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.
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: