1. 4.5 Using Congruent Triangles
Unit 1B Day 11

2. Do Now
What is the definition of triangle congruence?

3. 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

4. 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

5. 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

6. 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 ||

7. 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.

8. 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

9. 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.

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