Properties of a Trapezoid

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.

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 legs are supplemnetary.

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

Find the values of x and y. The measures of the interior angles: 60, x + 40, 8y, 108.

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 [8y] + [108] = 180.

Move +108 to the right side.

Then 8y = 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.

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

The median and the bases are parallel. [median] = (1/2)[b1 + b2]. b1, b2: Bases of a trapezoid.

There are two properties
of the median of a trapezoid.

1. The median and the bases are parallel.

2. [median] = (1/2)[b1 + b2]

b1, b2: Bases of a trapezoid.

Example 2

Find the value of x. Bases: 3, 7. Median: x.

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

Find the value of x. Bases: x, 12. Median: 9.

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.