle, the sum of all three measures is 180°. 7 x + 90 + y = 180 7 x + y = 90 Because the angle with the sum of the angle measures 5x° and 2 y° is a vertical angle to the right angle, the sum of their measures is 90°. 5 x + 2 y = 90 A system of equations is 7 x + y = 90 and 5 x + 2 y = 90. 7 x + y = 90 y = 90 − 7 x 5 x + 2 y = 90 2 y = 90 − 5 x y = 45 − 5 x 2 5 x 2 9 90 = 45 + x 2 9 45 = x 2 90 = x 9 10 = x 90 − 7 x = 45 − y = 90 − 7 x = 90 − 7(10) = 90 − 70 = 20 So, x = 10 and y = 20. Fair Game Review 26. x + 60 + 45 = 180 x + 105 = 180 x = 75 The value of x is 75. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 139 Chapter 5 27. x + 58.5 + 92.2 = 180 x + 150.7 = 180 x = 29.3 The value of x is 29.3. 28. x + x + 110 = 180 2 x + 110 = 180 2 x = 70 x = 35 The value of x is 35. 1 2 29. B; m = − ; (6, 4) 5 4 (2, 6) −2 1 3 (6, 4) −2 2 student will design an abstract art painting that includes triangles and count the different types of triangles used. 4. You can classify them as acute, obtuse, right, or equiangular depending on the measures of their angles. 5. Answer should include, but is not limited to: The student will find and name real-life triangles in architecture. 5.2 On Your Own (pp. 192–193) 1. x + 78 + 27 = 180 The value of x is 75. The triangle has all acute angles. So, it is an acute triangle. 2. x + 44 + 45 = 180 1 O b. Answer should include, but is not limited to: The x = 75 (4, 5) −2 1 right; yellow triangle: obtuse; blue triangle: obtuse x + 105 = 180 y (0, 7) 1 3. a. Sample answer: red triangle: acute; pink triangle: 1 2 3 4 5 6 7x Because the line crosses the y-axis at (0, 7), the 1 y-intercept is 7. So, the equation is y = − x + 7. 2 Section 5.2 5.2 Activity (pp. 190 –191) 1. a–f. Answer should include, but is not limited to: x + 89 = 180 x = 91 The value of x is 91. The triangle has an obtuse angle. So, it is an obtuse triangle. 3. x + x + 120 = 180 2 x + 120 = 180 2 x = 60 x = 30 Students will follow directions for steps (a)–(d) for four different triangles. Their conclusion should be that the sum of the angle measures of any triangle is 180°. 4. x + x + x = 180 2. a. A right triangle has one right angle. Because the green 3 x = 180 triangle has one right angle, it is a right triangle. b. An acute triangle has all acute angles. Because the angle measures in the blue, red, and yellow triangles are all acute, they are all acute triangles. c. An obtuse triangle has one obtuse angle. Because the purple triangle has an angle measure of 100°, which is obtuse, it is an obtuse triangle. d. An equiangular triangle has three congruent angles. Because the yellow triangle has three congruent angles, it is an equiangular triangle. e. An equilateral triangle has three congruent sides. Because the yellow triangle has three congruent sides, it is an equilateral triangle. f. An isosceles triangle has at least two sides that are congruent. Because the blue triangle has two sides that are congruent, it is an isosceles triangle. Because the yellow triangle has two sides that are congruent, it is also an isosceles triangle. 140 Big Ideas Math Blue Worked-Out Solutions The value of x is 30. Two of the sides are congruent and one angle is obtuse. So, it is an obtuse isosceles triangle. x = 60 The value of x is 60. All three angles are congruent and acute and all three sides are congruent. So, it is an acute, isosceles, equilateral, and equiangular triangle. 5. x + 63.9 + 61.8 = 180 x + 125.7 = 180 x = 54.3 The value of x is 54.3. 5.2 Exercises (pp. 194 –195) Vocabulary and Concept Check 1. An equilateral triangle has three congruent sides and an isosceles triangle has at least two congruent sides. So, an equilateral triangle is a specific type of isosceles triangle. Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 2. To find the missing angle of the triangle, write an equation such that the sum of the angles of the triangle is 180°. Next find the sum of the two known angles. Then subtract the sum of the angles from 180. x + 45 + 102 = 180 x + 147 = 180 x = 33 Because the value of x is 33, the measure of the missing angle is 33°. Practice and Problem Solving 3. Two of the sides are congruent and one angle is a right angle. So, it is a right isosceles triangle. 4. All three angles are acute. So, it is an acute triangle. 11. x + x + 132 = 180 2 x + 132 = 180 2 x = 48 x = 24 The value of x is 24. Two of the sides are congruent and one angle is obtuse. So, it is an obtuse isosceles triangle. 12. The solution incorrectly concludes that a triangle is acute if it has some acute angles. Because 98° > 90°, this angle is obtuse. The triangle is an obtuse triangle because it has one obtuse angle. 13. a. x + x + 40 = 180 2 x + 40 = 180 2 x = 140 x = 70 5. Two of the sides are congruent and one angle is an obtuse angle. So, it is an obtuse isosceles triangle. 6. x + 53 + 37 = 180 x + 90 = 180 x = 90 The value of x is 90. One of the angles is a right angle. So, it is a right triangle. 7. x + 73 + 13 = 180 x + 86 = 180 x = 94 The value of x is 94. One angle is obtuse. So, it is an obtuse triangle. 8. x + 48 + 84 = 180 x + 132 = 180 x = 48 The value of x is 48. Two of the sides are congruent and all of the angles are acute. So, it is an acute isosceles triangle. 9. x + x + 45 = 180 2 x + 45 = 180 2 x = 135 x = 67.5 The value of x is 67.5. Two of the sides are congruent and all of the angles are acute. So, it is an acute isosceles triangle. 10. x + x + 60 = 180 2 x + 60 = 180 2 x = 120 The value of x is 70. b. The triangle has two congruent sides, so it is an isosceles triangle. The triangle has all acute angles, so it is an acute triangle. 14. yes; Because 76.2 + 81.7 + 22.1 = 180, a triangle can have the given measures. 15. no; Because 115.1 + 47.5 + 93 = 255.6 and not 180, a triangle cannot have the given measures. x + 47.5 + 93 = 180 x + 140.5 = 180 x = 39.5 So, the angle measures 39.5°, 47.5°, and 93° can form a triangle. 2 1 + 64 + 87 = 157 and not 180, 3 3 a triangle cannot have the given measures. 16. no; Because 5 1 x + 64 + 87 = 180 3 1 x + 151 = 180 3 2 x = 28 3 So, the angle measures 28 2° 1° , 64 , and 87° can form 3 3 a triangle. 3 1 3 + 53 + 94 = 180, a triangle can 4 4 4 have the given measures. 17. yes; Because 31 x = 60 The value of x is 60. All three sides and angles are congruent and all of the angles are acute. So, it is an acute isosceles, equilateral, and equiangular triangle. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 141 Chapter 5 22. x + x + x + 3 + 9 + 5 + 2 = 28 18. a. Green triangle: x + 65 + 50 = 180 3x + 19 = 28 x + 115 = 180 3x = 9 x = 65 x = 3 The value of x is 65. The value of x is 3. Purple triangle: x + 25 + 130 = 180 x + 155 = 180 23. 1 ⎛ circumference ⎞ ⎜ ⎟ + three sides of rectangle = P 2 ⎝ of circle ⎠ 1 (2π r ) + x + (2 x − 4) + x = P 2 x = 25 The value of x is 25. Red triangle: x + 45 + 90 = 180 1 ( 2 • 3.14 • 3) + x + 2 x − 4 + x = 25.42 2 4 x + 5.42 = 25.42 x + 135 = 180 4 x = 20 x = 45 The value of x is 45. b. The angles opposite the congruent sides of the triangle are congruent. c. An isosceles triangle has at least two congruent angles. 19. If a triangle had two obtuse angles, then the sum of the angle measures would be greater than 180°. So, a triangle can have at most one obtuse angle, and at least two acute angles. 20. a. x + x + 36 = 180 x = 5 The value of x is 5. 24. A; Because you are starting out with $10, the y-intercept is a positive value. Each time you send a text message, your money decreases. So, the slope is negative. The equation is y = −0.25 x + 10. Section 5.3 5.3 Activity (pp. 196 –197) 1. b. Pentagon: n = 5 Sample answer: 2 x + 36 = 180 2 x = 144 x = 72 The value of x is 72. b. Sample answer: Because the length of each card is the same, the triangle formed must remain an isosceles triangle. So, the cards can be stacked such that the base of the triangle is shorter when the value of x is greater than 72, and longer when the value of x is less than 72. If x = 60, then the three cards form an equilateral triangle. This is not possible because the two upright cards would have to be exactly on the edges of the base card. So, x > 60. If x = 90, then the two upright cards would be vertical, which is not possible. The card structure would not be stable. So, x < 90. Fair Game Review 21. 2 x + x + 2 x + 8 + 5 = 48 5 x + 13 = 48 5 x = 35 Draw two lines that divide the pentagon into three triangles. Because the sum of the angle measures of each triangle is 180°, the sum of the angle measures of the pentagon is 3(180°) = 540°. c. Hexagon: n = 6 Sample answer: Draw three lines that divide the hexagon into four triangles. Because the sum of the angle measures of each triangle is 180°, the sum of the angle measures of the hexagon is 4(180°) = 720°. x = 7 The value of x is 7. 142 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 d. Heptagon: n = 7 e. S = 180n − 360 = 180(10) − 360 Sample answer: = 1800 − 360 = 1440 The sum of the angle measures of a polygon with 10 sides is 1440°. 3. yes; Because the equation from Activity 2 was derived Draw four lines that divide the heptagon into five triangles. Because the sum of the angle measures of each triangle is 180°, the sum of the angle measures of the heptagon is 5(180°) = 900°. e. Octagon: n = 8 using triangles, the equation can be used on a concave polygon because it can also be divided into triangles. The triangles must be formed such that each line segment connecting two vertices lies inside the polygon. At least one of the angle measures in a concave polygon will be greater than 180°, so you need to expand your definition of angles to include this type of angle. 4. Sample answer: Create examples of polygons and then Sample answer: draw lines to divide the polygon into triangles to find the sum of the angle measures. Then plot the number of sides and the sum of the angle measures and look for a pattern. Finally, write an equation to represent the data. 5.3 On Your Own (pp. 198 –200) 1. The spider web is in the shape of a heptagon. It has Draw five lines that divide the octagon into six triangles. Because the sum of the angle measures of each triangle is 180°, the sum of the angle measures of the octagon is 6(180°) = 1080°. 2. a. 7 sides. S = ( n − 2) • 180° = (7 − 2) • 180° = 5 • 180° Sides, n 3 4 5 6 7 8 Angle Sum, S 180 360 540 720 900 1080 = 900° The sum of the angle measures is 900°. 2. The honeycomb is in the shape of a hexagon. It has 6 sides. b. 1080 900 720 540 360 180 −180 −360 S (8, 1080) (7, 900) (6, 720) (5, 540) (4, 360) (3, 180) 1 2 3 4 5 6 7 8n c. A polygon with n sides can be divided into n − 2 triangles. Because the sum of the angle measures of each triangle is 180°, the sum of the measures of a polygon is ( n − 2) • 180°. So, the equation is S = ( n − 2) • 180 = 180n − 360. d. Because a polygon can have no fewer than 3 sides, the domain of the function is all integers greater than or equal to 3. S = ( n − 2) • 180° = (6 − 2) • 180° = 4 • 180° = 720° The sum of the angle measures is 720°. 3. The polygon has 6 sides. S = ( n − 2) • 180° = (6 − 2) • 180° = 4 • 180° = 720° The sum of the angle measures is 720°. 125 + 120 + 125 + 110 + 135 + x = 720 615 + x = 720 x = 105 The value of x is 105. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 143 Chapter 5 4. The polygon has 4 sides. 8. An 18-gon has 18 sides. S = ( n • 2) • 180° S = ( n − 2) • 180° = ( 4 − 2) • 180° = (18 − 2) • 180° = 2 • 180° = 16 • 180° = 360° = 2880° The sum of the angle measures is 360°. The sum of the angle measures is 2880°. 115 + 80 + 90 + x = 360 2880° ÷ 18 = 160° 285 + x = 360 The measure of each angle is 160°. x = 75 9. The value of x is 75. 5. The polygon has 5 sides. S = ( n − 2) • 180° = (5 − 2) • 180° = 3 • 180° = 540° No line segment connecting two vertices lies outside the polygon. So, the polygon is convex. The sum of the angle measures is 540°. 2 x + 145 + 145 + 2 x + 110 = 540 4 x + 400 = 540 10. 4 x = 140 x = 35 The value of x is 35. 6. An octagon has 8 sides. S = ( n − 2) • 180° A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. = (8 − 2) • 180° = 6 • 180° 11. = 1080° The sum of the angle measures is 1080°. 1080° ÷ 8 = 135° The measure of each angle is 135°. 7. A decagon has 10 sides. S = ( n − 2) • 180° = (10 − 2) • 180° = 8 • 180° = 1440° The sum of the angle measures is 1440°. 1440° ÷ 10 = 144° The measure of each angle is 144°. 144 Big Ideas Math Blue Worked-Out Solutions A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. 5.3 Exercises (pp. 201–203) Vocabulary and Concept Check 1. A three-sided polygon is a triangle. Because it is a regular polygon, all sides are congruent. So, the polygon is an equilateral triangle. Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 2. Because the second figure is not made up entirely of line 6. segments and the other three are, the second figure does not belong. 3. Because the first question asks for one angle measure and the other three ask for the sum of the angle measures, the first question is different. A regular pentagon has n = 5 sides. S = ( n − 2) • 180° = (5 − 2) • 180° = 3 • 180° = 540° 540° ÷ 5 = 108° The measure of one angle of a regular pentagon is 108°. S = ( n − 2) • 180° = (5 − 2) • 180° = 3 • 180° = 540° The sum of the measures of a regular, convex, or concave pentagon is 540°. Practice and Problem Solving 4. Draw two lines that divide the heptagon into 5 triangles. Because the sum of the measures of each triangle is 180°, the sum of the angle measures of a heptagon is 5(180°) = 900°. 7. The shape has 6 sides. S = ( n − 2) • 180° = (6 − 2) • 180° = 4 • 180° = 720° The sum of the angle measures is 720°. 8. The shape has 12 sides. S = ( n − 2) • 180° = (12 − 2) • 180° = 10 • 180° = 1800° The sum of the angle measures is 1800°. 9. The shape has 8 sides. S = ( n − 2) • 180° = (8 − 2) • 180° = 6 • 180° Draw one line that divides the quadrilateral into two triangles. Because the sum of the measures of each triangle is 180°, the sum of the angle measures of a quadrilateral is 2(180°) = 360°. = 1080° The sum of the angle measures is 1080°. 10. The formula is incorrect. The correct formula has the product of two less than the number of sides and 180°. S = ( n − 2) • 180° 5. = (13 − 2) • 180° = 11 • 180° = 1980° Draw six lines that divide the 9-gon into 7 triangles. Because the sum of the measures of each triangle is 180°, the sum of the angle measures of a 9-gon is 7(180°) = 1260°. 11. no; S = ( n − 2) • 180° = (5 − 2) • 180° = 3 • 180° = 540° Because the sum of the given angle measures is 120° + 105° + 65° + 150° + 95° = 535° and not 540°, a pentagon cannot have the given angle measures. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 145 Chapter 5 12. The polygon has 4 sides. 17. The polygon has 9 sides. S = ( n − 2) • 180° S = ( n − 2) • 180° = ( 4 − 2) • 180° = (9 − 2) • 180° = 2 • 180° = 7 • 180° = 360° = 1260° 155 + 25 + 137 + x = 360 1260° ÷ 9 = 140° 317 + x = 360 The measure of each angle is 140°. x = 43 18. The polygon has 12 sides. The value of x is 43. S = ( n − 2) • 180° 13. The polygon has 6 sides. = (12 − 2) • 180° S = ( n − 2) • 180° = 10 • 180° = (6 − 2) • 180° = 1800° = 4 • 180° 1800° ÷ 12 = 150° = 720° The measure of each angle is 150°. 90 + 90 + x + x + x + x = 720 180 + 4 x = 720 4 x = 540 x = 135 The value of x is 135. 19. The sum should have been divided by the number of angles, which is 20, not 18. 3240° ÷ 20 = 162° The measure of each angle is 162°. 20. a. The bolt has 5 sides. Find the sum of the angle 14. The polygon has 6 sides. measures. S = ( n − 2) • 180° S = ( n − 2) • 180° = (6 − 2) • 180° = (5 − 2) • 180° = 4 • 180° = 3 • 180° = 720° 3x + 45 + 135 + x + 135 + 45 = 720 4 x + 360 = 720 4 x = 360 x = 90 The value of x is 90. 15. Find the number of sides. S = ( n − 2) • 180 1260 = ( n − 2) • 180° 7 = n−2 9 = n Because the regular polygon has 9 sides, the measure of each angle is 1260° ÷ 9 = 140°. 16. A triangle has 3 sides. S = ( n − 2) • 180° = (3 − 2) • 180° = 180° = 540° Divide the sum by the number of angles, 5. 540° ÷ 5 = 108° The measure of each angle is 108°. b. Sample answer: Since the standard shape of a bolt is a hexagon, most people have tools to remove a hexagonal bolt and not a pentagonal bolt. So, a special tool is needed to remove the bolt from the fire hydrant. 21. The sum of the angles of the polygon is n • 165, where n is the number of sides. S = ( n − 2) • 180 n • 165 = ( n − 2) • 180 165n = 180n − 360 −15n = −360 n = 24 The polygon has 24 sides. 180° ÷ 3 = 60° The measure of each angle is 60°. 146 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 27. The shape is a regular octagon. 22. S = ( n − 2) • 180° = (8 − 2) • 180° = 6 • 180° = 1080° A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. 1080° ÷ 8 = 135° The measure of each angle is 135°. 28. Sample answer: 23. 45° 45° 270° 29. Find the sum of the angle measures of the heptagon. No line segment connecting two vertices lies outside the polygon. So, the polygon is convex. S = ( n − 2) • 180° = (7 − 2) • 180° = 5 • 180° 24. = 900° Let x represent the value of each of the three remaining angle measures, in degrees. 4 • 135 + 3 • x = 900 540 + 3x = 900 A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. 25. no; In a regular polygon, all the angles are congruent. Because a concave polygon has at least one angle that is greater than 180°, all of the angles are not congruent. So, a concave polygon cannot be regular. 26. Sample answer: 3x = 360 x = 120 The measure of each remaining angle is 120°. 30. a. The polygon has 11 sides. b. S = ( n − 2) • 180° = (11 − 2) • 180° = 9 • 180° = 1620° 1620° ÷ 11 ≈ 147° The measure of each angle is about 147°. Because not all of the angles are congruent, the polygon is not regular. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 147 Chapter 5 31. a = (n − 2) • 180 = 180n − 360 = 180 − 360 S = n n n n Sides of a Regular Polygon, n 180 − 360 n Measure of One Angle (degrees), a 3 180 − 360 3 60 4 180 − 360 4 90 5 180 − 360 5 108 6 180 − 360 6 120 c. Sample answer: The tessellation is formed using equilateral triangles and squares. Fair Game Review 33. 7 180 − 360 7 128.6 8 180 − 360 8 135 9 180 − 360 9 140 10 180 − 360 10 144 As the number of sides n increases by one, the measure of one angle a increases by 30°, 18°, 12°, 9°, 6°, 5°, and 4°, respectively. The rate of change is not constant. So, the table does not represent a linear function. 32. a. Sample answer: 3 x = 12 4 x • 4 = 12 • 3 34. 4 x = 36 42 = 21x x = 9 2 = x The value of x is 9. 35. x 14 = 21 3 14 • 3 = 21 • x x 2 = 9 6 x•6 = 9•2 The value of x is 2. 36. 4 x = 10 15 4 • 15 = 10 • x 6 x = 18 60 = 10 x x = 3 6 = x The value of x is 3. The value of x is 6. 37. 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, … . So, 16 is the only choice that could be the total number of tulips and daisies. Study Help (p. 204) Available at BigIdeasMath.com. b. Sample answer: Squares: Regular hexagons: Quiz 5.1–5.3 (p. 205) 1. Because 125° + 65° = 190°, the angles are neither complementary nor supplementary. 2. Because 63° + 27° = 90°, the angles are complementary. 3. Because 106° + 74° = 180°, the angles are supplementary. 4. The angles are supplementary. So, the sum of their measures is 180°. x + 34 = 180 x = 146 So, x is 146. 148 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 5. The angles are complementary. So, the sum of their 12. The polygon has 4 sides. measures is 90°. S = ( n − 2) • 180° x + 74 = 90 = ( 4 − 2) • 180° x = 16 = 2 • 180° So, x is 16. = 360° x + 122 + 134 + 46 = 360 6. The angles are vertical angles. Because vertical angles are congruent, the angles have the same measure. So, x is 59. x + 302 = 360 x = 58 7. x + 60 + 60 = 180 x + 120 = 180 x = 60 The value of x is 58. 13. The polygon has 7 sides. S = ( n − 2) • 180° The value of x is 60. All three angles are congruent and acute, and all three sides are congruent. So, it is an acute, isosceles, equilateral, and equiangular triangle. = (7 − 2) • 180° = 5 • 180° = 900° 8. x + 25 + 40 = 180 x + 130 + 140 + 120 + 115 + 154 + 115 = 900 x + 65 = 180 x + 774 = 900 x = 115 x = 126 The value of x is 115. The triangle has an obtuse angle. So, it is an obtuse triangle. 9. x + x + 90 = 180 The value of x is 126. 14. The polygon has 5 sides. S = ( n − 2) • 180° 2 x + 90 = 180 2 x = 90 = (5 − 2) • 180° x = 45 = 3 • 180° The value of x is 45. The triangle has a right angle and two angles are congruent. So, it is a right isosceles triangle. = 540° x + 40 + 4 x + 40 + 110 = 540 5 x + 190 = 540 10. The polygon has 8 sides. 5 x = 350 S = ( n − 2) • 180° x = 70 = (8 − 2) • 180° = 6 • 180° = 1080° The value of x is 70. 15. Because the 115° angle and ∠1 are supplementary, the sum of their measures is 180°. The sum of the angle measures is 1080°. x + 115 = 180 11. x = 65 So, the measure of ∠1 is 65°. Because the 115° angle and ∠ 2 are vertical angles, the angles have the same measure. So, the measure of ∠ 2 is 115°. Because ∠1 and ∠ 3 are vertical angles, the A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. measure of ∠ 3 is 65°. 16. S = ( n − 2) • 180° 4140 = ( n − 2) • 180° 23 = n − 2 25 = n The polygon has 25 sides. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 149 Chapter 5 17. x + x + 67.4 = 180 2. yes; 66 + 90 + x = 180 y + 90 + 24 = 180 2 x + 67.4 = 180 156 + x = 180 y + 114 = 180 2 x = 112.6 The triangles have the same angle measures, 66°, 24°, and 90°. So, they are similar. Two sides are congruent and all three angles are acute. So, it is an acute isosceles triangle. 3. Section 5.4 5.4 Activity (pp. 206 –207) 1. Answer should include, but is not limited to: Students will describe how to extend two sides of XYZ and then draw the third side of the new triangle parallel to the third side of XYZ . Students will measure the side lengths and calculate ratios of corresponding side lengths. Ratios may not be exactly equal due to rounding during measuring. Students should conclude that if two triangles have the same angle measures, then the triangles are similar. 40 x = 55 50 40 x 55 • = 55 • 55 50 x = 44 The distance across the river is 44 feet. 5.4 Exercises (pp. 210 –211) Vocabulary and Concept Check 1. Because the ratio of the corresponding side lengths in similar triangles are equal, a proportion can be used to find a missing measurement. ABC , DEF and GHI are 35°, 82°, and 63°, and the angle measures of JKL are 32°, 85°, and 63°. So, JKL does not belong with 2. a. true; By definition of similar. 2. The angle measures of true; By definition of similar. b. true; By definition of similar. false; A square and a rhombus with the same side lengths are not similar because their corresponding angles are not congruent. c. true; Shown in Activity 1. the other three. Practice and Problem Solving 3– 4. Answer should include, but is not limited to: true; The similar quadrilaterals will have the same shape. d. true; Shown in Activity 1 false; A square and a rectangle have congruent corresponding angles but the ratios of their corresponding side lengths are not equal. e. true; By definition of similar. false; A square and a rhombus with the same side lengths do not have identical shapes. 3. Sample answer: If corresponding side lengths of two triangles are proportional, then the triangles are similar. If corresponding angles of two triangles are congruent, then the triangles are similar. Sample answer: Construction and architecture use triangles to form buildings. 5.4 On Your Own (p. 209) 1. no; x + 28 + 80 = 180 y + 28 + 71 = 180 x + 108 = 180 y + 99 = 180 x = 72 y = 66 x = 24 x = 56.3 y = 81 Students should draw a triangle with the same angle measures as those in the textbook. The ratios of corresponding side lengths, student’s triangle length , book’s triangle length should be greater than 1, and should all be approximately equal. (Ratios may differ slightly due to rounding.) 5. yes; x + 34 + 39 = 180 y + 107 + 39 = 180 x + 73 = 180 y + 146 = 180 x = 107 y = 34 The triangles have the same angle measures, 34°, 39°, and 107°. So, they are similar. 6. no; x + 36 + 72 = 180 y + 72 + 75 = 180 x + 108 = 180 y + 147 = 180 x = 72 y = 33 The triangles do not have the same angle measures. So, they are not similar. The triangles do not have the same angle measures. So, they are not similar. 150 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 7. no; x + 64 + 85 = 180 y + 26 + 85 = 180 x + 149 = 180 y + 111 = 180 14. Sample answer: 10 ft y = 69 x = 31 The triangles do not have the same angle measures. So, they are not similar. 8. yes; x + 48 + 81 = 180 y + 48 + 51 = 180 x + 129 = 180 y + 99 = 180 y = 81 x = 51 The triangles have the same angle measures 48°, 51°, and 81°. So, the triangles are similar. 9. The proportion was set up incorrectly. If the first ratio is a length from the smaller triangle over a length from the larger triangle, the second ratio must be the same. 16 8 = x 18 16 x = 144 x 5 3 6 15. a. Because AG, GF, and FE are equal and AE = 9.48 feet, the length of segment AG is 9.48 ÷ 3 = 3.16 feet. AG BG = AE DE x 3.16 = 9.48 6 18.96 = 9.48 x 2 = x So, x is 2 feet. x = 9 b. The length of segment AF is 3.16 • 2 = 6.32 feet. 10. Because the corresponding angle measure is 50°, the value of x is 50. 11. Because the corresponding angle measure is 65°, the value of x is 65. 12. Find the missing dimension using indirect measurement. x 80 = 300 240 240 x = 24,000 Let y = CF . CF DE = AF AE y 6 = 6.32 9.48 9.48 y = 37.92 y = 4 So, CF is 4 feet. Fair Game Review x = 100 You take 100 steps from the pyramids to the treasure. 13. no; Consider the two similar triangles below. 16. Because the equation cannot be rewritten in slope-intercept form, it is nonlinear. 17. Because the equation is of the form y = mx + b, it is linear. 18. Because the equation is of the form y = mx + b, it is linear. 1.5y y 19. Because the equation cannot be rewritten in slope-intercept form, it is nonlinear. x 1.5x 1 A = bh 2 1 A = xy 2 1 A = bh 2 1 A = (1.5 x)(1.5 y ) 2 1 A = ( 2.25) xy 2 So, the area of the larger triangle is 2.25 times larger than the original triangle, which is a 125% increase. Copyright © Big Ideas Learning, LLC All rights reserved. 20. C; Find the slope of each line. Blue: m = rise 5 = run 2 Red: m = rise 4 = = 2 run 2 Green: m = rise 5 = run 2 Because the slopes of the blue line and the green line 5 are , the slopes are equal. 2 Big Ideas Math Blue Worked-Out Solutions 151 Chapter 5 Section 5.5 2. ∠1 and ∠ 2 are supplementary. ∠1 + ∠ 2 = 180° 5.5 Activity (pp. 212–213) 1. Two lines are parallel if they do not intersect. Sample answer: Draw one line. Then, draw two points that are the same distance from the line. Use these two points to draw a parallel line. Angles 1, 3, 5, and 7 are congruent. Angles 1 and 3 and angles 5 and 7 are vertical angles, which are congruent. Angles 3 and 7 are congruent because the corresponding angles the parallel lines form with the transversal are the same. Angles 2, 4, 6, and 8 are congruent using the same reasoning. 2. a. Sample answer: Measure the vertical angles and corresponding angles and make sure they are congruent. b. The studs are parallel lines and the diagonal support beam is a transversal. 3. a. Because the Sun’s rays are parallel, ∠ C ≅ ∠ F . Because ∠ A and ∠ D are both right angles, ABC are DEF , ∠ B ≅ ∠ E. ∠ A ≅ ∠ D. Because two angles of congruent to two angles of Therefore, b. Because ABC and DEF are similar triangles. ABC and DEF are similar triangles, the ratios of the corresponding side lengths are equal. So, write and solve a proportion to find the height of the flagpole. Height of flagpole Length of flagpole’s shadow = Height of boy Length of boy’s shadow x 36 = 5 3 3 x = 180 x = 60 The height of the flagpole is 60 feet. 4. Sample answer: A banister on a staircase has spindles that are parallel to each other. The base of the banister is a transversal to the spindles. Each angle created by a spindle and the base has the same measurement. 5. a. Because the flagpole is not being measured directly, the process is called “indirect” measurement. 63° + ∠ 2 = 180° ∠ 2 = 117° So, the measure of ∠ 2 is 117°. 3. The 59° angle is supplementary to both ∠1 and ∠ 3. ∠1 + 59° = 180° ∠1 = 121° So, the measures of ∠1 and ∠ 3 are 121°. ∠ 2 and the 59° angle are vertical angles. They are congruent. So, the measure of ∠ 2 is 59°. Find the remaining angle measures: Using corresponding angles, the measures of ∠ 4 and ∠ 6 are 121°, and the measures of ∠ 5 and ∠ 7 are 59°. 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 59° angle is congruent to the angle that is supplementary to ∠1, as shown. So, the measure of ∠1 is 180° − 65° = 115°. 5. ∠ 3 and ∠ 4 are supplementary angles. ∠ 3 + ∠ 4 = 180° ∠ 3 + 84° = 180° ∠ 3 = 96° So, 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 and ∠ 5 and ∠ 6 are supplementary. So, ∠ 4 and ∠ 6 are supplementary. ∠ 4 + ∠ 6 = 180° 84° + ∠ 6 = 180° ∠ 6 = 96° So, the measure of ∠ 6 is 96°. b–c. 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. 5.5 On Your Own (pp. 214 –216) 1. ∠1 and the 63° angle are corresponding angles. They are congruent. So, the measure of ∠1 is 63°. 152 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 11. Because ∠1 and ∠ 2 are corresponding angles, the 5.5 Exercises (pp. 217–219) measure of ∠ 2 is 60°. Vocabulary and Concept Check 1. Sample answer: p q t 2. Because ∠ 2 and ∠ 6 are corresponding angles and ∠ 6 and ∠ 8 are vertical angles, ∠ 2 ≅ ∠ 6 ≅ ∠ 8. Because ∠ 5 is supplementary to ∠ 2, ∠ 6, and ∠ 8, the statement “The measure of ∠ 5 ” does not belong with the other three. Practice and Problem Solving 12. Sample answer: The yard lines on a football field are parallel. The lampposts on a road are parallel. 13. ∠1, ∠ 3, ∠ 5, and ∠ 7 are congruent. ∠ 2, ∠ 4, ∠ 6 and ∠ 8 are congruent. 14. You need to know at least one angle measure. If you know the measure of ∠1, then ∠ 3, ∠ 5, and ∠ 7 can be found because they are congruent to ∠1. ∠ 2, ∠ 4, ∠ 6, and ∠ 8 can also be found because they are supplementary to ∠1. 15. ∠1 and the 61° angle are corresponding angles. They are 3. Lines m and n are parallel. congruent. So, the measure of ∠1 is 61°. 4. Line t is the transversal. ∠1 is supplementary to both ∠ 2 and ∠ 4. 5. 8 angles are formed by the transversal. 6. ∠1, ∠ 3, ∠ 5, and ∠ 7 are all congruent. ∠ 2, ∠ 4, ∠ 6, and ∠ 8 are all congruent. 7. ∠1 and the 107° angle are corresponding angles. They ∠1 + ∠ 2 = 180° 61° + ∠ 2 = 180° ∠ 2 = 119° So, the measures of ∠ 2 and ∠ 4 are 119°. ∠1 and ∠ 3 are vertical angles. They are congruent. are congruent. So, the measure of ∠1 is 107°. So, the measure of ∠ 3 is 61°. ∠1 and ∠ 2 are supplementary. Using corresponding angles, the measures of ∠ 5 ∠1 + ∠ 2 = 180° 107° + ∠ 2 = 180° ∠ 2 = 73° So, the measure of ∠ 2 is 73°. 8. ∠ 3 and the 95° angle are corresponding angles. They are congruent. So, the measure of ∠ 3 is 95°. and ∠ 7 are 119°, and the measure of ∠ 6 is 61°. 16. The 99° angle is supplementary to both ∠1 and ∠ 3. ∠1 + 99° = 180° ∠1 = 81° So, the measures of ∠1 and ∠ 3 are 81°. ∠ 2 and the 99° angle are vertical angles. They are ∠ 3 and ∠ 4 are supplementary. congruent. So, the measure of ∠ 2 is 99°. ∠ 3 + ∠ 4 = 180° Using corresponding angles, the measures of ∠ 4 and 95° + ∠ 4 = 180° ∠ 4 = 85° So, the measure of ∠ 4 is 85°. 9. ∠ 5 and the 49° angle are corresponding angles. They are congruent. So, the measure of ∠ 5 is 49°. ∠ 5 and ∠ 6 are supplementary. ∠ 5 + ∠ 6 = 180° 49° + ∠ 6 = 180° ∠ 6 = 131° So, the measure of ∠ 6 is 131°. ∠ 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° So, the measures of ∠1 and ∠ 3 are 90°. ∠ 2 and the right angle are vertical angles. They are congruent. So, the measure of ∠ 2 is 90°. Using corresponding angles, the measures of ∠ 4, ∠ 5, ∠ 6, and ∠ 7 are 90°. 10. The lines are not parallel, so corresponding angles ∠ 5 and ∠ 6 are not congruent. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 153 Chapter 5 18. Using corresponding angles, ∠1 is congruent to ∠ 8, which is supplementary to ∠ 4. ∠1 + ∠ 4 = 180° 124° + ∠ 4 = 180° ∠ 4 = 56° So, if the measure of ∠1 = 124°, then the measure of ∠ 4 = 56°. 26. Using vertical angles, ∠1 is congruent to ∠ 3, and ∠ 3 and ∠ 7 are congruent because they are alternate exterior angles. So, ∠1 is congruent to ∠ 7. Using corresponding angles, ∠1 is congruent to ∠ 5, and ∠ 5 and ∠ 7 are congruent because they are vertical angles. So, ∠1 is congruent to ∠ 7. 27. The 50° angle is congruent to the alternate interior angle ∠ 2 + ∠ 3 = 180° 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. So, the 50° angle is supplementary to the x° angle. 48° + ∠ 3 = 180° 50 + x = 180 ∠ 3 = 132° x = 130 19. Using corresponding angles, ∠ 2 is congruent to ∠ 7, which is supplementary to ∠ 3. So, 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. So, 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. So, 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. So, the value of x is 130. 28. The 115° angle is congruent to the corresponding angle formed by the intersection of line b and line d. This angle is congruent to the x° angle because they are alternate exterior angles. Because the 115° angle is congruent to the x° angle, the value of x is 115. 29. a. no; The lines look like they will intersect somewhere to the left of the illustration. b. Answer should include, but is not limited to: The student will draw an optical illusion using parallel lines. 30. a. m + 64° + m = 180° ∠ 7 + ∠ 6 = 180° 2m + 64° = 180° 50.5° + ∠ 6 = 180° 2m = 116° m = 58° ∠ 6 = 129.5° So, if the measure of ∠ 7 = 50.5°, then the measure of So, the value of m is 58. ∠ 6 = 129.5°. 23. Using alternate interior angles, ∠ 3 is congruent to ∠1, which is supplementary to ∠ 2. m° n° ∠ 3 + ∠ 2 = 180° 118.7° + ∠ 2 = 180° ∠ 2 = 61.3° So, 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°. Because the sides of the table are parallel and ∠ m and ∠ n are alternate interior angles, ∠ m is congruent to ∠ n. The measure of ∠ n is 58°. 58° + x° + n° = 180° 58° + x° + 58° = 180° x° + 116° = 180° x° = 64° So, the value of x is 64. 25. Because the lines are perpendicular, the lines intersect at right angles. So, all of the angles formed are right angles. 154 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 b. The goal is slightly wider than the hockey puck. So, there is some leeway allowed for the measure of x. By studying the diagram, you can see that x cannot be much greater. However, x can be a little less and still have the hockey puck go into the goal. Fair Game Review 31. 4 + 32 = 4 + 9 = 13 32. 5( 2) − 6 = 5 • 4 − 6 = 20 − 6 = 14 2 33. 11 + ( −7) − 9 = 11 + 49 − 9 = 60 − 9 = 51 2 6. Because the 82° angle and ∠ 6 are vertical angles, the angles are congruent. So, the measure of ∠ 6 is 82°. 7. Because the 82° angle and ∠ 4 are corresponding angles, the angles are congruent. So, the measure of ∠ 4 is 82°. 8. Using corresponding angles, the 82° angle is congruent to ∠ 4, which is supplementary to ∠1. Because the 82° angle and ∠1 are supplementary, the measure of ∠1 is 180° − 82° = 98°. 9. Using alternate exterior angles, ∠1 is congruent to ∠ 7. So, if the measure of ∠1 = 132°, then the measure of 34. 8 ÷ 2 2 + 1 = 8 ÷ 4 + 1 = 2 + 1 = 3 35. B; V = π r 2h 20π = π r 2 • 5 4 = r2 ∠ 7 = 123°. 10. Using corresponding angles, ∠ 2 is congruent to ∠ 6, which is supplementary to ∠ 5. Because ∠ 2 and ∠ 5 are supplementary, the measure of ∠ 5 is 180° − 58° = 122°. So, if the measure of ∠ 2 = 58°, 2 = r So, the radius of the base is 2 inches. Quiz 5.4–5.5 (p. 220) 1. x + 46 + 95 = 180 x + 141 = 180 x = 39 y + 39 + 46 = 180 y + 85 = 180 y = 95 The triangles have the same angle measures, 39°, 46°, and 95°. So, they are similar. then the measure of ∠ 5 = 122°. 11. Because ∠ 5 and ∠ 3 are alternate interior angles, ∠ 5 is congruent to ∠ 3. So, if the measure of ∠ 5 = 119°, then the measure of ∠ 3 = 119°. 12. Because ∠ 4 and ∠ 6 are alternate exterior angles, ∠ 4 is congruent to ∠ 6. So, if the measure of ∠ 4 = 60°, then the measure of ∠ 6 = 60°. 13. Using corresponding angles, the 72° angle is congruent to the angle that is supplementary to ∠1 and ∠ 2. So, the 72° angle is supplementary to both ∠1 and ∠ 2. 2. x + 40 + 51 = 180 ∠1 + 72° = 180° x + 91 = 180 ∠1 = 108° x = 89 y + 40 + 79 = 180 y + 119 = 180 y = 61 The triangles do not have the same angle measures. So, they are not similar. 3. Because the corresponding angle measure is 95°, the value of x is 95. 4. Because the corresponding angle measure is 26°, the value of x is 26. 5. Because the 82° angle and ∠ 2 are alternate exterior angles, the angles are congruent. So, the measure of ∠ 2 is 82°. So, the measures of ∠1 and ∠ 2 are 108. ABC have side lengths a, b, and c. The perimeter of ABC is a + b + c. Let A′B′C ′ have side lengths 2a, 2b, and 2c. The perimeter of A′B′C ′ is 14. yes; Let 2a + 2b + 2c = 2( a + b + c), which is 2 times the perimeter of ABC. Chapter 5 Review (pp. 221–223) 1. The angles are complementary angles. So, the sum of their measures is 90°. x + 69 = 90 x = 21 So, x is 21. 2. Because the angles are vertical angles, they are congruent. So, x is 84. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 155 Chapter 5 3. x + 49 + 90 = 180 8. x + 139 = 180 x = 41 The value of x is 41. The triangle has one right angle. So, it is a right triangle. No line segment connecting two vertices lies outside the polygon. So, the polygon is convex. 4. x + 35 + 110 = 180 x + 145 = 180 9. x = 35 The value of x is 35. The triangle has two congruent sides, so it is an isosceles triangle. The triangle has one obtuse angle. So, it is an obtuse isosceles triangle. A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. 5. The polygon has 4 sides. S = ( n − 2) • 180° 10. = ( 4 − 2) • 180° = 2 • 180° = 360° 60 + 128 + 95 + x = 360 283 + x = 360 A line segment connecting two vertices lies outside the polygon. So, the polygon is concave. x = 77 The value of x is 77. 6. The polygon has 7 sides. 11. yes; x + 68 + 90 = 180 y + 22 + 90 = 180 x + 158 = 180 y + 112 = 180 S = ( n − 2) • 180° x = 22 = (7 − 2) • 180° y = 68 The triangles have the same angle measures, 22°, 68°, and 90°. So, they are similar. = 5 • 180° 12. Because the corresponding angle measure is 50°, the = 900 135 + 125 + 135 + 105 + 150 + 140 + x = 900 790 + x = 900 x = 110 value of x is 50. 13. The 140° angle and ∠ 8 are alternate exterior angles. They are congruent. So, the measure of ∠ 8 is 140°. The value of x is 110. 14. The 140° angle and ∠ 5 are corresponding angles. 7. The polygon has 6 sides. They are congruent. So, the measure of ∠ 5 is 140°. S = ( n − 2) • 180° 15. The 140° angle and ∠ 3 are supplementary. So, the = (6 − 2) • 180° measure of ∠ 3 is 180° − 140° = 40°. ∠ 3 and ∠ 7 are = 4 • 180° = 720° 100 + 120 + 60 + 2 x + 65 + x = 720 3 x + 345 = 720 3 x = 375 x = 125 The value of x is 125. corresponding angles. They are congruent. So, the measure of ∠ 7 is 40°. 16. The 140° angle and ∠ 2 are supplementary. So, the measure of ∠ 2 is 180° − 140° = 40°. Chapter 5 Test (p. 224) 1. Because the angles are vertical angles, they are congruent. So, the value of x is 113. 156 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved. Chapter 5 2. The angles are complementary angles. So, the sum of their measures is 90°. 9. no; x + 61 + 70 = 180 y + 39 + 70 = 180 x + 131 = 180 y + 109 = 180 x + 56 = 90 y = 71 x = 49 x = 34 The triangles do not have the same angle measures. So, they are not similar. So, the value of x is 34. 3. The angles are supplementary angles. So, the sum of their measures is 180°. 10. Because the corresponding angle measure is 55°, the value of x is 55. x + 74 = 180 11. ∠1 and the 47° angle are supplementary. So, the measure x = 106 of ∠1 is 180° − 47° = 133°. So, the value of x is 106. 12. ∠1 and the 47° angle are supplementary. So, the measure 4. x + 23 + 129 = 180 of ∠1 is 180° − 47° = 133°. ∠1 and ∠ 8 are alternate x + 152 = 180 x = 28 The value of x is 28. The triangle has one obtuse angle. So, it is an obtuse triangle. exterior angles. They are congruent. So, the measure of ∠ 8 is 133°. 13. ∠ 4 and the 47° angle are supplementary. So, the measure of ∠ 4 is 180° − 47° = 133°. 5. x + 44 + 68 = 180 x + 112 = 180 14. ∠1 and the 47° angle are supplementary. So, the measure x = 68 of ∠1 is 180° − 47° = 133°. ∠1 and ∠ 5 are The value of x is 68. The triangle has two congruent angles and three acute angles. So, it is an acute isosceles triangle. corresponding angles. They are congruent. So, the measure of ∠ 5 is 133°. 15. The triangles are similar, so the ratios of the 6. x + x + x = 180 corresponding side lengths are equal. 3 x = 180 x = 60 The value of x is 60. The triangle has three congruent and acute angles and three congruent sides. So, it is an acute, isosceles, equilateral, and equiangular triangle. 7. d 80 = 105 140 140d = 8400 d = 60 The distance across the pond is 60 meters. Chapter 5 Standardized Test Practice (pp. 225 –227) 1. 147; Find the sum of the angle measures. S = ( n − 2) • 180° = (11 − 2) • 180° = 1620° No line segment connecting two vertices lies outside the polygon. So, the polygon is convex. 8. The polygon has 5 sides. Divide the sum by the number of angles, 11. 1620° ÷ 11 ≈ 147° The measure of each angle is about 147°. 2. B; S = ( n − 2) • 180° C = 11 + 1.6t C − 11 = 1.6t = (5 − 2) • 180° C − 11 = t 1.6 = 3 • 180° = 540° 2 x + 2 x + 125 + 125 + 90 = 540 4 x + 340 = 540 The formula in terms of t is t = C − 11 . 1.6 4 x = 200 x = 50 The value of x is 50. Copyright © Big Ideas Learning, LLC All rights reserved. Big Ideas Math Blue Worked-Out Solutions 157 Chapter 5 3. F; Find the slope using the points (0, −5) and (5, 0). slope = rise 5 = =1 run 5 The line crosses the y-axis at (0, −5), so the y-intercept is −5. So, the equation of the line is y = x − 5. 4. 152; The angles are supplementary. So, the sum of the angle measures is 180°. 9. G; Because the line crosses the y-axis at (0, 5), the y-intercept is 5. So, an equation of the line 2 is y = x + 5. 5 10. D; Because the temperature rises 98° − 95° = 3°F, the Heat Index is 3 • 4 + 122 = 12 + 122 = 134°F. 11. G; ∠1 + ∠ 2 = 180° 5 x − 3 = 11 28° + ∠ 2 = 180° 5 x = 14 ∠ 2 = 152° 5. D; Because the equation x + y = 1 is the only equation of the form Ax + By = C , the equation is linear. 6. H; Because there are 2000 computers, let A + d = 2000 represent one equation. Because each laptop weighs 8 pounds, 8A, and each desktop computer weighs 20 pounds, 20d, and the whole shipment is 34,000 pounds, let 8A + 2d = 34,000 represent the second equation. 7. A; The function is represented by the points ( −5, 2), x = 2.8 12. B; The ratios of the corresponding side lengths are equal. x 5 = 6 10 10 x = 30 x = 3 The length is 3 cm. 13. H; The lines intersect at the point ( 4, 2), so the solution of the system is ( 4, 2). (0, 0), and (5, −2). The domain is the set of all x-values, and the x-values are −5, 0, and 5. So, the domain is −5, 0, 5. 8. Part A: S = ( n − 2) • 180° Part B: S = ( n − 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: Sample answer: Divide the pentagon into 3 triangles. Because the sum of the angles of a triangle is 180°, the sum of the angles in a pentagon is 3 • 180° = 540°. 158 Big Ideas Math Blue Worked-Out Solutions Copyright © Big Ideas Learning, LLC All rights reserved.