Trapezoid

M 1312 Section 4.4 1 Trapezoids Definition: A trapezoid is a quadrilateral with exactly two parallel sides. Parts of a trapezoid: Leg Base  Leg Leg Base Base   Base Leg Isosceles Trapezoid: Every trapezoid contains two pairs of consecutive angles that are supplementary. Definition: An altitude of a trapezoid is a segment drawn from any point on one of the parallel sides (base) perpendicular to the opposite side (the other base). An infinite number of altitudes may be drawn in a trapezoid. M 1312 Section 4.4 2 Definition: A median of a trapezoid is the segment that joins the midpoints of the nonparallel sides (legs). Median Median Theorem: The median of a trapezoid is parallel to each base and the length of the median equals one-half the sum of the lengths of the two bases. M Definition: An isosceles trapezoid is a trapezoid in which the legs (nonparallel sides) are congruent. An isosceles trapezoid features some special properties not found in all trapezoids. Theorem 4.4.1: The base angles of an isosceles trapezoid are congruent. Theorem 4.4.2: The diagonals of an isosceles trapezoid are congruent. M 1312 Section 4.4 3 Properties of Isosceles Trapezoid 1. The legs are congruent. 2. The bases are parallel. 3. The lower base angles of an isosceles trapezoid are congruent. 4. The upper base angles of an isosceles trapezoid are congruent. 5. The lower base angle is supplementary to any upper base angle. 6. The diagonals of an isosceles trapezoid are congruent. 7. The median is parallel to the base. 8. The length of the median equals one-half the sum of the lengths of the two bases. Proving that Trapezoid is isosceles 1. If legs of a trapezoid are congruent then it is an isosceles trapezoid. 2. If two base angles of a trapezoid are congruent, then it is an isosceles trapezoid. 3. If the diagonals of a trapezoid are congruent, then it is an isosceles trapezoid. Example 1: Given the trapezoid HLJK H J If the mJ  65 and the mK  95 , the measure of angles H and L . L K M 1312 Section 4.4 Example 2: Use Isosceles Trapezoid ABCD with length of AD = BC. D C AB ll CD A B a. mDAB = 75. Find the mADC. b. AC = 40. Find BD . c. If mA  6x  25 and mB  8x  15 , find the measures of angle C and D. Definition: Am altitude is a line segment from one vertex of one base of the trapezoid and perpendicular to the opposite base. Theorem 4.4.3: The length of the median of a trapezoid equals one-half the sum of the bases. m 1 b1  b2  2 4 M 1312 Section 4.4 5 Example 3: B 7 I Find the missing measures of the given trapezoid. a. mIRD C A b. YR 75 c. DR D 3 X R Y d. AC Example 4: HJKL is an isosceles trapezoid with bases HJ and LK , and median RS . Use the given information to solve each problem. a. LK = 30 HJ = 42 find RS L K R b. RS = 17 HJ = 14 find LK c. RS = x + 5 HJ + LK = 4x + 6 find RS S H J M 1312 Example 5: Section 4.4 6 Given WXYZ is a trapezoid with WX || ZY , M N is the median W X M Z N Y a. If WX =19 and ZY = 31, find MN b. If WX = 4x ̶ 7, MN = 2x + 10 and ZY = 2x + 1, find x and the lengths of WX, MN and ZY. SUMMARY CHARTS: Special Quadrilateral Parallelogram Rectangle Rhombus Square Trapezoid Isosceles Trapezoid Diagonals Are Always Congruent Perpendicular No No Yes No No Yes Yes Yes No No Yes No There is an excellent chart in your book on page 205. Diagonals Always Bisect Each Other Angles Yes No Yes No Yes Yes Yes Yes No No No No