Login
Copyright

Congruency of Isosceles Triangles: Proving the Theorem

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Angles Formed by a Transversal

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:01 If, Then
  • 1:23 Theorem Proof
  • 2:32 Converse Proof
  • 4:24 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Create an account to start this course today
Try it free for 5 days!
Create An Account

Recommended Lessons and Courses for You

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.

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.

Triangle XYZ with angle bisector B

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

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher
What is your educational goal?
 Back

Unlock Your Education

See for yourself why 10 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account
Support