M 1312 Section Trapezoids

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1 M 1312 Section Trapezoids Definition: trapezoid is a quadrilateral with exactly two parallel sides. Parts of a trapezoid: Base Leg Leg Leg Base Base Base Leg Isosceles Trapezoid: Every trapezoid contains two pairs of consecutive angles that are supplementary. Definition: n 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). n infinite number of altitudes may be drawn in a trapezoid.

2 M 1312 Section Definition: 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: n isosceles trapezoid is a trapezoid in which the legs (nonparallel sides) are congruent. n 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.

3 M 1312 Section 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 L J K If the m J 65 and the m K 95, the measure of angles H and L.

4 M 1312 Section Popper 12 question 1: Given a kite BD. is the perpendicular bisector of BD. Find B if B = 5 and the perimeter of the kite is 24. D O B. 5 B D 14 E. None of these Example 2: Use Isosceles Trapezoid BD with length of D = B. D B ll D B a. mdb = 75. Find the md. b. = 40. Find BD. c. If m 6x 25 and m B 8x 15, find the measures of angle and D.

5 M 1312 Section Definition: m altitude is a line segment from one vertex of one base of the trapezoid and perpendicular to the opposite base. Popper 12 question 2: In parallelogram BD (not shown), the diagonals have the lengths = 7 and BD = 9. Which pair of angles have greater measures? and D B. B and. and D. and B E. None of these Theorem 4.4.3: The length of the median of a trapezoid equals one-half the sum of the bases. 1 m b 1 b 2 2 Example 3: Find the missing measures of the given trapezoid. B 7 I a. mird b. YR c. DR D X Y R 75 3 d.

6 M 1312 Section 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 R S K b. RS = 17 HJ = 14 find LK H J c. RS = x + 5 HJ + LK = 4x + 6 find RS Example 5: Given WXYZ is a trapezoid with WX ZY, MN is the median W X M N Z 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.

7 M 1312 Section SUMMRY HRTS: Special Diagonals re lways Diagonals lways Bisect Quadrilateral ongruent Perpendicular Each Other ngles Parallelogram No No Yes No Rectangle Yes No Yes No Rhombus No Yes Yes Yes Square Yes Yes Yes Yes Trapezoid No No No No Isosceles Trapezoid Yes No No No There is an excellent chart in your book on page 205. Use for Popper 12 questions 3-5: HJKL is an isosceles trapezoid with bases HJ and LK, and median RS. Use the given information to solve each problem. L K R S H J Popper 12 question 3: If LK =30 and HJ = 42 then find RS.. RS =36 B RS = 20. RS =7 D. RS =2 E. None of these Popper 12 question 4: If RS =17 and HJ = 14, find LK.. RS =36 B RS = 20. RS =7 D. RS =2 E. None of these Popper 12 question 5: If RS = x + 5 and HJ + LK = 4x + 6 find RS.. RS =36 B RS = 20. RS =7 D. RS =2 E. None of these

8 M 1312 Section Solutions to Popper 10 Solutions for Popper 10: Use for Popper 10 question 1 and question 2: H E 4x + 2 x + 8 G Popper 10 question 1: Find in parallelogram EFGH.. B.. D. D. None of these F Popper 10 question 2: Find in parallelogram EFGH.. B.. D. D. None of these Popper 10 question 3: ssume that X, Y, and Z are midpoints of the sides of RST. If RS = 28, ST = 12, and RT = 18, find XY.. XY=6 B. XY = 9. XY = 14 D. XY = 24 E. None of these Use for Popper 10 questions 4 and 5: Given : B =18, D = 9, E = 8 and B = 18 and D and E are midpoints.

9 M 1312 Section E D B Popper 10 question 4: Find the length of DE.. DE = 8 B. DE = 15. DE = 18 D. DE = 16 E None of these Popper 10 question 5: Find the length of E.. E = 8 B. E = 15. E = 18 D. E = 16 E None of these D y E x B 30

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