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 prove a certain argument in the exam.
The properties of the diagonals are very logical and besides, we are here to sort out the mess.

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.

Similarly, to remember the theorem by its logic, it is advisable to understand its underlying reasoning:
Let's look at the trapezoid $ABCD$ :

Diagonals of an Isosceles Trapezoid

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

2)  $AD=BC$

We have to prove: $AC=BD$

Solution:
Let's observe this trapezoid.
Given the trapezoid and also that $AD=BC$.

That is, 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 triangle 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 created by the diagonals:
$⊿ADC$ and $⊿BCD$

In fact, we have seen that we can superimpose the triangles that have been created from the diagonals 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

We must prove this property again and again in the exam. 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 example, 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.
We must also prove this property over and over again in the exam 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 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.
We will also need to prove this property over and over again in the exam 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 $⊿DEC$

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