Proving Triangles Congruent: SSS and SAS Proofs

1. LESSON 4–4 Proving Triangles Congruent – SSS, SAS

2. Five-Minute Check (over Lesson 4–3) TEKS Then/Now New Vocabulary Postulate 4.1: Side-Side-Side (SSS) Congruence Example 1: Use SSS to Prove Triangles Congruent Example 2: SSS on the Coordinate Plane Postulate 4.2: Side-Angle-Side (SAS) Congruence Example 3: Real-World Example: Use SAS to Prove Triangles are Congruent Example 4: Use SAS or SSS in Proofs Lesson Menu

3. Write a congruence statement for the triangles. A.ΔLMN ΔRTS B.ΔLMN ΔSTR C.ΔLMN  ΔRST D.ΔLMN  ΔTRS 5-Minute Check 1

4. Name the corresponding congruent angles for the congruent triangles. A.L  R, N  T, M  S B.L  R, M  S, N  T C.L  T, M  R, N  S D.L  R, N  S, M  T 5-Minute Check 2

5. ___ ___ ___ ___ ___ ___ A.LM  RT, LN  RS, NM  ST B.LM  RT, LN  LR, LM  LS C.LM  ST, LN  RT, NM  RS D.LM  LN, RT  RS, MN  ST ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ Name the corresponding congruent sides for the congruent triangles. 5-Minute Check 3

6. Refer to the figure. Find x. A. 1 B. 2 C. 3 D. 4 5-Minute Check 4

7. Refer to the figure.Find m A. A. 30 B. 39 C. 59 D. 63 5-Minute Check 5

8. A. A  E B. C  D C.AB  DE D.BC  FD ___ ___ ___ ___ Given that ΔABC ΔDEF, which of the following statements is true? 5-Minute Check 6

9. Targeted TEKS G.6(B) Prove two triangles are congruent by applying the Side- Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, mid-segments, and medians, and apply these relationships to solve problems. Also addresses G.5(B). Mathematical Processes G.1(E), G.1(G) TEKS

10. You proved triangles congruent using the definition of congruence. • Use the SSS Postulate to test for triangle congruence. • Use the SAS Postulate to test for triangle congruence. Then/Now

11. included angle Vocabulary

12. Concept 1

13. ___ ___ ___ ___ Given: QU AD, QD  AU Use SSS to Prove Triangles Congruent Write a flow proof. Prove: ΔQUD ΔADU Example 1

14. Use SSS to Prove Triangles Congruent Answer: Flow Proof: Example 1

15. Which information is missing from the flowproof?Given: AC ABD is the midpoint of BC.Prove: ΔADC  ΔADB ___ ___ ___ ___ A.AC  AC B.AB  AB C.AD  AD D.CB  BC ___ ___ ___ ___ ___ ___ Example 1 CYP

16. SSS on the Coordinate Plane Triangle DVW has vertices D(–5, –1), V(–1, –2), and W(–7, –4). Triangle LPM has vertices L(1, –5), P(2, –1), and M(4, –7).a. Graph both triangles on the same coordinate plane.b. Use your graph to make a conjecture as to whether the triangles are congruent. Explain your reasoning.c. Write a logical argument that uses coordinate geometry to support the conjecture you made in part b. Example 2A

17. Read the ItemYou are asked to do three things in this problem. In part a, you are to graph ΔDVW and ΔLPM on the same coordinate plane. In part b, you should make a conjecture that ΔDVW ΔLPM or ΔDVW ΔLPM based on your graph. Finally, in part c, you are asked to prove your conjecture. / SSS on the Coordinate Plane Solve the Itema. Graph both triangles on the same coordinate plane. Example 2B

18. SSS on the Coordinate Plane b. From the graph, it appears that the triangles have the same shapes, so we conjecture that they are congruent. c. Use the Distance Formula to show all corresponding sides have the same measure. Example 2C

19. SSS on the Coordinate Plane Example 2C

20. SSS on the Coordinate Plane Answer:WD = ML, DV = LP, and VW = PM. By definition of congruent segments, all corresponding segments are congruent. Therefore, ΔDVW  ΔLPM by SSS. Example 2 ANS

21. Determine whether ΔABCΔDEFfor A(–5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1). A. yes B. no C. cannot be determined Example 2A

22. Concept 2

23. ENTOMOLOGY The wings of one type of moth form two triangles. Write a two-column proof to prove that ΔFEG ΔHIG if EI FH, and G is the midpoint of both EI and FH. Use SAS to Prove Triangles are Congruent Example 3

24. Given:EIFH; G is the midpoint of both EI and FH. Proof: Statements Reasons 1. EI FH; G is the midpoint ofEI; G is the midpoint of FH. 1. Given 2. 2. Midpoint Theorem 3. FGE  HGI 4. ΔFEG ΔHIG 3. Vertical Angles Theorem 4. SAS Use SAS to Prove Triangles are Congruent Prove:ΔFEGΔHIG Example 3

25. The two-column proof is shown to prove that ΔABG ΔCGB if ABG  CGB and AB  CG. Choose the best reason to fill in the blank. Proof: Statements Reasons 1. 1. Given 2. ? Property 2. 3. SSS 3.ΔABGΔCGB A. Reflexive B. Symmetric C. Transitive D. Substitution Example 3

26. Use SAS or SSS in Proofs Write a paragraph proof. Prove: Q  S Example 4

27. Use SAS or SSS in Proofs Answer: Example 4

28. Choose the correct reason to complete the following flow proof. A. Segment Addition Postulate B. Symmetric Property C. Midpoint Theorem D. Substitution Example 4

29. LESSON 4–4 Proving Triangles Congruent – SSS, SAS