Isosceles Trapezoid: Property

How to use the properties of an isosceles trapezoid to solve the related problems: definition, 2 properties (angles, diagonals), 2 examples, and their solutions.

Definition

Definition

An isosceles trapezoid is a trapezoid
whose legs are congruent.

Just like an isosceles triangle,
its base angles are also congruent.

An isosceles trapezoid is also a trapezoid.

So an isosceles trapezoid
has all the properties of a trapezoid.

Property: Angles

Property

For an isosceles trapezoid,
two interior angles
that inscribe the same base
are congruent.

m∠1 = m∠1'
m∠2 = m∠2'

And two interior angles
that inscribe the same leg
are supplementary.

m∠1 + m∠2 = 180
m∠1' + m∠2' = 180

(This is also true for a trapezoid.)

Example 1

Example

Solution

The given quadrilateral is an isosceles trapezoid.

∠A and ∠D are the interior angles
that inscribe the same left leg.

m∠D = 60

So m∠A + [60] = 180.

Move +60 to the right side.

Then m∠A = 120.

Write 120º on ∠A.

∠A and ∠B are the interior angles
that inscribe the same top base.

∠A is 120º.

So ∠B is 120º.

∠D and ∠C are the interior angles
that inscribe the same bottom base.

∠D is 60º.

So ∠C is 60º.

Write m∠A, m∠B, and m∠C.

So
m∠A = 120
m∠B = 120
m∠C = 60
is the answer.

Property: Diagonals

Property

For an isosceles trapezoid,
four segments are formed by the diagonals.

The top two segments are congruent (blue).
And the bottom two segments are congruent. (brown)

Example 2

Example

Solution

The given quadrilateral is an isosceles trapezoid.

By the diagonals AC and BD,
four segments are formed.

The top two segments are congruent:
AP = BP = 6.

The bottom two segments are congruent:
PD = PC = 11.

See BD.

BP = 6
PD = 11

So BD = 6 + 11.

6 + 11 = 17

So BD = 17.