# Geometry: Properties of Trapezoids

## Properties of Trapezoids

A *trapezoid* is a quadrilateral with exactly two parallel sides. Figure 15.1 shows trapezoid ABCD. Remember the naming conventions for polygons. You must list the vertices in consecutive order. In trapezoid ABCD, ¯BC ¯AD. The parallel sides ¯BC and ¯AD are called the *bases*, and the nonparallel sides ¯AB and ¯CD are *legs*. *Base angles* are a pair of angles that share a common base. In Figure 15.1, ∠A and ∠D form one set of base angles.

When the midpoints of the two legs of a trapezoid are joined together, the resulting segment is called the *median* of the trapezoid. In Figure 15.2, R and S are the midpoints of ¯AB and ¯CD, and ¯RS is the median of trapezoid ABCD. The median of a trapezoid is parallel to each base. Strangely enough, the length of the median of a trapezoid equals one-half the sum of the lengths of the two bases. Accept these statements as theorems (without proof), and use them when needed.

**Theorem 15.1**: The median of a trapezoid is parallel to each base.**Theorem 15.2**: The length of the median of a trapezoid equals one-half the sum of the lengths of the two bases.**Example 1**: In trapezoid ABCD, ¯BC ¯AD, R is the midpoint of ¯AB and S is the midpoint of ¯CD, as shown in Figure 15.3. Find AD, BC, and RS if BC = 2x, RX = 4x − 25 and AD = 3x − 5.

**Solution**: Because RS =^{1}/_{2}(AD + BC), you can substitute the values for each segment length:- 4x − 25 =
^{1}/_{2}(3x − 5 + 2x) - Rearranging and simplifying gives:
- 4x − 25 =
^{5}/_{2}x -^{5}/_{2} - 4x −
^{5}/_{2}x = 25 -^{5}/_{2} ^{3}/_{2}x =^{45}/_{2}- x = 15
- So, x = 15, BC = 30, RS = 35, and AD = 40.

An *altitude* of a trapezoid is a perpendicular line segment from a vertex of one base to the other base (or to an extension of that base). In Figure 15.4, ¯BT is an altitude of trapezoid ABCD.

##### Solid Facts

A **trapezoid** is a quadrilateral with exactly two parallel sides.

The **bases** of a trapezoid are the parallel sides.

The **legs** of a trapezoid are the nonparallel sides.

The **median** of a trapezoid is the line segment joining the midpoints of the two legs.

An **altitude** of a trapezoid is a perpendicular line segment from a vertex of one base to the other base (or to an extension of that base).

**Base angles** of a trapezoid are a pair of angles that share a common base.

Built into the trapezoid are two parallel lines (the bases ¯BC and ¯AD) cut by a transversal (one of the legs, either ¯AB or ¯CD). You know that the two interior angles on the same side of the transversal are supplementary angles (Theorem 10.5), so ∠A and ∠B are supplementary angles, as are ∠C and ∠D.

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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