Name An Angle Pair That Satisfies Each Condition.

inSubstituteFindSubstitute.in.1-5 Angle RelationshipsSo,32. If m JNP 2x 3, m KNL 3x – 17, and mKNJ 3x 34, find the measure of each angle.SOLUTION:In the figure,.FindSubstituteFindSubstitute33. PHYSICS As a ray of light meets a mirror, the lightis reflected. The angle at which the light strikes themirror is the angle of incidence. The angle at whichthe light is reflected is the angle of reflection. Theangle of incidence and the angle of reflection arecongruent. In the diagram below, if m RMI 106,find the angle of reflection and m RMJ.inSOLUTION:The angle of reflection and the angle of incidence arecongruent.So,.in.In the figure,FindSubstitute.in.Substitute.33. PHYSICS As a ray of light meets a mirror, the lightis reflected. The angle at which the light strikes themirror is the angle of incidence. The angle at whichthe light is reflected is the angle of reflection. Theangle of incidence and the angle of reflection arecongruent. In the diagram below, if m RMI 106,find the angle of reflection and m RMJ.The angle of reflection measures 53 .In the figure,34. ALGEBRA Rays AB and BC are perpendicular.Point D lies in the interior of ABC. If m ABD 3r 5 and m DBC 5r – 27, find m ABD andm DBC.SOLUTION:The angle of reflection and the angle of incidence arecongruent.So,.eSolutions Manual - Powered by CogneroIn the figure,SOLUTION:. Here,Page 9.

In the figure,1-5 Angle Relationships34. ALGEBRA Rays AB and BC are perpendicular.Point D lies in the interior of ABC. If m ABD 3r 5 and m DBC 5r – 27, find m ABD andm DBC.35. ALGEBRAandintersect at point V. Ifm WVY 4a 58 and m XVY 2b – 18, find thevalues of a and b so thatis perpendicular to.SOLUTION:SOLUTION:. ndicular tointersect at point V and,andisSo, a is 8 and b is 54.Determine whether each statement can beassumed from the figure. Explain.35. ALGEBRAandintersect at point V. Ifm WVY 4a 58 and m XVY 2b – 18, find thevalues of a and b so thatis perpendicular to.SOLUTION:Sinceandperpendicular tointersect at point V and,andeSolutions Manual - Powered by Cognerois36.4 and7 are vertical angles.SOLUTION:Page 10andare nonadjacent angles and formed bytwo intersecting lines.

SOLUTION:In the figure,1-5 Angle RelationshipsSo,.The answer is “Yes”.So, a is 8 and b is 54.Determine whether each statement can beassumed from the figure. Explain.36.4 and7 are vertical angles.SOLUTION:andare nonadjacent angles and formed bytwo intersecting lines.So,andare vertical angles.The answer is “Yes”.37.4 andand8 are supplementary.SOLUTION:Sinceandfrom a linear pair, they aresupplementary.The answer is “Yes”.38.SOLUTION:Since the intersection of the lines p and t is a rightangle, they are perpendicular.The answer is “Yes”.39.41.5 and7 form a linear pair.SOLUTION:A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays.anddo not form a linear pair, since they arenot adjacent angles.42. CCSS ARGUMENTS In the diagram of thepruning shears shown, m 1 m 3. Whatconclusion can you reach about the relationshipbetween 4 and 2? Explain.SOLUTION:Vertical angles are two nonadjacent angles formedby two intersecting lines. Here,andarevertical angles, andandare vertical angles.So,.andWe are given thatSo,.SubstituteNow, substituteSOLUTION:From the figure, 3 and 6 are adjacent. Since 5is a right angle, 3 and 6 will be complementary.This determines that both angles are acute.However, unless we know that the larger angle wasbisected to form 3 and 6, the measures ofandare unknown. So, we cannot say.The answer is “No”.in.in.So,FLIGHT The wings of the aircraft shown canpivot up to 60º in either direction from theperpendicular position.40.SOLUTION:In the figure,andSo,.The answer is “Yes”.41.5 and7 form a linear pair.eSolutions Manual - Powered by CogneroSOLUTION:A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays.43. Identify a pair of vertical angles.SOLUTION:Page 11Vertical angles are two nonadjacent angles formedby two intersecting lines.and.are vertical angles. So areand

Now, substitute1-5 Angle RelationshipsSo,in.FLIGHT The wings of the aircraft shown canpivot up to 60º in either direction from theperpendicular position.46. What is the minimum possible value for mmaximum?2? theSOLUTION:When the wing is in its normal position, it isperpendicular to the body of the plane.The wing can be pivoted up to 60 in either direction.If the wing to the left side of the plane is pivoted 60ºforward, then m 1 90 60 or 150 and m 2 180– 150 or 30.If the wing to the left side of the plane is pivoted 60ºbackwards, then m 1 90 – 60 or 30 and m 2 180 – 30 or 150.43. Identify a pair of vertical angles.SOLUTION:Vertical angles are two nonadjacent angles formedby two intersecting lines.and.are vertical angles. So areand44. Identify two pairs of supplementary angles.SOLUTION:Sample answer: 1 and 2 form a linear pair, sothe angles are supplementary. 3 and 4 form a linear pair, so the angles aresupplementary.So, two pairs of supplementary angles are 1 and 2; 3 and 4.45. If m1 110, what is m3? m4?SOLUTION:In the figure, 1 and 3 are vertical angles. 1 3, since vertical angles are congruent.By the definition of congruent angles, m 1 m 3.Given that m 1 110, then m 3 110.Therefore, the minimum possible value for m 2 is 30and the maximum possible value is 150.47. Is there a wing position in which none of the anglesare obtuse? Explain.SOLUTION:Sample answer: If the wing is not rotated at all, thenthe wing is perpendicular to the body of the plane.So, all of the angles are right angles, which areneither acute nor obtuse. So, the answer is “Yes”.48. MULTIPLE REPRESENTATIONS In thisproblem, you will explore the relationship betweenthe sum of the interior angles of a triangle and theangles vertical to them.a. GEOMETRIC Draw three sets of threeintersecting lines and label each as shown.b. TABULAR For each set of lines, measure andrecord m 1, m 2, and m 3 in a table. Recordm 1 m 2 m 3 in a separate column.c. VERBAL Explain how you can find m 4, m 5,and m 6 when you know m 1, m 2, and m 3.d. ALGEBRAIC Write an equation that relatesm 1 m 2 m 3 to m 4 m 5 m 6.Then use substitution to write an equation that relatesm 4 m 5 m 6 to an integer.Since 1 and 4 form a linear pair, 1 and 4 aresupplementary angles.46. What is the minimum possible value for mmaximum?eSolutions Manual - Powered by CogneroSOLUTION:When the wing is in its normal position, it is2? theSOLUTION:Sample answers:a. Draw three intersecting lines and label them.Page 12

SOLUTION:1-5 AngleRelationshipsSample answers:a. Draw three intersecting lines and label them.Angles that have a measure greater than or equal to90 can not have a complement, since the addition ofany other angle measure will produce a sum greaterthan 90. Therefore, right angles and obtuse angles donot have a complement.50. OPEN ENDED Draw a pair of intersecting linesthat forms a pair of complementary angles. Explainyour reasoning.SOLUTION:Complementary angles are two angles with measuresthat have a sum of 90.b. Assume the measures for ,, and, thenfind. Record those values in atable.Twoc. Vertical angles are two nonadjacent angles formedby two intersecting lines.,, and, sincethey are pairs of vertical angles.d. The sum of the measures of angles in a triangle is180. So,Refer part c.Therefore,49. CCSS REASONING Are there angles that do nothave a complement? Explain.SOLUTION:Complementary angles are two angles with measuresthat have a sum of 90. By definition, the measure ofan angle must be greater than 0. So, each angle musthave a measure less than 90. Thus, each angle in acomplementary pair is an acute angle.Angles that have a measure greater than or equal to90 can not have a complement, since the addition ofany other angle measure will produce a sum greaterthan 90. Therefore, right angles and obtuse angles donot have a complement.50. OPEN ENDED Draw a pair of intersecting linesthat forms a pair of complementary angles. Explainyour reasoning.SOLUTION:eSolutionsManual - Poweredby CogneroComplementaryanglesare twothat have a sum of 90.angles with measuresangles are complementary, since51. CHALLENGE If a line, line segment, or ray is aperpendicular to a plane, it is perpendicular to everyline, line segment, or ray in the plane that intersectsit.a. If a line is perpendicular to each of twointersecting lines at their point of intersection, thenthe line is perpendicular to the plane determined bythem. If line a is perpendicular to line and line m atpoint X, what must also be true?b. If a line is perpendicular to a plane, then any lineperpendicular to the given line at the point ofintersection with the given plane is in the given plane.If line a is perpendicular to plane P and line m atpoint X, what must also be true?c. If a line is perpendicular to a plane, then everyplane containing the line is perpendicular to the givenplane. If line a is perpendicular to plane P, what mustalso be true?SOLUTION:a. Since line and line m are contained in plane P,line a is perpendicular to plane P.b. Since line m is perpendicular to line a at point X,line m is in plane P.c. Since line a is perpendicular to plane P, anyplane containing line a is perpendicular to plane P.52. WRITING IN MATH Describe three differentways you can determine that an angle is a rightangle.Page 13SOLUTION:Sample answer: We can determine a right angle

line a is perpendicular to plane P.b. Since line m is perpendicular to line a at point X,line m is in plane P.line a is perpendicular to plane P, anyc. Since1-5 AngleRelationshipsplane containing line a is perpendicular to plane P.52. WRITING IN MATH Describe three differentways you can determine that an angle is a rightangle.SOLUTION:Sample answer: We can determine a right angleusing three different ways. You can determine if anangle is right if it is marked with a right angle symbol,if the angle is a vertical pair with a right angle, or ifthe angle forms a linear pair with a right angle.53. What is mRMS in the figure below?So, the correct option is B.54. EXTENDED RESPONSE For a fundraiser, atheater club is making 400 cookies. They want tomake twice as many chocolate chip as peanut buttercookies and three times as many peanut butter asoatmeal raisin cookies. Determine how many of eachtype of cookie the theater club will make. Show yourwork.SOLUTION:Let x be the number of oatmeal raisin cookies. Thenthe number of peanut butter cookies isand thenumber of chocolate chip cookies is.The total number of cookies isSolve for x.A 26B 38C 52D 128SOLUTION:andDivide each side by 10.So, they need 40 oatmeal raisin, 3(40) or 120 peanutbutter and 6(40) or 240 chocolate chip cookies.are supplementary, since55. ALGEBRA Which inequality is graphed below?Given thatIn the figure,SubstituteandFGSo, the correct option is B.54. EXTENDED RESPONSE For a fundraiser, atheater club is making 400 cookies. They want tomake twice as many chocolate chip as peanut buttercookies and three times as many peanut butter asoatmeal raisin cookies. Determine how many of eachtype of cookie the theater club will make. Show yourwork.SOLUTION:Let x be the number of oatmeal raisin cookies. Thenthe number of peanut butter cookies isand theeSolutions Manual - Powered by Cogneronumber of chocolate chip cookies is.HJSOLUTION:The graph has a solid line, so the inequality shouldhave either the or symbol, which rules outoptions F and G. The inequality symbol in theequation should be , since the graph has beenshaded below the -axis. So, the correct option is J.56. SAT/ACT One third of a number is three more thanPage 14one fourth the same number. What is the number?A 3B 12

The graph has a solid line, so the inequality shouldhave either the or symbol, which rules outoptions F and G. The inequality symbol in theequationshould be , since the graph has been1-5 AngleRelationshipsshaded below the -axis. So, the correct option is J.56. SAT/ACT One third of a number is three more thanone fourth the same number. What is the number?A 3B 12C 36D 42E 48figure appears to be a right angle, soPoint C on anglelies on the exterior angle ofright angle, sois an obtuse angle.Using a protractor, you will find that58.DBCSOLUTION:SOLUTION:Let x be the number.If the corner of a sheet of paper in set on B and onedge of the paper is aligned with, then point Cis in the interior of the right angle formed by thepaper. So, the measure of DBC must be less than90. Therefore,is an acute angle. Use a.protractor to find thatThe correct option is C.Copy the diagram shown and extend each ray.Classify each angle as right, acute, or obtuse.Then use a protractor to measure the angle tothe nearest degree.59.ABDSOLUTION:If the corner of a sheet of paper is placed at B andone edge of the paper is aligned with57.ABCSOLUTION:, the otheredge of the paper appears to be aligned with.This would mean thatis probably a rightangle Using a protactor,Find the coordinates of the midpoint of asegment with the given endpoints.60. P(3, –7), Q(9, 6)SOLUTION:Using the corner of a sheet of paper,in thefigure appears to be a right angle, soPoint C on anglelies on the exterior angle ofright angle, sois an obtuse angle.Using a protractor, you will find that58.DBCIfhas endpoints atandinisthe coordinate plane, then the midpoint M of.Substitute,,, andinSOLUTION:.eSolutions Manual - Powered by CogneroPage 15

one edge of the paper is aligned with, the otheredge of the paper appears to be aligned with.This wouldmean thatis probably a right1-5 AngleRelationshipsangle Using a protactor,SOLUTION:IfSOLUTION:has endpoints atandin.,has endpoints at, andinis.Substitute,andthe coordinate plane, then the midpoint M ofisthe coordinate plane, then the midpoint M ofSubstitute.61. A(–8, –5), B(1, 7)Find the coordinates of the midpoint of asegment with the given endpoints.60. P(3, –7), Q(9, 6)IfisThe midpoint of,,, andin.in.isThe midpoint of.62. J(–7, 4), K(3, 1)isThe midpoint ofSOLUTION:.Ifhas endpoints atSOLUTION:is.has endpoints atandinisthe coordinate plane, then the midpoint M ofSubstitute,,,, andin.Substituteinthe coordinate plane, then the midpoint M of61. A(–8, –5), B(1, 7)Ifand,, andin.The midpoint ofeSolutions Manual - Powered by CogneroThe midpoint ofis.is.63. SNOWBOARDING In the design on thePage 16snowboard shown,bisectsat R. If SN 163 centimeters, find RN.

1-5 AngleRelationshipsThe midpointofiscongruent to each other. Similarly, the angles markedwith the same symbol are congruent to each other.Congruent segments:Congruent angles:.63. SNOWBOARDING In the design on thesnowboard shown,bisectsat R. If SN 163 centimeters, find RN.65.In the figure,SOLUTION:Segments that have the same measure are calledcongruent segments. Angles that have the sameangle measures are called congruent angles.Here the segments marked with the same symbol arecongruent to each other. Similarly, the angles markedwith the same symbol are congruent to each other.Congruent segments:Given thatCongruent angles:SOLUTION:Sincebisectsat R,.Substitute.Divide each side by 2.So,66.SOLUTION:Segments that have the same measure are calledcongruent segments. Angles that have the sameangle measures are called congruent angles.Here the segments marked with the same symbol arecongruent to each other. Similarly, the angles markedwith the same symbol are congruent to each other.Congruent segments:Congruent angles:centimeters.Name the congruent sides and angles in eachfigure.64.SOLUTION:Segments that have the same measure are calledcongruent segments. Angles that have the sameangle measures are called congruent angles.Here the segments marked with the same symbol arecongruent to each other. Similarly, the angles markedwith the same symbol are congruent to each other.Congruent segments:Congruent angles:67.SOLUTION:Segments that have the same measure are calledcongruent segments. Angles that have the sameangle measures are called congruent angles.Here the segments marked with the same symbol arecongruent to each other. Similarly, the angles markedwith the same symbol are congruent to each other.Congruent segments:Congruent angles:65.SOLUTION:Segments that have the same measure are calledcongruent segments. Angles that have the sameangle measures are called congruent angles.eSolutions Manual - Powered by CogneroPage 17

two acute vertical angles 62/87,21 Vertical angles are two nonadjacent angles formed by two intersecting lines. You can use the corner of a piece of paper to see that Ø ZVY and Ø WVU are less than right angles. 7KHUHIRUH DQG DUHDFXWHYHUWLFDO angles. two obtuse adjacent angles 62/87,21 Adjacent angles are two angles that lie in the same