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

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

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

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

Name the corresponding congruent sides for the congruent triangles. 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 ___ 5-Minute Check 3

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

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

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

Mathematical Processes G.1(E), G.1(G) 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

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

included angle Vocabulary

Concept 1

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

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

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

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

Solve the Item a. Graph both triangles on SSS on the Coordinate Plane Read the Item You 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. / Solve the Item a. Graph both triangles on the same coordinate plane. Example 2B

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

SSS on the Coordinate Plane Example 2C

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

Determine whether ΔABC  ΔDEF for 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

Concept 2

Use SAS to Prove Triangles are Congruent 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. Example 3

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

A. Reflexive B. Symmetric C. Transitive D. Substitution 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. 1. Reasons Proof: Statements 1. Given 2. ? Property 2. 3. SSS 3. ΔABG ΔCGB A. Reflexive B. Symmetric C. Transitive D. Substitution Example 3

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

Use SAS or SSS in Proofs Answer: Example 4

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

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