and R Given: Quadrilateral PQST with midpoints A, B, C, and

Trapezoid Bases Legs Base Angles Median Isosceles Trapezoid The Trapezoid 4.4 KEY CONCEPTS A trapezoid is a quadrilateral with exactly two parallel sides. DEFINITION An altitude of a trapezoid is a line segment from one vertex of one base of the trape- zoid perpendicular to the opposite base or to an extension of that base. DEFINITION Figure 4.32 shows trapezoid HJKL, in which . The parallel sides and are bases, and the nonparallel sides and are legs. Because and both have for a side, they are a pair of base angles of the trapezoid; and are also a pair of base angles because is a base. When the midpoints of the two legs of a trapezoid are joined, the resulting line segment is known as the median of the trapezoid. Given that M and N are the midpoints of the legs and in trapezoid HJKL, is the median of the trapezoid. [See Figure 4.33a]. If the two legs of a trapezoid are congruent, the trapezoid is known as an isosceles trapezoid. In Figure 4.33b, RSTV is an isosceles trapezoid because and . RS ‘ VT RV ⬵ ST MN LK HJ HL ∠L ∠H JK ∠K ∠J LK HJ JK HL HL ‘ JK J K L H base base leg leg Figure 4.32 K J H L M N a median Figure 4.33 T V R S b K J H L c Every trapezoid contains two pairs of consecutive interior angles that are supple- mentary. Each of these pairs of angles is formed when parallel lines are cut by a trans- versal. In Figure 4.33c, angles H and J are supplementary, as are angles L and K. See the “Reminder” at the left. Reminder If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. EXAMPLE 1 In Figure 4.32, suppose that and . Find and . Solution Because , H and J are supplementary angles, as are L and K. Then and . Substitution leads to and , so and . m ∠L = 122° m ∠J = 73° m ∠L + 58 = 180 107 + m ∠J = 180 m ∠L + m∠K = 180 m ∠H + m∠J = 180 ∠s ∠s HL ‘ JK m ∠L m ∠J m ∠K = 58° m ∠H = 107° 쮿 In Figure 4.34, , , and are altitudes of trapezoid HJKL. The length of any altitude of HJKL is called the height of the trapezoid. KQ JP , LY HX Exs. 1–6 P H L Q J X Y K Figure 4.34 Discover Using construction paper, cut out two trapezoids that are copies of each other. To accomplish this, hold two pieces of paper together and cut once left and once right. Take the second trapezoid and turn it so that a pair of congruent legs coincide. What type of quadrilateral has been formed? Parallelogram ANSWER The preceding activity may provide insight for a number of theorems involving the trapezoid. The base angles of an isosceles trapezoid are congruent. THEOREM 4.4.1 EXAMPLE 2 Study the picture proof of Theorem 4.4.1. PICTURE PROOF OF THEOREM 4.4.1 GIVEN: Trapezoid RSTV with and [See Figure 4.35a]. PROVE: and PROOF: By drawing and we see that Theorem 4.1.6. By HL, so CPCTC. in Figure 4.35a because these angles are supplementary to congruent angles and . ∠T ∠V ∠R ⬵ ∠S ∠V ⬵ ∠T 䉭RYV ⬵ 䉭SZT RY ⬵ SZ SZ ⬜ VT , RY ⬜ VT ∠R ⬵ ∠S ∠V ⬵ ∠T RS ‘ VT RV ⬵ ST T V R S a Figure 4.35 T V Y Z R S b Some of the glass panels and trim pieces of the light fixture are isosceles trapezoids. Other glass panels are pentagons. 쮿 Geometry in the Real World