# Properties of a Trapezoid

How to solve the trapezoid problems by using its properties: definition, properties of its angles and median, examples, and their solutions.

## Definition

A trapezoid is a quadrilateral

that have a pair of parallel sides.

Base: Parallel sides

Legs: Non-parallel sides

## Property: Angles

For a trapezoid,

two interior angles that inscribe the leg

are supplementary:

m∠[blue] + m∠[green] = 180

Supplementary angles

This is true because

these two angles are

the consecutive interior angles in parallel lines.

Consecutive interior angles in parallel lines

## Example 1

The given figure is a trapezoid.

So the left two angles are supplementary.

So [*x* + 40] + [60] = 180.

40 + 60 = 100

Move +100 to the right side.

Then *x* = 80.

The right two angles are also supplementary.

So [8*y*] + [108] = 180.

Move +108 to the right side.

Then 8*y* = 72.

Divide both sides by 8.

Then *y* = 9.

So *x* = 80, *y* = 9 is the answer.

## Median of a Trapezoid

The median of a trapezoid

is a line segment

that connects the midpoints of the legs.

Its like the midsegment of a triangle.

Midsegment of a Triangle

## Property: Median

There are two properties

of the median of a trapezoid.

1. The median and the bases are parallel.

2. [median] = (1/2)[*b*_{1} + *b*_{2}]*b*_{1}, *b*_{2}: Bases of a trapezoid.

## Example 2

The given figure is a trapezoid.

And the blue segment is the median.

Median: *x*

Bases: 3, 7

So *x* = (1/2)⋅(3 + 7).

3 + 7 = 10

(1/2)⋅10 = 5

So *x* = 5.

## Example 3

The given figure is a trapezoid.

And the blue segment is the median.

Median: 9

Bases: *x*, 12

So 9 = (1/2)⋅(*x* + 12).

Switch both sides.

Multiply 2 on both sides.

Then *x* + 12 = 18

Move +12 to the right side.

Then *x* = 6.