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LESSON7.2?Angle Theoremsfor TrianglesESSENTIAL QUESTIONEXPLORE ACTIVITY 1Expressions,equations, andrelationships—8.8.D Use informalarguments to establishfacts about the anglesum and exterior angleof triangles, What can you conclude about the measures of the anglesof a triangle?8.8.DSum of the Angle Measures in a TriangleThere is a special relationship between the measures of the interiorangles of a triangle.A Draw a triangle and cut it out. Label the angles A, B, and C.B Tear off each “corner” of the triangle. Each corner includes thevertex of one angle of the triangle.C Arrange the vertices of the triangle around a point so that noneof your corners overlap and there are no gaps between them.ABCBD What do you notice about how the angles fit together around a point?ACE What is the measure of a straight angle? Houghton Mifflin Harcourt Publishing CompanyF Describe the relationship among the measures of the angles of ABC.The Triangle Sum Theorem states that for ABC, m A m B m C .Reflect1. Justify Reasoning Can a triangle have two right angles? Explain.2. Analyze Relationships Describe the relationship between the twoacute angles in a right triangle. Explain your reasoning.Lesson 7.2199
EXPLORE ACTIVITY 28.8.DJustifying the Triangle Sum TheoremYou can use your knowledge of parallel lines intersected by a transversalto informally justify the Triangle Sum Theorem.Follow the steps to informally prove the Triangle Sum Theorem. You shoulddraw each step on your own paper. The figures below are provided for youto check your work.A Draw a triangle and label the angles as 1, 2, and 3 as shown.b1B Draw line a through the base of the triangle.C The Parallel Postulate states that through a point not ona line ℓ, there is exactly one line parallel to line ℓ. Drawline b parallel to line a, through the vertex opposite thebase of the triangle.D Extend each of the non-base sides of the triangle toform transversal s and transversal t. Transversals s and tintersect parallel lines a and b.ab2st4 1 5E Label the angles formed by line b and the transversalsas 4 and 5.F Because 4 andare alternate interiorangles, they are3a23.Label 4 with the number of the angle to which it is congruent.G Because 5 andare alternate interior angles,.Label 5 with the number of the angle to which it is congruent.H The three angles that lie along line b at the vertex of the triangle are 1, 4, and 5. Notice that these three angles lie along a line.So, m 1 m 4 m 5 .Because angles 2 and 4 are congruent and angles 3 and 5 arecongruent, you can substitute m 2 for m 4 and m 3 for m 5 inthe equation above.So, m 1 m 2 m 3 .This shows that the sum of the angle measures in a triangle isalways200Unit 3. Houghton Mifflin Harcourt Publishing Companythey are
Reflect3. Analyze Relationships How can you use the fact that m 4 m 1 m 5 180 to show that m 2 m 1 m 3 180 ?Finding Missing Angle Measuresin TrianglesIf you know the measures of two angles in a triangle, you can use the TriangleSum Theorem to find the measure of the third angle.Math On the Spotmy.hrw.comEXAMPL 1EXAMPLE8.8.DMy NotesFind the missing angle measure.STEP 1Write the Triangle Sum Theorem forthis triangle.DE100 m D m E m F 180 STEP 2Substitute the given angle measures.F55 100 m E 55 180 STEP 3Solve the equation for m E. Houghton Mifflin Harcourt Publishing Company100 m E 55 180 155 m E 180 -155 -155 m E 25 Subtract 155 from both sides.Simplify.So, m E 25 .YOUR TURNFind the missing angle measure.4.5. SKJ71 m K 56 R29 61 Lm R TPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 7.2201
EXPLORE ACTIVITY 38.8.DExterior Angles and Remote Interior AnglesAn interior angle of a triangle is formed by two sides of the triangle. An exterior angleis formed by one side of the triangle and the extension of an adjacent side. Each exteriorangle has two remote interior angles. A remote interior angle is an interior angle that isnot adjacent to the exterior angle. 1, 2, and 3 are interior angles.2 4 is an exterior angle. 1 and 2 are remote interior angles to 4.34There is a special relationship between the measure of anexterior angle and the measures of its remote interior angles.2A Extend the base of the triangle and label the exterior angleas 4.B The Triangle Sum Theorem states:m 1 m 2 m 3 ,so m 3 m 4 .BandCto complete the following equation:m 1 m 2 E Use properties of equality to simplify the equation in m 4D:The Exterior Angle Theorem states that the measure of anis equal to the sum of the measure of itsReflect6. Sketch a triangle and draw all of its exterior angles.How many exterior angles does a triangle haveat each vertex?7. How many total exterior angles does a triangle have?202Unit 33.C 3 and 4 form aD Use the equations in1angleangles. Houghton Mifflin Harcourt Publishing Company1
Using the Exterior Angle TheoremYou can use the Exterior Angle Theorem to find the measures of the interiorangles of a triangle.EXAMPL 2EXAMPLE8.8.DFind m A and m B.STEP 1AWrite the Exterior AngleTheorem as it applies to thistriangle.Dmy.hrw.com(4y - 4) 3y 52 BCm A m B m ACDSTEP 2Math On the SpotSubstitute the given angle measures.(4y - 4) 3y 52 Math TalkMathematical ProcessesSTEP 3Solve the equation for y.Describe two waysto find m ACB.(4y - 4) 3y 52 4y - 4 3y 52 7y - 4 52 4 4 7y 56 7y 56 77y 8 Houghton Mifflin Harcourt Publishing CompanySTEP 4Remove parentheses.Simplify.Add 4 to both sides.Simplify.Divide both sides by 7.Simplify.Use the value of y to find m A and m B.m A 4y - 44(8) - 432 - 428m B 3y 3(8) 24So, m A 28 and m B 24 .YOUR TURN8. Find m M and m N.Mm M (5y 3) m N N(4y 8) 146 QPPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 7.2203
Guided PracticeFind each missing angle measure. (Explore Activity 1 and Example 1)1.2.MR126 L31 78 24 QNm M Sm Q Use the Triangle Sum Theorem to find the measure of each anglein degrees. (Explore Activity 2 and Example 1)3.4.T (7x 4) XYn 1n 2(5x 3) V(2x 5) UZm T , m U 1n 2,m X m V , m Y ,m Z Use the Exterior Angle Theorem to find the measure of each anglein degrees. (Explore Activity 3 and Example 2)6. JD(7y 6) C4y m C 116 E, m D L,m L , m M m LKM ESSENTIAL QUESTION CHECK-IN7. Describe the relationships among the measures of the angles of a triangle.204Unit 3(18z 3) (5z - 3) Fm DEC ?K161 M, Houghton Mifflin Harcourt Publishing Company5.
NameClassDate7.2 Independent PracticePersonalMath Trainer8.8.Dmy.hrw.comOnlineAssessment andInterventionFind the measure of each angle.8.9. TD2x 98 Ex x Fx Wm E m T m F m V G10.Q11.5x (3y 5) (2y - 7) S 153 RHV3x 4x JPm Q m G m P m H m QRP Houghton Mifflin Harcourt Publishing Companym J 12.13.B3x D58 85 A 78 LCK2x x M60 ENm K m ACB m L m BCD m KML m DCE m LMN 14. Multistep The second angle in a triangle is five times as large as the first.The third angle is two-thirds as large as the first. Find the angle measures.Lesson 7.2205
15. Analyze Relationships Can a triangle have two obtuse angles? Explain.FOCUS ON HIGHER ORDER THINKINGWork Area16. Critical Thinking Explain how you can use the Triangle Sum Theorem tofind the measures of the angles of an equilateral triangle.17. a.Draw Conclusions Find the sum of theAmeasures of the anglesinquadrilateralABCD.(Hint: Draw diagonal AC. How can you use thefigures you have formed to find the sum?)DBCSum 18. Communicate Mathematical Ideas Describe two ways that an exteriorangle of a triangle is related to one or more of the interior angles.206Unit 3 Houghton Mifflin Harcourt Publishing Companyb. Make a Conjecture Write a “Quadrilateral Sum Theorem.” Explainwhy you think it is true.
Math Trainer Online Assessment and Intervention Personal my.hrw.com Math On the Spot my.hrw.com Using the Exterior Angle Theorem You can use the Exterior Angle Theorem to find the measures of the interior angles of a triangle. Find m A and m B. Write the Exterior Angle
