Proofs for Isosceles Trapezoids

Proofs for Isosceles Trapezoids
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  • 0:04 Isosceles Trapezoids
  • 0:50 Base Angles
  • 2:47 Diagonals
  • 3:41 Opposite Angles
  • 4:51 Lesson Summary
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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.


Isosceles Trapezoid
Isosceles Trapezoid


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.


Base angles are equal


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


Parallelogram ADCE
Parallelogram ADCE


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.


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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.


null


So,


null


Therefore, the angles DAB and CBA are equal.


null


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


null


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


null


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


null


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.


Diagonals of an isosceles trapezoid
Diagonals of an isosceles trapezoid


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