# Proofs for Isosceles Trapezoids

Coming up next: Proofs for Triangles

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

Replay
Your next lesson will play in 10 seconds
• 0:04 Isosceles Trapezoids
• 0:50 Base Angles
• 2:47 Diagonals
• 3:41 Opposite Angles
• 4:51 Lesson Summary

Want to watch this again later?

Timeline
Autoplay
Autoplay
Speed

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, you will learn what an isosceles trapezoid is and the important theorems related to it. You will also understand the proofs of these theorems with clear steps.

## Isosceles Trapezoids

Irene has just bought a house and is very excited about the backyard. She's a bit of math nerd, and plans to create a garden in the shape of an isosceles trapezoid. She paints the lawn white where her future raised garden bed will be.

In geometry, a trapezoid is a quadrilateral that has at least one pair of parallel sides. A trapezoid in which non-parallel sides are equal is called an isosceles trapezoid.

ABCD is an isosceles trapezoid with AB parallel to CD (we write it like this in math: AB||CD), and line AD is equal to line BC.

But how can Irene prove her garden is an isosceles trapezoid? Let's see some important theorems related to isosceles trapezoid to help her out.

## Base Angles

The angles formed between non-parallel sides and parallel sides, called base angles, are equal in an isosceles trapezoid. In Irene's lawn trapezoid ABCD, angles C and D are equal.

To prove this theorem, let's let's draw a line CE parallel to AD such that ADCE becomes a parallelogram.

In this parallelogram, we know that line AD = line CE. We also know that line AD = line BC, so we also know that line BC = line CE.

Now, as line BC and line CE are equal, the triangle BCE becomes an isosceles triangle. Therefore, angles CBE and CEB are equal.

We understand that line AD and line CE are parallel, and the line AE is the transversal. So, the sum of interior angles on the same side, angle DAE and angle CEA is 180 degrees.

So,

Therefore, the angles DAB and CBA are equal.

Next, we know that ADCE is a parallelogram, so the opposite angles would be equal.

Now, angles CBE and BCD would be equal because they are alternate interior angles for the parallel lines AE and CD.

We already know that the angles CEB and CEB are equal. Therefore,

Thus, it is proved that the base angles in an isosceles trapezoid are equal.

## Diagonals

Diagonals of an isosceles trapezoid are equal in length. So, in Irene's isosceles trapezoid ABCD, the diagonals AC and BD are equal.

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

### Register to view this lesson

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 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.