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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.

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.

Here's an 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?

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

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.

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

We could say 'if I race a tortoise, I'll always win the race.' That's probably true, especially since I learned something from that hare about not underestimating our tortoise friend. But the converse of that statement is 'if I win the race, then I raced the tortoise.' That's not necessarily true, right? Maybe I can outrace all sorts of slow animals, like three-toed sloths.

Let's consider the converse of our triangle theorem. That would be 'if two angles of a triangle are congruent, then the sides opposite these angles are also congruent.'

Okay, here's triangle XYZ.

We know that angle Y is congruent to angle Z. Can we prove that XY is congruent to XZ?

Again, let's start by stating what we know. Angle Y is congruent to angle Z. Now let's add an angle bisector from X to YZ. If we add point B, we can call this line XB. An angle bisector line divides the angle into two equal parts. So we can state that angle YXB is congruent to angle ZXB. Next, let's state that XB is congruent to XB. That's the reflexive property.

Now we can state that triangle YXB is congruent to triangle ZXB. Why? This is the angle-angle-side theorem, or AAS. If you don't remember AAS, you can determine that our final two angles, XBY and XBZ, are congruent since they're the last two remaining angles of the triangles.

But let's use AAS. And now we can state that XY is congruent to XZ because of CPCTC. Therefore, the converse of our theorem is also true. That's like finding out that the only animal I can outrace is a tortoise. Did I get bested by a sloth? Maybe I did.

In summary, we proved two 'if, then' statements that relate to isosceles triangles. We proved the theorem that states that *if two sides of a triangle are congruent, then the angles opposite these sides are also congruent*. We also proved its converse, which states that *if two angles of a triangle are congruent, then the sides opposite these angles are also congruent*.

Finishing this video lesson could heighten your ability to do the following:

- Illustrate an isosceles triangle
- Prove the congruency of isosceles triangles theorem and its inverse
- Use the angle-side-angle theorem

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Lesson
14 in chapter 5 of the course:

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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
- Converse of a Statement: Explanation and Example 5:09
- 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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