LESSON 4 4 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 (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

Over Lesson 4– 3 Write a congruence statement for the triangles. A. ΔLMN ΔRTS B. ΔLMN ΔSTR C. ΔLMN ΔRST D. ΔLMN ΔTRS

Over Lesson 4– 3 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

Over Lesson 4– 3 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

Over Lesson 4– 3 Refer to the figure. Find x. A. 1 B. 2 C. 3 D. 4

Over Lesson 4– 3 Refer to the figure. Find m A. A. 30 B. 39 C. 59 D. 63

Over Lesson 4– 3 Given that ΔABC ΔDEF, which of the following statements is true? A. A E B. C D ___ ___ C. AB DE D. BC FD

Targeted TEKS G. 6(B) Prove two triangles are congruent by applying the Side. Angle-Side, Angle-Side-Angle, 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)

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.

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

Use SSS to Prove Triangles Congruent Answer: Flow Proof:

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

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.

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.

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.

SSS on the Coordinate Plane

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.

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

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.

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

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

Use SAS or SSS in Proofs Write a paragraph proof. Prove: Q S

Use SAS or SSS in Proofs Answer:

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

LESSON 4– 4 Proving Triangles Congruent – SSS, SAS