# Diagonals of an isosceles trapezoid - Examples, Exercises and Solutions

## 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.

## Examples with solutions for Diagonals of an isosceles trapezoid

### Exercise #1

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 #2

$∢D=50°$

The isosceles trapezoid

What is $∢B$?

### Step-by-Step Solution

Let's recall that in an isosceles trapezoid, the sum of the two angles on each of the trapezoid's legs equals 180 degrees.

In other words:

$A+C=180$

$B+D=180$

Since angle D is known to us, we can calculate:

$180-50=B$

$130=B$

130°

### Exercise #3

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 #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:

$22=3+3+7+CD$

$22=CD+13$

$22-13=CD$

$9=CD$

9

### Exercise #6

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 #7

Below is an isosceles trapezoid.

$∢B=2y+20$

$∢D=60$

Find $∢B$.

### Step-by-Step Solution

To answer the exercise, certain information is needed:

1. In a quadrilateral the sum of the interior angles is 180.

2. The isosceles trapezoid has equal angles.

3. From here it is we know that the sum of the angles adjacent to a side of the trapezoid is 180°.

We turn this conclusion into an exercise:

2y+20+60=180

We add up the relevant angles

2y+80=180

We move the sections:

2y=180-80

2y=100

Divided by 2

y=50

When we substitute Y we get:

2(50)+20=120

And this is the solution!

120°

### Exercise #8

Shown below is the isosceles trapezoid ABCD.

Given in cm:
BC = 7

Height of the trapezoid (h) = 5

Perimeter of the trapezoid (P) = 34

Calculate the area of the trapezoid.

### Step-by-Step Solution

Since ABCD is a trapezoid, one can determine that:

$AD=BC=7$

Thus the formula to find the area will be

$S_{ABCD}=\frac{(AB+DC)\times h}{2}$

Since we are given the perimeter of the trapezoid, we can find$AB+DC$

$P_{ABCD}=7+AB+7+DC$

$34=14+AB+DC$

$34-14=AB+DC$

$20=AB+DC$

Now we will place the data we obtained into the formula in order to calculate the area of the trapezoid:

$S=\frac{20\times5}{2}=\frac{100}{2}=50$

50

### Exercise #9

Given: $∢A=120°$

The isosceles trapezoid

Find a: $∢C$

60°

### Exercise #10

Look at the polygon in the diagram.

What type of shape is it?

Trapezoid

### Exercise #11

$(ΔABE,ΔCED)\text{ }$

True

### Exercise #12

Given: $∢A=y+20$

$∢D=50$

trapecio isósceles.

Find a $∢A$

130

### Exercise #13

Given that: the perimeter of the trapezoid is equal to 75 cm

AB= X cm

AC= 15 cm

BD= 15 cm

CD= X+5 cm

Find the size of AB.

20

### Exercise #14

Given that: the perimeter of the trapezoid is equal to 35 cm

AB= 10 cm

CD= 15 cm

The isosceles trapezoid

Find the sum of the sizes of the sides.

10

### Exercise #15

The trapezoid ABCD is isosceles.

Calculate angle $\alpha$.
A,B=110.5 | C,D=69.5 | $\alpha=110.5$