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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.

Figure 15.1The trapezoid ABCD.

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.

Figure 15.2R and S are the midpoints of AB and CD, and RS is the median of trapezoid ABCD.

  • 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.

Figure 15.3Trapezoid ABCD, BC ? ? AD AB has midpoint R and CD has midpoint S.

  • 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 = 5/2x - 5/2
  • 4x ? 5/2x = 25 - 5/2
  • 3/2x = 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.

Figure 15.4The trapezoid ABCD, with altitude BT.

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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