Diagonals of an isosceles trapezoid

🏆Practice isosceles trapezoid

The first property

The diagonals of an isosceles trapezoid have the same length.
This theorem also holds true in reverse, meaning, 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 need to prove it.

The second property:

Given the isosceles trapezoid $ABCD$ it holds that:

$⊿ADC=⊿BCD$

Diagonals of an Isosceles Trapezoid

Observe- You will need to prove this property in the exam. We can prove it based on the SSS congruence theorem.

The second property

Given the isosceles trapezoid $ABCD$ it holds that:

$⊿ADC=⊿BCD$

Angles of the Diagonals of an Isosceles Trapezoid

Observe- You will need to prove this property in the exam. We can prove it based on the congruence theorem $SSS$.

The third property

The diagonals of an isosceles trapezoid create, along with the bases,$4$ equal angles.
We will mark them.

Observe- You will need to prove this property on the exam.

The fourth property

The diagonals of an isosceles trapezoid create, along with the bases, two isosceles triangles.
We will notice that these are $⊿AEB$ and $⊿DEC$.
We will mark in orange the triangles that are created from the diagonals and the bases:

Isosceles Triangles in a Trapezoid

Observe- You will need to prove this property on the exam.

Test yourself on isosceles trapezoid!

$$∢D=50°$$

The isosceles trapezoid

What is $$∢B$$?

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 first property

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 $⊿BCD$

 Argument Explanation $AD=BC$ (Side) Dato $∢BCD=∢ADC$  (Angle) ABCD is an isosceles trapezoid. In an isosceles trapezoid, the base angles have the same measure. $DC=DC$  (Side) A shared side is equal to itself $⊿ADC=⊿BCD$ According to Side Angle Side Therefore- $AC=BD$ Equal sides in congruent triangles

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.

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The second property

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:

 Argument Explanation $AD=BC$ (Side) Given - In an isosceles trapezoid, the oblique sides have the same length. $AC=BD$ (Side) ABCD is an isosceles trapezoid. In an isosceles trapezoid, the diagonals are equal. $DC=DC$ (Side) A shared side is equal to itself $⊿ADC=⊿BCD$ According to Side Side Side

The third property

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:

 Argument Explanation $∢ACD=∢BDC$ Corresponding angles between congruent triangles are equivalent. $∢ACD=∢CAB$ Alternate angles between congruent triangles are equivalent. $∢BDC=∢DBA$ Alternate angles between congruent triangles are equivalent. $∢ACD=∢BDC=∢CAB=∢DBA$ Transitive Relation
Do you know what the answer is?

The fourth and last property

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:

 Argument Explanation $∢ACD=∢BDC$ Corresponding angles between congruent triangles are equivalent. This is proven with the third property. Therefore $ED=EC$ Opposite equivalent angles there are equivalent sides. Therefore $⊿DEC$ isosceles A triangle with equal sides is equilateral. $∢BAE=∢ABE$ It is verified with the third property. Therefore $AE=BE$ Opposite equivalent angles there are equivalent sides. Por lo tanto $⊿AEB$ isosceles A triangle with equal sides is equilateral.

Examples and exercises with solutions of diagonals of an isosceles trapezoid

Exercise #1

Given: $∢C=2x$

$∢A=120°$

isosceles trapezoid.

Find x.

Step-by-Step Solution

Given that the trapezoid is isosceles and the angles on both sides are equal, it can be argued that:

$∢C=∢D$

$∢A=∢B$

We know that the sum of the angles of a quadrilateral is 360 degrees.

Therefore we can create the formula:

$∢A+∢B+∢C+∢D=360$

We replace according to the existing data:

$120+120+2x+2x=360$

$240+4x=360$

$4x=360-240$

$4x=120$

We divide the two sections by 4:

$\frac{4x}{4}=\frac{120}{4}$

$x=30$

30°

Exercise #2

True OR False:

In all isosceles trapezoids the bases are equal.

Step-by-Step Solution

True: in every isosceles trapezoid the base angles are equal to each other.

True

Exercise #3

In an isosceles trapezoid ABCD

$∢B=3x$

$∢D=x$

Calculate the size of angle $∢B$.

Step-by-Step Solution

To answer the question, we must know an important rule about isosceles trapezoids:

The sum of the angles that define each of the trapezoidal sides (not the bases) is equal to 180

Therefore:

∢B+∢D=180

3X+X=180

4X=180

X=45

It's important to remember that this is still not the solution, because we were asked for angle B,

Therefore:

3*45 = 135

And this is the solution!

135°

Exercise #4

Do the diagonals of the trapezoid necessarily bisect each other?

Step-by-Step Solution

The diagonals of an isosceles trapezoid are always equal to each other,

but they do not necessarily bisect each other.

(Reminder, "bisect" means that they meet exactly in the middle, meaning they are cut into two equal parts, two halves)

For example, the following trapezoid ABCD, which is isosceles, is drawn.

Using a computer program we calculate the center of the two diagonals,

And we see that the center points are not G, but the points E and F.

This means that the diagonals do not bisect.

No

Exercise #5

The perimeter of the trapezoid equals 22 cm.

AB = 7 cm

AC = 3 cm

BD = 3 cm

What is the length of side CD?

Step-by-Step Solution

Since we are given the perimeter of the trapezoid and not the length of CD, we can calculate:

$22=3+3+7+CD$

$22=CD+13$

$22-13=CD$

$9=CD$