Diagonals of an isosceles trapezoid

The diagonals of an isosceles trapezoid are very special and have $4$ main properties that will help us a lot when we want to solve problems and prove geometric properties on exams. The properties of the diagonals are very logical, and we are here to make them clear and easy to understand.

Shall we start?

The diagonals of an isosceles trapezoid have the same length.
This theorem also holds true in reverse, that is, we can determine that a certain trapezoid is isosceles if we know that its diagonals are equal.
You can use this theorem in the exam as you see it and you will not have to prove it.

However, understanding why this property is true helps you remember it. Let's look at the trapezoid $ABCD$ :

Diagonals of an Isosceles Trapezoid

The only data we have are:
1) ABCD is a trapezoid with
$AB∥DC$
$AD∦BC$

2)  $AD=BC$

We have to prove: $AC=BD$

Solution:
Given that $AD=BC$, we can deduce that the trapezoid is isosceles.

In the exam, you will already be able to deduce that $AC=BD$ since the diagonals of an isosceles trapezoid have the same length.
But, to give an example, let's see how this theorem is proven.
We can prove it based on the congruence of two triangles formed by the diagonals and the sides of the trapezoid:
$⊿ADC$ and $⊿BCD$

In fact, we have seen that we can superimpose the triangles that have been formed by the diagonals with the base and non-parallel sides, and in this way demonstrate the first property of the diagonals of an isosceles trapezoid.

In the isosceles trapezoid $ABCD$ the following is true:
$⊿ADC≅⊿BCD$

Diagonals of an Isosceles Trapezoid

You may need to prove this property on exams. But, do not worry, we can do it easily and quickly by superimposing the triangles according to SSS.
Observe: It is true that we have proven the congruence of these triangles in the first property, but now we will do it based on the first property of the diagonals of an isosceles trapezoid.
Let's see the proof:

The diagonals of an isosceles trapezoid create, along with the bases, $4$ equal angles.
You may need to prove this property on exams to use it We know that this property might seem a bit complicated at first glance, so we think it's very important that you start by observing in the illustration which angles we are talking about:
We will mark all the angles that are between a diagonal and the base, as follows:

Isosceles Triangles in a Trapezoid

Great! Now that we know which angles we are dealing with, we can demonstrate that they are equal based on the congruence of the triangles we made above. That is, congruence between 
$⊿ADC≅⊿BCD$
Observe- To demonstrate this property, you should know what the corresponding angles and alternate interior angles are.
After having demonstrated the congruence of the triangles, we can claim that:

The diagonals of an isosceles trapezoid create, along with the bases, two isosceles triangles. You may need to prove this property on exams to use it.
First, let's see which triangles we are talking about.
We will mark in orange the triangles that are created with the diagonals and the bases:

Note that these are $⊿AEB$ and $⊿DEC$, where E is the intersection point of the diagonals.

In the previous property, we have proven that the green angles that arise from the diagonals of the isosceles trapezoid and the bases are equal .
We already know that the in a triangle, sides opposite equal angles have equal length Therefore, the demonstration will look as follows:
After we prove that the green angles are equal, we can claim that:

Great! Now you know the important properties of the diagonals of an isosceles trapezoid.
You will be able to prove them without any problem and use them as arguments whenever you need to demonstrate something.