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Sample answer: Vertical angles are two nonadjacent angles formed by two intersecting lines. There are many acute vertical angles in the figure. and are acute vertical angles. 1-5 Angle Relationships Name an angle pair that satisfies each condition. 13. two supplementary adjacent angles SOLUTION: Sample answer: If the sum of the measures of two adjacent angles is 180, then they are supplementary adjacent angles. There are many supplementary adjacent angles in the figure. and share a common side and vertex, also . So, and are supplementary adjacent angles. 1. two acute vertical angles SOLUTION: Vertical angles are two nonadjacent angles formed by two intersecting lines. 15. an angle complementary to FDG SOLUTION: Complementary angles are two angles with measures that have a sum of 90. Since is complementary to 17. an angle supplementary to JAE SOLUTION: Supplementary angles are two angles with measures that have a sum of 180. Since , is supplementary to . You can use the corner of a piece of paper to see that ∠ZVY and ∠WVU are less than right angles. Therefore, and are acute vertical angles. Find the value of each variable. Name an angle or angle pair that satisfies each condition. 19. SOLUTION: In the figure, the angle and the angle are vertical angles. Vertical angles are congruent. 9. two acute vertical angles SOLUTION: Sample answer: Vertical angles are two nonadjacent angles formed by two intersecting lines. There are many acute vertical angles in the figure. and are acute vertical angles. 13. two supplementary adjacent angles SOLUTION: Sample answer: If the sum of the measures of two eSolutions Manual - Powered by Cognero adjacent angles is 180, then they are supplementary adjacent angles. There are many supplementary adjacent angles in the figure. 21. Page 1 SOLUTION: Since (2x + 25)° and (3x – 10)° are vertical angles, they are congruent. Solve for y. 1-5 Angle Relationships 23. 21. SOLUTION: Since (2x + 25)° and (3x – 10)° are vertical angles, they are congruent. SOLUTION: Supplementary angles have measures that sum to 180. So, and . Consider . Solve for y. 25. ALGEBRA E and F are supplementary. The measure of E is 54 more than the measure of F. Find the measures of each angle. 23. SOLUTION: Supplementary angles have measures that sum to 180. So, and . Consider SOLUTION: Supplementary angles are two angles with measures that have a sum of 180. Then, . It is given that . Substitute. . Substitute in . Manual - Powered by Cognero eSolutions 25. ALGEBRA E and F are supplementary. The 27. ALGEBRA The measure of the supplement of an angle is 40 more than two times the measure of the complement of the angle. Find the measure of the angle. Page 2 SOLUTION: Let x be the measure of an angle. Substitute in 1-5 Angle Relationships . 27. ALGEBRA The measure of the supplement of an angle is 40 more than two times the measure of the complement of the angle. Find the measure of the angle. SOLUTION: Let x be the measure of an angle. The measure of an angle which is complementary to angle is The measure of an angle which is supplementary to angle is 31. If m LNM = 8x + 12 and m m JNP. JNL = 12x – 32, find SOLUTION: The angles in a linear pair are supplementary. So, . and are vertical angles. Since the vertical angles are congruent, The measure of an angle is 40. Substitute in ALGEBRA Use the figure below. So, 29. If m KNL = 6x – 4 and m LNM = 4x + 24, find the value of x so that KNM is a right angle. SOLUTION: In the figure, Since is a right angle, 33. PHYSICS As a ray of light meets a mirror, the light is reflected. The angle at which the light strikes the mirror is the angle of incidence. The angle at which the light is reflected is the angle of reflection. The angle of incidence and the angle of reflection are congruent. In the diagram below, if m RMI = 106, find the angle of reflection and m RMJ. . 31. If m LNM = 8x + 12 and m m JNP. JNL = 12x – 32, find SOLUTION: The angles in a linear pair are supplementary. So, . eSolutions Manual - Powered by Cognero SOLUTION: The angle of reflection and the angle of incidence are congruent. So, . In the figure, . Substitute. Page 3 In the figure, Substitute in 1-5 Angle Relationships So, 33. PHYSICS As a ray of light meets a mirror, the light is reflected. The angle at which the light strikes the mirror is the angle of incidence. The angle at which the light is reflected is the angle of reflection. The angle of incidence and the angle of reflection are congruent. In the diagram below, if m RMI = 106, find the angle of reflection and m RMJ. 35. ALGEBRA and intersect at point V. If m WVY = 4a + 58 and m XVY = 2b – 18, find the values of a and b so that is perpendicular to . SOLUTION: Since and perpendicular to intersect at point V and , and is SOLUTION: The angle of reflection and the angle of incidence are congruent. So, . In the figure, . Substitute. The angle of reflection measures 53°. In the figure, So, a is 8 and b is 54. Determine whether each statement can be assumed from the figure. Explain. 35. ALGEBRA and intersect at point V. If m WVY = 4a + 58 and m XVY = 2b – 18, find the values of a and b so that is perpendicular to . SOLUTION: Since and perpendicular to intersect at point V and , and eSolutions Manual - Powered by Cognero is 37. 4 and 8 are supplementary. SOLUTION: Since and supplementary. Page 4 form a linear pair, they are 1-5 Angle Relationships So, a is 8 and b is 54. Determine whether each statement can be assumed from the figure. Explain. 37. 4 and 8 are supplementary. SOLUTION: Since and form a linear pair, they are supplementary. The answer is “Yes”. 39. SOLUTION: From the figure, ∠3 and ∠6 are adjacent. Since ∠5 is a right angle, ∠3 and ∠6 will be complementary. This determines that both angles are acute. However, unless we know that the larger angle was bisected to form ∠3 and ∠6, the measures of and are unknown. So, we cannot say . The answer is “No”. 41. 5 and 7 form a linear pair. SOLUTION: A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays. and do not form a linear pair, since they are not adjacent angles. 49. JUSTIFY ARGUMENTS Are there angles that do not have a complement? Explain. SOLUTION: Complementary angles are two angles with measures that have a sum of 90. By definition, the measure of an angle must be greater than 0. So, each angle must have a measure less than 90. Thus, each angle in a complementary pair is an acute angle. Angles that have a measure greater than or equal to 90 can not have a complement, since the addition of any other angle measure will produce a sum greater than 90. Therefore, right angles and obtuse angles do not have a complement. eSolutions Manual - Powered by Cognero Page 5