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 an**isosceles trapezoid.** Figure

**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* 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* AB *and

*,*CD

*E*the midpoint of

*, and*AD

*F*the midpoint of

*, by*BC

*Theorem 55:*

**Figure 3 **A trapezoid with its median.

**Example 1:** In Figure 4*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*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: