4.5 Using Congruent Triangles Unit 1B Day 11

Do Now What is the definition of triangle congruence?

Corresponding Parts Recall the fact that if two figures are congruent, then their ___________________ parts are congruent. We sum up this idea with the phrases “_____________________________________________,” abbreviated _____________. We can use it to prove things about congruent triangles. corresponding corresponding parts of congruent triangles are congruent CPCTC

Ex. 1: Planning a proof Suppose you want to prove that PQS ≅ RQS in the diagram shown at the right. Plan: Show that ∆PQS ≅ ∆RQS by ________. Use ___________ to conclude that PQS ≅ RQS. SSS CPCTC

Ex. 2: Corresponding Parts Given: AB ║ CD, BC ║ DA Prove: AB ≅ CD Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent. two pairs of parallel lines — given BD = DB — reflexive <ADB = <CBD — if lines (AD and BC) ||, alt. int. <‘s congruent <ABD = <CDB — if lines (BA and CD) ||, alt. int. <‘s congruent ΔABD = ΔCDB — ASA congruence post. AB = CD — CPCTC

Ex. 3: Proving Lines Parallel Given: A is the midpoint of MT; A is the midpoint of SR. Prove: MS ║TR. Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that M ≅ T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR. … — given MA = TA; SA = RA — definition of midpoint <MAS = <TAR — vertical angles are congruent ΔMAS = ΔTAR — SAS <M = <T — CPCTC MS || TR — if alt. int. <‘s congruent, lines ||

Ex. 4: Using more than one pair of triangles. Given: 1 ≅ 2, 3 ≅ 4. Prove ∆BCE ≅ ∆DCE The only information you have about ∆BCE and ∆DCE is that 1≅2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE. … — given AC = AC — reflexive prop. ΔABC = ΔADC — ASA congruence post. BC = DC — CPCTC CE = CE — reflexive prop. ΔBCD = ΔDCE — SAS congruence post.

Ex. 5: Corresponding Parts Plan: Prove ΔSCT = ΔTDS by SSS. Then prove <SCT = <TDS by CPCTC . … — given ST = TS — reflexive prop. ΔSCT = ΔTDS — SSS congruence post. <SCT = <TDS — CPCTC

Corresponding Parts Use the marked diagram to state the method used to prove the triangles congruent. Name the additional corresponding parts that could then be concluded to be congruent.

Closure What does CPCTC stand for? How do you use it?