6.2 AAS Triangle Congruence

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1 Name lass ate 6. S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? xplore G.6. Prove two triangles are congruent b appling the ngle-ngle-side congruence conditions. lso G.., G.5. xploring ngle-ngle-side ongruence If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, are the triangles congruent? Resource Locker In this activit ou ll be coping a side and two angles from a triangle. Use a compass and straightedge to cop segment. Label it as segment. op using _ as a side of the angle. Houghton Mifflin Harcourt Publishing ompan On a separate transparent sheet or a sheet of tracing paper, cop. Label its vertex G. Make the ras defining G longer than their corresponding sides on. G Now overla the ra from with the ra from G to form a triangle. Make sure that side _ maintains the length ou defined for it. Module Lesson

2 How man triangles can ou construct? op all of G to the transparenc. Then overla it on. re the triangles congruent? How do ou know? Reflect 1. Suppose ou had started this activit b coping segment and then angles and. Would our results have been the same? Wh or wh not?. ompare our results to those of our classmates. oes this procedure work with an triangle? xplain 1 Justifing ngle-ngle-side ongruence The following theorem summarizes the previous activit. ngle-ngle-side (S) ongruence Theorem If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent. Prove the S ongruence Theorem. Given:,, _ _ Prove: Statements Reasons 1.,, _ _ 1. Given. m + m + m = m = m - m 3. Subtraction Propert of qualit. m + m + m = 180. Triangle Sum Theorem 5. m = m - m 5. Subtraction Propert of qualit 6. m = m, m = m m = m - m 7. Houghton Mifflin Harcourt Publishing ompan 8. m m 8. Transitive Propert of qualit 9. m Triangle ongruence Theorem Module 6 30 Lesson

3 Reflect 3. iscussion The Third ngles Theorem sas If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. How could using this theorem simplif the proof of the S ongruence Theorem?. ould the S ongruence Theorem be used in the proof? xplain. xplain Using ngle-ngle-side ongruence xample Use the S Theorem to prove the given triangles are congruent. Given: _ _ and m n m Prove: n Given m ǁ n Given lt. Int Thm. S ong. Thm. Houghton Mifflin Harcourt Publishing ompan lt. Int Thm. Given: _ _, _ _, and _ _. Prove: Module 6 31 Lesson

4 Your Turn 5. Given:, _ _, _ _. Use the S Theorem to prove the triangles are congruent. Write a paragraph proof. xplain 3 ppling ngle-ngle-side ongruence xample 3 The triangular regions represent plots of land. Use the S Theorem to explain wh the same amount of fencing will surround either plot. Given: It is given that. lso, because both are right angles. ompare and using the istance ormula. = ( x - x 1 ) + ( - 1 ) = (-1-(-)) + ( - 0) = 3 + = 5 = 5 = ( x - x 1 ) + ( - 1 ) = ( - 0) + (1 - ) = + (-3) = x Houghton Mifflin Harcourt Publishing ompan = 5 ecause two angles and a non-included side are congruent, b S. Therefore the triangles have the same perimeter and the same amount of fencing is needed. Module 6 3 Lesson

5 Given: P Z, Q X It is given that P Z and Q X. X ompare YZ and using the distance formula. - O Q 0 x Z Y - P YZ = ( x - x 1 ) + ( - 1 ) = ( x - x 1 ) + ( - 1 ) = = ( -1- ) + ( -- ) = ( - 0) + ( - 0) ( ) + ( ) = ( ) + ( ) = + = + = = ecause two angles and a non-included side are congruent, XYZ b S. Therefore the triangles have the same perimeter and the same amount of fencing is needed. Reflect 6. xplain how ou could have avoided using the distance formula in xample. Houghton Mifflin Harcourt Publishing ompan Module 6 33 Lesson

6 Your Turn Refer to the diagram to answer the questions. Given: and x - 7. Show that the two triangles are congruent using the S Theorem. Use the distance formula to compare and. 8. Show that the two triangles are congruent using the S Theorem. Use the distance formula to compare and. Houghton Mifflin Harcourt Publishing ompan Module 6 3 Lesson

7 laborate 9. Two isosceles triangles share a side. With which diagram can the S Theorem be used to show the triangles are congruent? xplain. 10. What must be true of the right triangles in the roof truss to use the S ongruence Theorem to prove the two triangles are congruent? xplain. Houghton Mifflin Harcourt Publishing ompan 11. ssential Question heck-in You know that a pair of triangles has two pairs of congruent corresponding angles. What other information do ou need to show that the triangles are congruent? Module 6 35 Lesson

8 Houghton Mifflin Harcourt Publishing ompan valuate: Homework and Practice ecide whether ou have enough information to determine that the triangles are congruent. If the are congruent, explain wh ach diagram shows two triangles with two congruent angles or sides. Identif one additional pair of corresponding angles or sides such that, if the pair were congruent, the two triangles could be proved congruent b S Online Homework Hints and Help xtra Practice Module 6 36 Lesson

9 13. omplete the proof. Given:, bisects. Prove: Statements 1. _ _ 1.. bisects.. Given Reasons efinition of angle bisector.. Given Write a two-column proof or a paragraph proof. Given: _ _, _ _. Prove: Statements Reasons Houghton Mifflin Harcourt Publishing ompan ach diagram shows and on the coordinate plane, with, and. Identif whether the two triangles are congruent. If the are not congruent, explain how ou know. If the are congruent, find the length of each side of each triangle x x Module 6 37 Lesson

10 x x x x Which theorem or postulate can be used to prove that the triangles are congruent? Select all that appl.. S. SS. SSS. S H.O.T. ocus on Higher Order Thinking. nalze Relationships XYZ and KLM have two congruent angles: X K and Y L. an it be concluded that Z M? an it be concluded that the triangles are congruent? xplain. Houghton Mifflin Harcourt Publishing ompan 3. ommunicate Mathematical Ideas GHJ and PQR have two congruent angles: G P and H Q. If _ HJ is congruent to one of the sides of PQR, are the two triangles congruent? xplain. Module 6 38 Lesson

11 . Make a onjecture ombine the theorems of S ongruence and S ongruence into a single statement that describes a condition for congruenc between triangles. 5. Justif Reasoning Triangles and are constructed with the following angles: m = 35, m = 5, m = 65, m = 5. lso, = = 1 units. re the two triangles congruent? xplain. 6. Justif Reasoning Triangles and are constructed with the following angles: m = 65, m = 60, m = 65, m = 55. lso, = = 7 units. re the two triangles congruent? xplain. 7. lgebra biccle frame includes VSU and VTU, which lie in intersecting planes. rom the given angle measures, can ou conclude that VSU VTU? xplain. m VUS = (7 - ) m VUT = ( 5 1_ x - 1_ ) m USV = 5 _ m UTV = (x + 8 ) 3 m SVU = (3-6) m TVU = x Houghton Mifflin Harcourt Publishing ompan S V U T Module 6 39 Lesson

12 Lesson Performance Task mapmaker has successfull mapped arlisle Street and River venue, as shown in the diagram. The last step is to map eacon Street correctl. To save time, the mapmaker intends to measure just one more angle or side of the triangle. alle arlisle 0. mi eacon Street 8 River venue a. Which angle(s) or side(s) could the mapmaker measure to be sure that onl one triangle is possible? or each angle or side that ou name, justif our answer. b. Suppose that instead of measuring the length of arlisle Street, the mapmaker measured and along with. Would the measures of the three angles alone assure a unique triangle? xplain. Houghton Mifflin Harcourt Publishing ompan Module Lesson