Congruency of Isosceles Triangles: Proving the Theorem

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  • 0:01 If, Then
  • 1:23 Theorem Proof
  • 2:32 Converse Proof
  • 4:24 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.

Isosceles triangles have two equal sides. Are the base angles also equal? In this lesson, we'll prove how this is true. We'll also prove the theorem's converse.

If, Then

Here's an isosceles triangle.

Isosceles triangle

We know it's an isosceles triangle because it has two equal sides. That's the definition of an isosceles triangle. But if it's an isosceles triangle, what else can we prove?

Geometry is full of these if, then statements, just like life. Some of these are simple. If a shape has four equal sides and four right angles, then it's a square. That's like saying if you go swimming, you're going to get wet. There's really no ambiguity there.

With an isosceles triangle, there are some 'if, then' statements that seem logical, but we need to test them to be sure. It's like saying if you make guacamole, then it's going to be awesome. We can't be sure of this until you make some guacamole, right? Then we need to test it by sampling some. And maybe we aren't so sure with just one taste. Why don't we try a whole bowl? Then we'll know for sure.

Anyway, an isosceles triangle has parts we can label. We call the equal sides of isosceles triangles the legs. The third side is called the base. The angles across from the legs are called the base angles.

We know our triangle has equal sides, or legs, but let's try to prove a theorem. There's a theorem that states that if two sides of a triangle are congruent, then the angles opposite these sides are also congruent. Is this 'if, then' statement true?

Theorem Proof

Let's prove the theorem. Here's triangle ABC.

Isosceles triangle ABC with equal sides marked

We're given that AB is congruent to AC. It's 'isosceles-iness' is therefore established. We want to prove that angle B is congruent to angle C.

First, let's state what we know. AB is congruent to AC. That's given. Now, let's add a midpoint on BC and call it M and a line from A to M. This is a median line.

We can then state that BM is congruent to MC. Next, let's state that AM is congruent to AM because of the reflexive property, also known as, well, it's the same line.

AB and AC, BM and MC, and AM and AM. That's three sides of the two triangles formed when we added the median. So triangle ABM is congruent to triangle ACM because of the side-side-side postulate.

That allows us to state that angle B is congruent to angle C because corresponding parts of congruent triangles are congruent, or CPCTC.

So our theorem is true! That's almost as satisfying as figuring out that your guacamole is awesome.

Converse Proof

We proved our theorem, but what about its converse? The converse of an 'if, then' statement is tricky.

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