Properties of Trapezoids

Recall that a trapezoid is a quadrilateral with only one pair of opposite sides parallel and that the parallel sides are called bases and the nonparallel sides are called legs. If the legs of a trapezoid are equal, it is called anisosceles trapezoid. Figure is an isosceles trapezoid.

Figure 1 An isosceles trapezoid.

A pair of angles that share the same base are called base angles of the trapezoid. In Figure 1, ∠ A and ∠ B or ∠ C and ∠ D are base angles of trapezoid ABCD. Two special properties of an isosceles trapezoid can be proven.

Theorem 53: Base angles of an isosceles trapezoid are equal.

Theorem 54: Diagonals of an isosceles trapezoid are equal.

In isosceles trapezoid ABCD (Figure 2) with bases AB and CD :

  • By Theorem 53, m ∠ DAB = m ∠ CBA, and m ∠ ADC = m ∠ BCD.
  • By Theorem 54, AC = BD.

Figure 2 An isosceles trapezoid with its diagonals.

Recall that the median of a trapezoid is a segment that joins the midpoints of the nonparallel sides.

Theorem 55: The median of any trapezoid has two properties: (1) It is parallel to both bases. (2) Its length equals half the sum of the base lengths.

In trapezoid ABCD (Figure 3) with bases AB and CD E the midpoint of AD , and F the midpoint of BC , by Theorem 55:

Figure 3 A trapezoid with its median.

Example 1: In Figure 4, find m ∠ ABC and find BD.

Figure 4 An isosceles trapezoid with a specified angle and a specified diagonal.

m ∠ ABC = 120°, because the base angles of an isosceles trapezoid are equal.

BD = 8, because diagonals of an isosceles trapezoid are equal.

Example 2: In Figure 5, find TU.

Figure 5 A trapezoid with its two bases given and the median to be computed.

Because the median of a trapezoid is half the sum of the lengths of the bases: