5.3 Proving Triangle Congruence by SAS Essential Question
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5.3 Proving Triangle Congruence by SAS Essential Question
5.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.5.A G.6.B Proving Triangle Congruence by SAS Essential Question What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? Drawing Triangles Work with a partner. Use dynamic geometry software. a. Construct circles with radii of 2 units and 3 units centered at the origin. Construct a 40° angle with its vertex at the origin. Label the vertex A. 4 3 2 1 40° 0 SELECTING TOOLS To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data. b. Locate the point where one ray of the angle intersects the smaller circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw △ABC. −4 −3 −2 −1 A 0 1 2 3 4 5 4 5 −1 −2 −3 4 3 c. Find BC, m∠B, and m∠C. B 2 1 d. Repeat parts (a)–(c) several times, redrawing the angle in different positions. Keep track of your results by copying and completing the table below. Write a conjecture about your findings. A B C 0 −4 −3 −2 −1 A C 40° 0 1 2 3 −1 −2 −3 AB AC BC m∠A 1. (0, 0) 2 3 40° 2. (0, 0) 2 3 40° 3. (0, 0) 2 3 40° 4. (0, 0) 2 3 40° 5. (0, 0) 2 3 40° m∠B m∠C Communicate Your Answer 2. What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? 3. How would you prove your conjecture in Exploration 1(d)? Section 5.3 Proving Triangle Congruence by SAS 249 5.3 Lesson What You Will Learn Use the Side-Angle-Side (SAS) Congruence Theorem. Solve real-life problems. Core Vocabul Vocabulary larry Using the Side-Angle-Side Congruence Theorem Previous congruent figures rigid motion Theorem Theorem 5.5 Side-Angle-Side (SAS) Congruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. STUDY TIP The included angle of two sides of a triangle is the angle formed by the two sides. — ≅ DE —, ∠A ≅ ∠D, and AC — ≅ DF —, If AB then △ ABC ≅ △DEF. E B F C Proof p. 250 A D Side-Angle-Side (SAS) Congruence Theorem — ≅ DE —, ∠A ≅ ∠D, AC — ≅ DF — Given AB E B Prove △ABC ≅ △DEF F C A D First, translate △ABC so that point A maps to point D, as shown below. E B E B′ F C A F D C′ D This translation maps △ABC to △DB′C′. Next, rotate △DB′C′ counterclockwise through ∠C′DF so that the image of ⃗ DC′ coincides with ⃗ DF, as shown below. E E B′ F F D C′ D B″ — ≅ DF —, the rotation maps point C′ to point F. So, this rotation maps Because DC′ △DB′C′ to △DB″F. Now, reflect △DB″F in the line through points D and F, as shown below. E E F F D B″ D Because points D and F lie on ⃖⃗ DF, this reflection maps them onto themselves. Because a reflection preserves angle measure and ∠B″DF ≅ ∠EDF, the reflection maps ⃗ DB″ — ≅ DE —, the reflection maps point B″ to point E. So, this reflection to ⃗ DE. Because DB″ maps △DB″F to △DEF. Because you can map △ABC to △DEF using a composition of rigid motions, △ABC ≅ △DEF. 250 Chapter 5 Congruent Triangles Using the SAS Congruence Theorem B Write a proof. STUDY TIP Make your proof easier to read by identifying the steps where you show congruent sides (S) and angles (A). — ≅ DA —, BC — AD — Given BC C Prove △ABC ≅ △CDA A D SOLUTION STATEMENTS REASONS S 1. Given — — 1. BC ≅ DA — — 2. BC AD 2. Given A 3. ∠BCA ≅ ∠DAC — ≅ CA — S 4. AC 3. Alternate Interior Angles Theorem (Thm. 3.2) 4. Reflexive Property of Congruence (Thm. 2.1) 5. △ABC ≅ △CDA 5. SAS Congruence Theorem Using SAS and Properties of Shapes — and RP — pass through the center M of the circle. What can you In the diagram, QS conclude about △MRS and △MPQ? S R P M Q SOLUTION Because they are vertical angles, ∠PMQ ≅ ∠RMS. All points on a circle are the same —, MQ —, MR —, and MS — are all congruent. distance from the center, so MP So, △MRS and △MPQ are congruent by the SAS Congruence Theorem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com In the diagram, ABCD is a square with four congruent sides and four right — ⊥ SU — angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT — ≅ VU —. and SV S B R A V U C T D 1. Prove that △SVR ≅ △UVR. 2. Prove that △BSR ≅ △DUT. Section 5.3 Proving Triangle Congruence by SAS 251 Copying a Triangle Using SAS C Construct a triangle that is congruent to △ABC using the SAS Congruence Theorem. Use a compass and straightedge. A SOLUTION Step 1 Step 2 Step 3 Step 4 F D E Construct a side — so that it Construct DE —. is congruent to AB D E Construct an angle Construct ∠D with vertex D and side ⃗ DE so that it is congruent to ∠A. B D F E D Construct a side — so that Construct DF —. it is congruent to AC E Draw a triangle Draw △DEF. By the SAS Congruence Theorem, △ABC ≅ △DEF. Solving Real-Life Problems Solving a Real-Life Problem You are making a canvas sign to hang on the triangular portion of the barn wall shown in the picture. You think you can use two identical triangular sheets of — ⊥ QS — and canvas. You know that RP — ≅ PS —. Use the SAS Congruence PQ Theorem to show that △PQR ≅ △PSR. R S Q P SOLUTION — ≅ PS —. By the Reflexive Property of Congruence (Theorem 2.1), You are given that PQ — — RP ≅ RP . By the definition of perpendicular lines, both ∠RPQ and ∠RPS are right angles, so they are congruent. So, two pairs of sides and their included angles are congruent. △PQR and △PSR are congruent by the SAS Congruence Theorem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. You are designing the window shown in the photo. You want to make △DRA — ≅ DG — and congruent to △DRG. You design the window so that DA ∠ADR ≅ ∠GDR. Use the SAS Congruence Theorem to prove △DRA ≅ △DRG. D A 252 Chapter 5 Congruent Triangles R G Exercises 5.3 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. WRITING What is an included angle? 2. COMPLETE THE SENTENCE If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then ___________. Monitoring Progress and Modeling with Mathematics In Exercises 15–18, write a proof. (See Example 1.) In Exercises 3–8, name the included angle between the pair of sides given. J — — — 15. Given PQ bisects ∠SPT, SP ≅ TP L Prove △SPQ ≅ △TPQ P K P S — and KL — 3. JK — and LK — 4. PK 5. — and LK — LP 6. — and JK — JL 7. — and JL — KL 8. — and PL — KP Q — ≅ CD —, AB — CD — 16. Given AB Prove △ABC ≅ △CDA A In Exercises 9–14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Theorem 5.5). Explain. 9. △ABD, △CDB A 1 2 M L B Q N B C 11. △YXZ, △WXZ Prove △ABC ≅ △EDC D 12. △QRV, △TSU R S A Q 13. △EFH, △GHF V U K B T —≅ RT —, QT — ≅ ST — 18. Given PT 14. △KLM, △MNK E E C Y F C — and BD —. 17. Given C is the midpoint of AE P Z X D 10. △LMN, △NQP D W T L Prove △PQT ≅ △RST P Q T G H N M S Section 5.3 R Proving Triangle Congruence by SAS 253 27. PROOF The Navajo rug is made of isosceles In Exercises 19–22, use the given information to name two triangles that are congruent. Explain your reasoning. (See Example 2.) 19. ∠SRT ≅ ∠URT, and 20. ABCD is a square with R is the center of the circle. S four congruent sides and four congruent angles. B D C B T R triangles. You know ∠B ≅ ∠D. Use the SAS Congruence Theorem (Theorem 5.5) to show that △ABC ≅ △CDE. (See Example 3.) A C E U A D — ⊥ MN —, KL — ⊥ NL —, 22. MK 21. RSTUV is a regular 28. HOW DO YOU SEE IT? and M and L are centers of circles. pentagon. T B What additional information do you need to prove that △ABC ≅ △DBC? K S U 10 m M R A L 10 m V △ABC ≅ △DEC. Then find the values of x and y. CONSTRUCTION In Exercises 23 and 24, construct a triangle that is congruent to △ABC using the SAS Congruence Theorem (Theorem 5.5). A 4y − 6 3y + 1 D 2x + 6 C B 24. B B D 29. MATHEMATICAL CONNECTIONS Prove that N 23. C 4x E 30. THOUGHT PROVOKING There are six possible subsets A A C C of three sides or angles of a triangle: SSS, SAS, SSA, AAA, ASA, and AAS. Which of these correspond to congruence theorems? For those that do not, give a counterexample. 25. ERROR ANALYSIS Describe and correct the error in finding the value of x. ✗ Y 5x − 1 4x + 6 X W 5x − 5 Z 3x + 9 31. MAKING AN ARGUMENT Your friend claims it is 4x + 6 = 3x + 9 x+6=9 x=3 possible to construct a triangle congruent to △ABC by first — and AC —, and constructing AB then copying ∠C. Is your friend correct? Explain your reasoning. 26. WRITING Describe the relationship between your conjecture in Exploration 1(d) on page 249 and the Side-Angle-Side (SAS) Congruence Theorem (Thm. 5.5). Intersecting Lines Theorem (Theorem 4.3). Reviewing what you learned in previous grades and lessons Classify the triangle by its sides and by measuring its angles. (Section 5.1) 254 34. Chapter 5 Congruent Triangles A 32. PROVING A THEOREM Prove the Reflections in Maintaining Mathematical Proficiency 33. C 35. 36. B