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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 an isosceles trapezoid. Figure 1 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, mDAB = mCBA, and mADC = mBCD.

  • 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 mABC and find BD.





Figure 4

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


mABC = 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:




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