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Geometry: High School15 chapters | 160 lessons

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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!

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

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!

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

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.

Next, angle *AEB* is congruent to angle *CED*. They're vertical angles, which are always congruent. So we've got angle-included side-angle, and that's the ASA postulate. So triangle *AEB* is congruent to triangle *DEC*. That's a relief. If they weren't congruent, our bowtie would be uneven. Fashion faux pas! Anyway, now we can say that *BE* is congruent to *CE* because of CPCTC. We did it! Let's try another.

Here's a four-sided shape.

Let's say it undergoes mitosis. Wait, when did this turn into a biology lesson? With this added line, it's divided into two triangles.

All we're told is that *AB* is parallel to *DC* and *AD* is parallel to *BC*. So pre-mitosis, it's a parallelogram. We need to prove that angle *A* is congruent to angle *C*.

First, *AB* is parallel to *DC*. That means that angle *ABD* is congruent to *BDC*; they're alternate interior angles. We know that *BD* is congruent to *BD* because of the reflexive property. That's the fancy way of saying I'm congruent to me because I'm me.

We're given that *AD* is parallel to *BC*. And then angle *ADB* is congruent to angle *DBC*. Again, because they're alternate interior angles. So, angle-side-angle again. That's the ASA postulate. So triangle *ABD* is congruent to triangle *CDB*. Therefore, angle *A* is congruent to angle *C* because of, wait for it, CPCTC.

In summary, we learned about **CPCTC**. This acronym stands for corresponding parts of congruent triangles are congruent. We practiced this with a few proofs. In each problem, we use the given information to determine that our triangles were congruent. Once that was established, we just needed to find the corresponding angles or sides of the triangles and we knew they must be congruent.

When you're done studying this lesson, you may be able to:

- Dissect the acronym,
*CPCTC* - Use this theorem to prove triangle congruency when solving practice problems

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Geometry: High School15 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Similarity Transformations in Corresponding Figures 7:28
- How to Prove Relationships in Figures using Congruence & Similarity 5:14
- Practice Proving Relationships using Congruence & Similarity 6:16
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Go to High School Geometry: Triangles, Theorems and Proofs

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