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LESSON 5.2 Name ASA Triangle Congruence 5.2 Explore 1 G.6.B Mathematical Processes G.1.E Create and use representations to organize, record, and communicate mathematical ideas. Drawing Triangles Given Two Angles and a Side Draw a segment that is 4 inches long. Label the endpoints A and B. _ Use a protractor to draw a 30° angle so that one side is AB and its vertex is point A. 1.A, 1.B, 1.F, 2.C.4, 3.H, 4.D © Houghton Mifflin Harcourt Publishing Company Have students work in pairs to label and color code congruent angles and a side in pairs of triangles. C _ Use a protractor to draw a 40° angle so that one side is AB and its vertex is point B. Label the point where the sides of the angles intersect as point C. Language Objective A 1. In a polygon, the side that connects two consecutive angles is the included side of those two angles. Describe the triangle you drew using the term included side. Be as precise as possible. It is a triangle with a 30° angle, a 40° angle, and an included side that is 4 inches long. 2. Discussion Based on your results, how can you decide whether two triangles are congruent without checking that all six pairs of corresponding sides and corresponding angles are congruent? Possible answer: If two angles and the included side of one triangle are 40° B congruent to two angles and the included side of another triangle, then the triangles are congruent. Module 5 ges EDIT--Chan DO NOT Key=TX-A Correction must be Lesson 2 261 gh "File info" made throu Date Class nce gle Congrue Name ASA Trian about em tell you uence Theor le Congr ASA Triang ... congruence does the Side-Angle ion: What g the ... Angletriangles? ent by applyin es are congru les two triangl G.5.A, G.5.C G.6.B Prove n Two Ang Also G.3.B, conditions. ngles Give 5.2 Resource Locker Quest Essential HARDCOVER PAGES 217226 Tria Drawing and a Side ore 1 Expl ent g of congru of correspondin have six pairs ent if they all three pairs for les are congru possible to check shortcuts two triang there are always seen that er, it is not . Fortunately, You have g parts. Howevcorresponding angles of ent. correspondin all three pairs triangles are congru B. sides and whether two ints A and the endpo determining nt that is Draw a segme 4 inches long. a ctor to draw Use a protra is point A. its vertex 30° angle Label so that one _ and side is AB Turn to these pages to find this lesson in the hardcover student edition. C _ and side is AB A so that one of the a 40° angle the sides ctor to draw point where Use a protra Label the is point B. C. its vertex ct as point Is there a angles interse each other. this triangle beside What does a classmate’s one to the other? and le triang ns that maps Put your of rigid motio les? sequence . triang the ruent are cong tell you about triangles Yes; the 30° 40° 4 in. B ed side of is the includ side. Be as utive angles included cts two consec using the term that conne le you drew n, the side is ibe the triang side that In a polygo included angles. Descr , and an those two possible. , a 40° angle precise as 30° angle le with a It is a triang long. s inche 4 triangles whether two and you decide ng sides pondi , how can corres results on your six pairs of are ing that all ssion Based triangle 2. Discu uent without check uent? side of one le, then included are congr are congr er triang s and the ng angles correspondi side of anoth If two angle answer: included Possible s and the to two angle congruent ruent. les are cong the triang y g Compan Reflect © Houghto n Mifflin Harcour t Publishin 1. Lesson 2 261 Module 5 5L2 261 86_U2M0 ESE3538 GE_MTX Lesson 5.2 4 in. Reflect GE_MTXESE353886_U2M05L2 261 261 30° Put your triangle and a classmate’s triangle beside each other. Is there a sequence of rigid motions that maps one to the other? What does this tell you about the triangles? Yes; the triangles are congruent. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Explain that the flags of many countries incorporate geometric objects such as triangles. Then preview the Lesson Performance Task. Resource Locker You have seen that two triangles are congruent if they have six pairs of congruent corresponding parts. However, it is not always possible to check all three pairs of corresponding sides and all three pairs of corresponding angles. Fortunately, there are shortcuts for determining whether two triangles are congruent. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions. Also G.3.B, G.5.A, G.5.C If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. ASA Triangle Congruence G.6.B Prove two triangles are congruent by applying the ... Angle-Side-Angle ... congruence conditions. Also G.3.B, G.5.A, G.5.C The student is expected to: Essential Question: What does the ASA Triangle Congruence Theorem tell you about triangles? Date Essential Question: What does the ASA Triangle Congruence Theorem tell you about triangles? Texas Math Standards ENGAGE Class 2/21/14 4:49 AM 2/21/14 4:49 AM Explore 2 Justifying ASA Triangle Congruence EXPLORE 1 Explain the results of Explore 1 using transformations. A Use tracing paper to make two copies of the triangle from Explore 1 as shown. Identify the corresponding parts you know to be congruent and mark these congruent parts on the figure. F ∠D ∠A ≅ A D ∠E E ∠B ≅ B _ _ DE AB ≅ C Drawing Triangles Given Two Angles and a Side INTEGRATE TECHNOLOGY B What can you do to show that these triangles are congruent? Find a sequence of rigid motions that maps one triangle onto the other triangle. C translate △ABC so that point A maps to point D. What translation vector did you use? ⇀ the vector with initial point A and terminal point D AD D E F G ( ) D A Use a rotation to map point B to point E. What is the center of the rotation? What is the angle of the rotation? The center of the rotation is point D (or A); the angle of the rotation is m∠EDB. How do you know_ the image _ of point B is point E? It is given that AB ≈ DE, so the image of point B must be point E. What rigid motion do you think will map point C to point F ? ‹ › − reflection across DE QUESTIONING STRATEGIES F B C F D A E B C EXPLORE 2 ‹ › − to show that the image of point C is point F, notice that ∠A is reflected across DE, so the measure of the angle is preserved. Since ∠A ≅ ∠D you can conclude that the image → → _ ‾ ‾ of AC lies on DF . In particular, the image of point C must lie on DF . By similar → → _ ‾ ‾ reasoning, the image of BC lies on EF and the image of point C must lie on EF . _ _ the only point that lies on both DF and EF is point F . Describe the sequence of rigid motions used to map △ABC to △DEF. a translation followed by a rotation followed by a reflection Reflect 3. Discussion Arturo said the argument in the activity works for any triangles with two pairs of congruent corresponding angles, and it is not necessary for the included sides to be congruent. Do you agree? Explain. No; the included sides must be congruent to conclude that the image of point B is point E after a rotation around point D. Module 5 262 How can you check whether the triangles you draw are congruent to the triangles your classmates draw? Place one student’s page on top of the other student’s page and check to see if the triangles can be made to coincide exactly. E Justifying ASA Triangle Congruence © Houghton Mifflin Harcourt Publishing Company H Have students explore the Angle-Side-Angle theorem using geometry software. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students respond to this prompt in their math journals. “If I know that two pairs of corresponding angles and the included sides of two triangles are congruent, I know ________. I know this because ________.” QUESTIONING STRATEGIES Lesson 2 PROFESSIONAL DEVELOPMENT GE_MTXESE353886_U2M05L2 262 Math Background Students know that when triangles are congruent, all pairs of corresponding sides and corresponding angles are congruent. As an extension of that, students to begin to develop converses of the statement that Corresponding Parts of Congruent Triangles Are Congruent in which they do not need to know that all six pairs of corresponding parts are congruent in order to prove that triangles are congruent. In this lesson, they explore the Angle-Side-Angle (ASA) Theorem. They find that if they can prove that two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. 1/22/15 3:04 AM What is the benefit of using the Angle-SideAngle Theorem instead of CPCTC? You need to find only three pairs of congruent corresponding parts with ASA, as opposed to six pairs with CPCTC. ASA Triangle Congruence 262 Explain 1 EXPLAIN 1 You can state your findings about triangle congruence as a theorem. This theorem can help you decide whether two triangles are congruent. Deciding Whether Triangles Are Congruent Using ASA Triangle Congruence ASA Triangle Congruence Theorem If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Example 1 CONNECT VOCABULARY Remind students that the ASA Triangle Congruence Theorem is a shortened form of its full name, the Angle-Side-Angle Triangle Congruence Theorem. When you write ASA, it is helpful to read it aloud as Angle-Side-Angle and have students do the same to reinforce what it means. Give each pair pictures of congruent and non-congruent triangles, highlighters, protractors, and rulers. Instruct them to prove which pairs are congruent by using Angle-Side-Angle to prove it. Have students highlight the angles and the side they used to show congruence, and write notes explaining why the triangles are congruent. E B 74° m∠D + m∠E + m∠F = 180° m∠D + 74° + 61° = 180° m∠D + 135° = 180° A 61° 45° 2.3 cm m∠D = 45° D C 61° 2.3 cm F Step 2 Compare the angle measures and side lengths. m∠A = m∠D = 45°, AC = DF = 2.3 cm, and m∠C = m∠F = 61° _ _ So, ∠A ≅ ∠D, AC ≅ DF, and ∠C ≅ ∠F. _ _ ∠A and ∠C include side AC, and ∠D and ∠F include side DF. So, △ABC ≅ △DEF by the ASA Triangle Congruence Theorem. J Step 1 Find m∠P. m∠M + m∠N + m∠P = 180° 69 ° + m∠P = 180° m∠P = 111 31° K 62 in. 31 ° + 38 ° + m∠P = 180° © Houghton Mifflin Harcourt Publishing Company LANGUAGE SUPPORT Determine whether the triangles are congruent. Explain your reasoning. Step 1 Find m∠D. QUESTIONING STRATEGIES Why do you need to find the measure of the missing angle to use the ASA Triangle Congruence Theorem? The two sides that you know are congruent need to be the included sides, so you need to know the measures of the angles at their endpoints. Deciding Whether Triangles Are Congruent Using ASA Triangle Congruence 110° L P ° Step 2 Compare the angle measures and side lengths. 62 in. N 38° 31° M None of the angles in △MNP has a measure of 110° . Therefore, there is/is not a sequence of rigid motions that maps △MNP onto △JKL, and △MNP is/is not congruent to △JKL. Reflect 4. In Part B, do you need to find m∠K? Why or why not? No; you only need to know that △JKL has an angle (∠L) that is not congruent to any angle of △MNP. At that point, you can conclude that the triangles are not congruent. Module 5 263 Lesson 2 COLLABORATIVE LEARNING GE_MTXESE353886_U2M05L2 263 Small Group Activity Have students experiment with congruent triangles and triangles that are not congruent but do have some congruent parts. Instruct them to draw a pair of congruent triangles and a pair of non-congruent triangles that meet the following criteria: • at least two pairs of congruent sides • at least one pair each of congruent sides and congruent angles • all three pairs of congruent angles • at least two pairs of congruent sides and one pair of congruent angles 263 Lesson 5.2 24/02/14 6:08 AM Your Turn EXPLAIN 2 Determine whether the triangles are congruent. Explain your reasoning. 5. 6. A P Q B D C _ _ ∠B ≅ ∠C, BD ≅ CD, and ∠ADB ≅ ∠ADC since both are right angles. _ BD and ∠B and ∠ADB include side_ ∠ADC and ∠C include side DC. So, △ADB ≅ △ADC by the ASA Triangle Congruence Theorem. Explain 2 72° 38° 1 in. S R Proving Triangles Are Congruent Using ASA Triangle Congruence 38° 67° U 1 in. T 72° + 38° + m∠R = 180° AVOID COMMON ERRORS None of the angles in △PQR has a measure of 67°. So, △PQR is not congruent to △STU. Some students may forget to include in their proofs the information that is given in the diagram. Remind them to start the proof by listing the given information. m∠R = 70° Proving Triangles Are Congruent Using ASA Triangle Congruence QUESTIONING STRATEGIES The ASA Triangle Congruence Theorem may be used as a reason in a proof. Example 2 Write each proof. How do you know when you have enough information to complete the proof? To complete the proof, you need to show that two angles and the included side of one triangle are congruent to the corresponding angles and side of the other triangle. M Given: ∠MQP ≅ ∠NPQ, ∠MPQ ≅ ∠NQP Prove: △MQP ≅ △NPQ Q P N Reasons 1. ∠MQP ≅ ∠NPQ 1. Given 2. ∠MPQ ≅ ∠NQP 2. Given _ _ 3. QP ≅ QP 3. Reflexive Property of Congruence 4. △MQP ≅ △NPQ 4. ASA Triangle Congruence Theorem Module 5 264 CONNECT VOCABULARY © Houghton Mifflin Harcourt Publishing Company Statements Have students review the different units used to measure triangles to show congruence. Length is measured in linear units such as cm, mm, or inches, while angles are measured in degrees. Lesson 2 DIFFERENTIATE INSTRUCTION GE_MTXESE353886_U2M05L2 264 Modeling 1/22/15 3:04 AM Instruct students to draw and label three triangles according to the following specifications. • There is just enough information to prove that they are congruent using the ASA Triangle Congruence Theorem. • There is enough information to prove that they are not congruent. • There is some of the information required to prove that they are congruent, but not enough. ASA Triangle Congruence 264 D _ Given: ∠A ≅ ∠C, E is the midpoint of AC. B A Prove: △AEB ≅ △CED E C B Statements 1. ∠A ≅ ∠C Reasons 1. Given _ 2. E is the midpoint of AC . 2. Given _ _ 3. AE ≅ CE 3. Definition of midpoint 4. ∠AEB ≅ ∠CED 4. Vertical angles are congruent. 5. △AEB ≅ △CED 5. ASA Triangle Congruence Theorem Reflect 7. _ In Part B, suppose the length of AB is 8.2 centimeters. Can you determine the length of any other segments in the figure? _ _ Explain. Yes; CD = 8.2 cm because AB ≅ CD by CPCTC. Your Turn Write each proof. 8. Given: ∠JLM ≅ ∠KML, ∠JML ≅ ∠KLM J Prove: △JML ≅ △KLM K © Houghton Mifflin Harcourt Publishing Company L Statements M Reasons 1. ∠JLM ≅ ∠KML 1. Given 2. ∠JML ≅ ∠KLM _ _ 3. LM ≅ LM 2. Given 4. △JML ≅ △KLM 4. ASA Triangle Congruence Theorem Module 5 3. Reflexive Property of Congruence 265 Lesson 2 LANGUAGE SUPPORT GE_MTXESE353886_U2M05L2 265 Vocabulary Development Make sure students understand the meaning of included side. Sketch 4BER, _ 4MAT, and 4SQU on the board. Use color to highlight SQ and define what it means for it to be included between ∠S and ∠Q. Ask questions such as “What _ side is included between ∠E and ∠R?” and “Between which two angles is MT ?” Continue to drill students until they can recognize and name included sides fluently. 265 Lesson 5.2 1/22/15 3:04 AM 9. _ _ Given: ∠S and ∠U are right angles, RV bisects SU. R ELABORATE Prove: △RST ≅ △VUT S T U VISUAL CUES Use colored pencils to label congruent sides using ticks and congruent angles using arcs to help students better visualize the angles and sides that are congruent V Statements Reasons 1. ∠S and ∠U are right angles. 1. Given 2. ∠S ≅ ∠U _ _ 3. RV bisects SU. _ _ 4. ST ≅ UT 2. All right angles are congruent. 5. ∠RTS ≅ ∠VTU 5. Vertical angles are congruent. 6. △RST ≅ △VUT 6. ASA Triangle Congruence Theorem 3. Given SUMMARIZE THE LESSON 4. Definition of bisector Why would you use the ASA Triangle Congruence Theorem? What do you need to know to use it? You would use the ASA Triangle Congruence Theorem to prove that two triangles are congruent by using only three pairs of congruent parts. You need to know that two pairs of corresponding angles are congruent and the included sides between those angles are also congruent. Elaborate 10. Discussion Suppose you and a classmate both draw triangles with a 30° angle, a 70° angle, and a side that is 3 inches long. How will they compare? Explain your reasoning. The triangles will be congruent by the ASA Triangle Congruence Theorem if the 3 inch side is the included side. Otherwise, the triangles will have the same shape but not necessarily the same size. © Houghton Mifflin Harcourt Publishing Company 11. Discussion How can a diagram show you that corresponding parts of two triangles are congruent without providing specific angle measures or side lengths? Possible answer: Vertical angles are congruent. Overlapping sides are congruent. Right angles are congruent. Angles or sides marked with congruence symbols are congruent. 12. Essential Question Check-In What must be true in order for you to use the ASA Triangle Congruence Theorem to prove that triangles are congruent? Two angles and the included side of one triangle must be congruent to two angles and the included side of another triangle. Module 5 GE_MTXESE353886_U2M05L2 266 266 Lesson 2 2/21/14 4:49 AM ASA Triangle Congruence 266