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.

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.