Congruence Proofs: Corresponding Parts of Congruent Triangles

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  • 0:01 CPCTC
  • 0:59 Practice Problem #1
  • 2:19 Practice Problem #2
  • 3:36 Practice Problem #3
  • 4:48 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Congruent triangles have congruent sides and angles, and the sides and angles of one triangle correspond to their twins in the other. In this lesson, we'll try practice with some geometric proofs based around this theorem.


I don't have a twin, but I bet there are fun things about having one. It's like having a spare you. Wish you had a friend nearby? You got one! Need a shirt? Your twin wears the same size! Need a kidney? Hey, brother...

This is the joy of congruent triangles. Once we determine triangles are congruent, we know they're twins. They have the same sides and the same angles. This allows us to use a fun theorem: CPCTC, or corresponding parts of congruent triangles are congruent.

CPCTC doesn't tell us the triangles are congruent. But once we've established their congruency, CPCTC is our reason for explaining why matching angles or sides are congruent. It's like having a rule that doesn't say this guy's my twin, but once I know that, I can totally borrow his shoes and know they'll fit. Thanks, brother!

Practice Problem #1

Let's see CPCTC in action. Here are two triangles.

two triangles, ABC and DEF

They sure look like twins, but we're not quite sure. They're wearing matching outfits, which might be a tip-off. Let's say we're given that AB is congruent to DE, AC is congruent to DF and BC is congruent to EF. Can we prove that angle B is congruent to angle E?

Let's set up a proof. We have our statements on the left and our reasons on the right. Let's start with AB is congruent to DE. Why? It's given. And AC is congruent to DF. That's also given. And BC is congruent to EF. Again, that's given. Now we can say that triangle ABC is congruent to triangle DEF.

Why? That's the SSS postulate, or the side-side-side postulate. If three sides of one triangle are congruent to three sides of another triangle, then those triangles are congruent. So they are twins! Since the triangles are twins, or congruent, we can say angle B is congruent to angle E using CPCTC. And that's it!

Practice Problem #2

Let's try a trickier one. Here's a bowtie.

If a bowtie was split into two triangles
bowtie triangles

Bowties are cool, sure, but what else? Does this bowtie make twin triangles? More importantly, are twins in bowties extra awesome? Let's say we're given that AB is parallel to CD and that E is the midpoint of AD. Can we prove that BE is congruent to CE?

To the proof! Well, first of all, AB is parallel to CD. That's given. And angle A is congruent to angle D because they're alternate interior angles. We're given that E is the midpoint of AD. That means that AE is congruent to ED. That's the definition of a midpoint.

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