Geometry 4.2 - Trapezoids

A trapezoid is a quadrilateral with only one pair of parallel sides. As shown in the picture below, the parallel sides of trapezoid ABCD are called bases and the sides that are not parallel are called legs.

In trapezoid ABCD:

Quadrilateral -- Trapezoid

(1) The degree measure of the four angles add up to 360 degrees. This is true of any quadrilateral.

Let lower case letters a, b, c and d = the angles of trapezoid ABCD.
Then: a + b + c + d = 360 degrees.

(2) The lower and upper base angles are supplementary (add up to 180 degrees).

angle a + angle b = 180 degrees
angle c + angle d = 180 degrees

Isosceles Trapezoid

There is a special kind of trapezoid called an isosceles trapezoid. An isosceles trapezoid is a trapezoid in which the legs are equal. You'll notice that in the first trapezoid in this lesson (above), the legs are NOT equal.

In isosceles trapezoid ABCD:

Isosceles Trapezoid

(1) The two lower base angles have the same measure, and the two upper base angles have the same measure.

angle a = angle d
angle b = angle c

(2) The diagonals have the same lengths.

diagonal AC = diagonal BD

Sample:

In isosceles trapezoid MATH, side HT is parallel to side MA, line segment MH is congruent to line segment AT. The degree measure of angle MHT = 60 degrees. What are the measures of the other 3 angles?

Sample Trapezoid

Solution:

We know the two legs are congruent, so this is an isosceles trapezoid. Given that, we know the two base angles have the same measure. That tells us that

Angle T = 60

Since the top and bottom angles are supplementary, we know that
Angle M = 180 - 60
Angle M = 120

By the same logic, Angle A = 120 degrees as well.