Essential Question: What does the AAS Triangle Congruence Theorem tell you about two triangles? Explore Exploring Angle-Angle-Side A C E F

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1 OMMON OR G G Locker LSSON 6. S Triangle ongruence Name lass ate 6. S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? ommon ore Math Standards The student is expected to: OMMON OR G-SRT..5 Use congruence... criteria for triangles to solve problems and to prove relationships in geometric figures. Mathematical Practices OMMON OR MP.3 Logic Language Objective xplain in our own words the difference between the S and S congruence theorems. NGG ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, the triangles are congruent. PRVIW: LSSON PRORMN TSK View the ngage section online. iscuss the figure, making sure students understand wh a triangle with the given side lengths cannot be drawn. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing ompan xplore 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? 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. 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 op using _ as a side of the angle. Resource Locker 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 6 83 Lesson G Name lass ate 6. S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? G-SRT..5 Use congruence... criteria for triangles to solve problems and to prove relationships in geometric figures. xplore xploring ngle-ngle-side Houghton Mifflin Harcourt Publishing ompan 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? 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. 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. op using _ as a side of the angle. Resource 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. HROVR PGS 5 5 Turn to these pages to find this lesson in the hardcover student edition. Module 6 83 Lesson 83 Lesson 6.

2 How man triangles can ou construct? Onl one triangle is possible. Reflect op all of G to the transparenc. Then overla it on. re the triangles congruent? How do ou know? Yes; possible answer: ll sides are congruent, so the are congruent b SSS. XPLOR xploring ngle-ngle-side ongruence 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? Yes; no specific side lengths and angle measures were involved.. ompare our results to those of our classmates. oes this procedure work with an triangle? Yes. ll triangles with congruent angles and a congruent non-included side are congruent. INTGRT THNOLOGY Students can use geometr software to explore how much the know about a triangle when given two angle measures and the length of a nonincluded side. 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 1.,, _ _ 1. Given Reasons. m + m + m = 180. Triangle Sum Theorem 3. 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 6. efinition of congruent angles 7. m = m - m 7. Substitution 8. m = m 8. Transitive Propert of qualit Houghton Mifflin Harcourt Publishing ompan QUSTIONING STRTGIS If ou know the measures of two angles in a triangle, and the length of a side that is not included, does this describe a unique triangle? What is the relationship between the information necessar to describe a unique triangle and the information necessar to prove two triangles congruent? Yes; the same sides and angles that must be known to be congruent to prove two triangles congruent must be given to describe a unique triangle (two angles and the included side, two sides and the included angle, etc.). XPLIN 1 Justifing ngle-ngle-side ongruence efinition of congruent angles S Triangle ongruence Theorem Module 6 8 Lesson PROSSIONL VLOPMNT Integrate Mathematical Practices This lesson provides an opportunit to address Mathematical Practice MP.3, which calls for students to construct viable arguments and critique the reasoning of others. Students learn to justif the ngle-ngle-side ongruence Theorem b using a paragraph proof. Mathematical proofs must use precise language to ensure their validit. s students continue to explore congruent triangles, ask them to justif their conclusions and communicate them with the class. Promoting this tpe of dialogue in the classroom is an essential aspect of the standard. QUSTIONING STRTGIS If ou know the measures of two angles of a triangle, what theorem gives ou a wa to determine the measure of the third angle? Third ngles Theorem VOI OMMON RRORS fter finding the measure of the third angle, some students ma want to sa the triangles are congruent b. Remind them that there is no congruence theorem because having three pairs of congruent angles does not ensure two triangles have the same size. S Triangle ongruence 8

3 XPLIN Using ngle-ngle-side ongruence 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? Steps 8 could be simplified to one step,, making the whole QUSTIONING STRTGIS proof onl three steps. How does the S ongruence Theorem differ from the S ongruence Theorem? While both the S and S ongruence Theorems require ou to know that two pairs of corresponding angles are congruent between two triangles, the S ongruence Theorem requires ou to also know that a pair of corresponding included sides are congruent. The S ongruence Theorem requires ou to know also that a pair of corresponding non-included sides are congruent. If S is used as the method of proof, can the triangles also be proved congruent using S? Yes; ou can use the Third ngles Theorem to show the third pair of angles is congruent, and then ou can use S.. ould the S ongruence Theorem be used in the proof? xplain. No, that is the theorem being proved, so it cannot be used until it has been proven. xplain xample Using ngle-ngle-side ongruence Use the S Theorem to prove the given triangles are congruent. Given: _ _ and m n Prove: m ǁ n Given Given lt. Int Thm. S ong. Thm. m n lt. Int Thm. Houghton Mifflin Harcourt Publishing ompan Given: _ _, _ _, and _ _. Prove: ǁ Given Given orr. Thm. S ong. Thm. ǁ Given orr. Thm. Module 6 85 Lesson OLLORTIV LRNING Small Group ctivit Students ma benefit from exploring conditions that do not guarantee congruent triangles. Have students work in small groups and use a protractor to draw triangles with angles that measure 35, 50, and 95. Then then compare their triangles within each group. Students will see that all the triangles have the same shape, but not necessaril the same size. Since the triangles are not all congruent, the activit shows that there is no ongruence riteria. 85 Lesson 6.

4 Your Turn 5. Given:, _ _, _ _. Use the S Theorem to prove the triangles are congruent. Write a paragraph proof. XPLIN 3 ppling ngle-ngle-side ongruence ecause is parallel to, this means that is congruent to, using the orresponding ngles Theorem. Since is congruent to and is congruent to, then one pair of corresponding angles and two pairs of non-included corresponding sides are congruent. This means that is congruent to using the S Triangle ongruence Theorem. xplain 3 ppling ngle-ngle-side ongruence VOI OMMON RRORS The real-world situation ma distract some students. Make sure the are certain what the are solving for before the begin. 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 pairs of angles and a pair of non-included sides are congruent, b S. Therefore the triangles have the same perimeter and the same amount of fencing is needed. Module 6 86 Lesson IRNTIT INSTRUTION uditor ues s students work through the lesson, have them identif orall whether the triangles are congruent b S, S, SS, or SSS. Then have them work with a partner to identif the congruent parts and to determine if the congruent pairs of angles or sides are corresponding parts. S Triangle ongruence 86

5 QUSTIONING STRTGIS When is the distance formula useful for proving two triangles to be congruent? How would ou use it? It is useful when the triangles are on a coordinate grid and their side lengths are not given; ou would use it to find the lengths of corresponding sides and determine whether or not the sides are congruent. Given: P Z, Q X It is given that P Z and Q X. ompare YZ and OP using the distance formula. YZ = ( x - x 1 ) + ( - 1 ) = = ((-1) - -5 ) + ((-) - - ) ( ) + ( ) = 16 + = 0 OP = ( x - x 1 ) + ( - 1 ) ( - 0) + ( - 0) = - = ( ) + ( - ) = + 16 = 0 ecause two pairs of angles and a pair of non-included sides are congruent, X - Y O Q 0 x Z - P Houghton Mifflin Harcourt Publishing ompan XYZ QOP 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. I could have used XY and QO as corresponding sides. counting gridlines, it is eas to see that XY = QO = 7. These are also not the included sides between the angles, so S applies. Module 6 87 Lesson LNGUG SUPPORT ommunicate Math ivide students into pairs. Give each pair pictures of congruent and non-congruent triangles, highlighters, protractors, and rulers. Instruct them to prove which pairs are congruent b angle-angle-side. Have students highlight the angles and the side the used to show congruence, and write notes explaining wh the triangles are congruent. 87 Lesson 6.

6 Your Turn Refer to the diagram to answer the questions. Given: and ONNT VOULRY Students can create a chart, or add to a chart, proving triangle congruence, defining angle/angle/side, and showing a picture that demonstrates this condition x - 7. Show that the two triangles are congruent using the S Theorem. Use the distance formula to compare and. It is given that and. = (-5 - (-1) ) + (5 - ) = (1-5) + ( - (-1) ) = 5 = 5 Sides and are equal. Therefore, because two angles and a non-included side are congruent, b the S Theorem. 8. Show that the two triangles are congruent using the S Theorem. Use the distance formula to compare and. = = It is given that and. = (- - (-1) ) + (0 - ) = ( - 5) + (-3 - (-1) ) = 9 + = 9 + = 13 = 13 Sides and are equal. Therefore, because two angles and a non-included side are congruent, b the S Theorem. Houghton Mifflin Harcourt Publishing ompan Module 6 88 Lesson S Triangle ongruence 88

7 LORT INTGRT MTHMTIL PRTIS ocus on Modeling MP. Students ma benefit from seeing the proof of the S ongruence riterion presented as a flowchart proof. laborate 9. Two isosceles triangles share a side. With which diagram can the S Theorem be used to show the triangles are congruent? xplain. SUMMRIZ TH LSSON When do ou use the S Triangle ongruence Theorem? What information do ou need in order to use this theorem? You use the S Triangle ongruence Theorem to prove that two triangles are congruent b using three pairs of congruent corresponding parts. You need to know that two pairs of corresponding angles and one pair of corresponding non-included sides are congruent in order to use it. The one on the right. The shared side is the non-included side. 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 ecause and are right angles,. Since _ is a shared side for both triangles, it must be the non-included side. So to use the S Theorem, must be congruent to. This gives two angles and a non-included side. 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? You need a pair of congruent sides that are either both included or both not included. Module 6 89 Lesson 89 Lesson 6.

8 valuate: Homework and Practice ecide whether ou have enough information to determine that the triangles are congruent. If the are congruent, explain wh. 1.. Online Homework Hints and Help xtra Practice VLUT ongruent, b S ongruence 3.. ongruent, b S ongruence annot be determined. 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 _ _, or _ _ _ _, _ _, _ _, or _ _ ongruent, b S ongruence annot be determined. ongruent, S ongruence Houghton Mifflin Harcourt Publishing ompan SSIGNMNT GUI oncepts and Skills xplore xploring ngle-ngle-side ongruence xample 1 Justifing ngle-ngle-side ongruence xample Using ngle-ngle-side ongruence xample 3 ppling ngle-ngle-side ongruence QUSTIONING STRTGIS Practice xercises 1 1 xercises 13 1 xercises 13 1 xercises 15 1 If ou know the measures of two angles of a triangle, how can ou find the measure of the third angle? ind the difference between 180 and the sum of the two known angle measures. _ _, or _ _ Module 6 90 Lesson xercise epth of Knowledge (.O.K.) OMMON OR Mathematical Practices Recall of information MP.6 Precision 13 1 Skills/oncepts MP. Modeling 3 Strategic Thinking MP. Reasoning 3 3 Strategic Thinking MP. Reasoning 3 Strategic Thinking MP. Reasoning S Triangle ongruence 90

9 VISUL US Have students each make a poster on which the describe and illustrate the S theorem. The illustrations should show both an example and a nonexample. 13. omplete the proof. Given:, bisects. Prove: Statements Reasons _ 1. _ 1. Reflexive Propert of ongruence. bisects.. Given efinition of angle bisector.. Given S Triangle ongruence Theorem 1. Write a two-column proof or a paragraph proof. Given: _ _, _ _. Prove: Statements Reasons 1. _ _, _ _ 1. Given.,. lternate Interior ngles Theorem S Triangle ongruence Theorem 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 ongruent; = = ; = = ; = =.7 ongruent. = = 3, = =, = = 5 Module 6 91 Lesson xercise epth of Knowledge (.O.K.) OMMON OR Mathematical Practices 5 3 Strategic Thinking MP.3 Logic 6 3 Strategic Thinking MP.3 Logic 7 3 Strategic Thinking MP. Modeling 91 Lesson 6.

10 x Not congruent. Segments and are corresponding, but the have different lengths ( =, = 5) x 8 1 x 0 8 ongruent; = =.8; = = 8; = = x VOI OMMON RRORS Some students ma think that two pairs of congruent angles and an pair of non-included sides is enough to use S. The need to be sure the non-included sides are corresponding sides, otherwise the ma be looking at triangles that are not congruent Not congruent. Segments and are corresponding, but the have different lengths ( = 9, = 10). ongruent. = = 10.9, = = 9.9, = =.8 1. 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. It can be concluded that Z M, because the sum of the three angles of a triangle equals 180. While the two triangles are similar, it cannot be concluded that the triangles are congruent without identifing at least one congruent side. 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. The are not necessaril congruent. Onl if _ HJ is congruent to the corresponding side, which is _ QR, will the triangles be congruent. Module 6 9 Lesson S Triangle ongruence 9

11 JOURNL Have students write a journal entr in which the explain and illustrate the S Theorem.. Make a onjecture ombine the theorems of S ongruence and S ongruence into a single statement that describes a condition for congruenc between triangles. If two triangles have two congruent angles and one corresponding congruent side, then the two triangles are congruent. 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. The two triangles cannot be congruent. ecause the sum of the three angles of a triangle equals 180, it can be calculated that m = 100 and that m = 70. The triangles do not have matching angles, so the are not congruent. 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. The two triangles are congruent b either S or S ongruence. The triangles each have angles of 65, 60, and 55, and each has a side of length 7 that is opposite the angle of 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 U V Yes, the sum of the measures in each triangle must be 180, which makes it possible to solve for x and. The value of x is 15, and the value of is 1. ach triangle has angles measuring 8, 68, and 30. _ VU _ VU b the Reflexive Propert of ongruence. So VSU VTU b S or S. T Module 6 93 Lesson 93 Lesson 6.

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. 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. arlisle Street 0. mi a., b the S Triangle ongruence Theorem;, b the S Triangle ongruence Theorem; _, b the SS Triangle ongruence Theorem b. No; there is no Triangle ongruence Theorem. There are man similar triangles with the same angle measures. Houghton Mifflin Harcourt Publishing ompan INTGRT MTHMTIL PRTIS ocus on ritical Thinking MP.3 raw attention to the figure in the Lesson Performance Task. State that m = 85. sk students to imagine that the 8 angle where arlisle Street and River venue intersect increases in size until arlisle Street and eacon Street no longer intersect. sk: What is the smallest measure of for which this would be possible? xplain our reasoning. 133 ; Sample answer: The smallest measure of for which this would be possible would be when the streets became parallel. and would be supplementar interior angles on the same side of the River venue transversal: m = 180 ( ) = 7. So, m = = 133. INTGRT MTHMTIL PRTIS ocus on Modeling MP. sk students to explain how the could use equilateral triangles to show that ongruence does not hold. Sample answer: ll equilateral triangles from small ones on a compan s logo to large Yield traffic signs share three pairs of congruent (60 ) angles. Yet the triangles are clearl not congruent. Module 6 9 Lesson XTNSION TIVITY surveor has a contract to la out the boundaries of a triangular park. Here are the instructions given to the surveor. (ll of the streets are straight.) The vertex of one angle of the park will be the 30 angle where harles Street and arton Street intersect. One side of the park will start at the harles-arton vertex and extend 100 ards along harles Street. The side of the park opposite the 30 angle will connect arton Street and harles Street and measure 60 ards. What problem did the surveor run into? xplain and draw a diagram. There are two possible triangles that meet the given conditions. Scoring Rubric points: Student correctl solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. 60 d 100 d 60 d 30 S Triangle ongruence 9