Diagonals of an isosceles trapezoid - Examples, Exercises and Solutions

Come discover the 4 properties of the diagonals of an 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 ABCD it holds that:

ADC=BCD⊿ADC=⊿BCD

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 ABCD it holds that:

ADC=BCD⊿ADC=⊿BCD

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


The third property

The diagonals of an isosceles trapezoid create, along with the bases,4 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⊿AEB and DEC⊿DEC.
We will mark in orange the triangles that are created from the diagonals and the bases:

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

Practice Diagonals of an isosceles trapezoid

Exercise #1

Given: C=2x ∢C=2x

A=120° ∢A=120°

isosceles trapezoid.

Find x.

AAABBBDDDCCC120°2x

Video Solution

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 ∢C=∢D

A=B ∢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 ∢A+∢B+∢C+∢D=360

We replace according to the existing data:

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

 240+4x=360 240+4x=360

4x=360240 4x=360-240

4x=120 4x=120

We divide the two sections by 4:

4x4=1204 \frac{4x}{4}=\frac{120}{4}

x=30 x=30

Answer

30°

Exercise #2

True OR False:

In all isosceles trapezoids the bases are equal.

Video Solution

Step-by-Step Solution

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

Answer

True

Exercise #3

The perimeter of the trapezoid equals 22 cm.

AB = 7 cm

AC = 3 cm

BD = 3 cm

What is the length of side CD?

AAABBBDDDCCC733

Video Solution

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=3+3+7+CD

22=CD+13 22=CD+13

2213=CD 22-13=CD

9=CD 9=CD

Answer

9

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.

 

 

Answer

No

Exercise #5

In an isosceles trapezoid ABCD

B=3x ∢B=3x

D=x ∢D=x


Calculate the size of angle B ∢B .

Video Solution

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!

Answer

135°

Exercise #1

Below is an isosceles trapezoid.

B=2y+20 ∢B=2y+20

D=60 ∢D=60

Find B ∢B .

AAABBBDDDCCC2y+2060°

Video Solution

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!

Answer

120°

Exercise #2

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.

777h=5h=5h=5AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Since ABCD is a trapezoid, it can be argued that:

AD=BC=7 AD=BC=7

The formula to find the area will be

SABCD=(AB+DC)×h2 S_{ABCD}=\frac{(AB+DC)\times h}{2}

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

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

34=14+AB+DC 34=14+AB+DC

3414=AB+DC 34-14=AB+DC

20=AB+DC 20=AB+DC

Now we will place the data we received in the formula to calculate the area of the trapezoid:

S=20×52=1002=50 S=\frac{20\times5}{2}=\frac{100}{2}=50

Answer

50

Exercise #3

D=50° ∢D=50°

The isosceles trapezoid

What is B ∢B ?

AAABBBDDDCCC50°

Video Solution

Answer

130°

Exercise #4

Given: A=120° ∢A=120°

The isosceles trapezoid

Find a: C ∢C

AAABBBDDDCCC120°

Video Solution

Answer

60°

Exercise #5

Look at the polygon in the diagram.

What type of shape is it?

1354578

Video Solution

Answer

Trapezoid

Exercise #1

¿Los triángulos marcados son isósceles?

(ΔABE,ΔCED)  (ΔABE,ΔCED)\text{ }

AAABBBDDDCCCEEE

Video Solution

Answer

True

Exercise #2

Given: A=y+20 ∢A=y+20

D=50 ∢D=50

trapecio isósceles.

Find a A ∢A

AAABBBDDDCCCy+2050°

Video Solution

Answer

130

Exercise #3

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.

AAABBBDDDCCCX1515X+5

Video Solution

Answer

20

Exercise #4

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.

AAABBBDDDCCC1015

Video Solution

Answer

10

Exercise #5

The trapezoid ABCD is isosceles.

AD = AE

Calculate angle α \alpha .

7267ABCDE

Video Solution

Answer

A,B=110.5 | C,D=69.5 | α=110.5 \alpha=110.5

Topics learned in later sections

  1. Trapezoids
  2. Types of trapezoids
  3. Isosceles Trapezoid