Technology ctivity 6.5 idsegment of a Trapezoid Question What are some properties of the midsegment of a trapezoid? Explore 1 raw. raw a point not on and construct a line parallel to through point. onstruct a point on the new line and draw &* and &*, as shown below. onstruct the midpoints of &* and &*. Label the points E and F. raw EF &*. E F tudent Help VOULY TI Quadrilateral is a trapezoid, a quadrilateral with exactly one pair of parallel sides. EF& is called a midsegment of trapezoid. Think bout It 1. easure the distances,, and EF.. alculate ( ).. rag points,,, and. o not allow &* to cross &*. What do you notice about EF and ( )? 4. easure the slopes of,, and EF &*. What do you notice? What do the slopes tell you about,, and EF &*? 5. Extension rag the points so that &* crosses &*. is no longer a polygon. Write an expression for EF in terms of and. Technology ctivity 1
6.5 Trapezoids Goal Use properties of trapezoids. trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. The nonparallel sides are the legs. leg base leg Key Words trapezoid bases, legs, and base angles of a trapezoid isosceles trapezoid midsegment of a trapezoid trapezoid has two pairs of base angles. In trapezoid, a and a are one pair of base angles. a and a are the other pair. If the legs of a trapezoid are congruent, then the trapezoid is an. isosceles trapezoid base isosceles trapezoid THEOE 6.1 and 6.1 Theorem 6.1 Words If a trapezoid is isosceles, then each pair of base angles are congruent. ymbols In the isosceles trapezoid, a ca and a ca. Theorem 6.1 Words If a trapezoid has a pair of congruent base angles, then it is isosceles. ymbols In trapezoid, if a ca then is isosceles. Visualize It! 50 50 10 10 Q In Example 1, fill in the missing angle measures as you find them. EXLE 1 F d ngl ea re of Trapez d Q is an isosceles trapezoid. Find the missing angle measures. olution 1 Q is an isosceles trapezoid and a and a are a pair of base angles. o, ma ma 50. ecause a and a are same-side interior angles formed by parallel lines, they are supplementary. o, ma 180 50 10. ecause aq and a are a pair of base angles of an isosceles trapezoid, maq ma 10. 50 hapter 6 Quadrilaterals
Find le e re f Tra ids is an isosceles trapezoid. Find the missing angle measures. 1... 100 70 75 tud t H lp VOULY TI The midsegment of a trapezoid is sometimes called the median of a trapezoid. idsegments The midsegment of a trapezoid is the segment that connects the midpoints of its legs. The midsegment of a trapezoid is parallel to the bases. The length of the midsegment midsegment of a trapezoid is half the sum of the lengths of the bases. 1 ( ) EXLE d egm t of a Trapez d Find the length of the midsegment G &** of trapezoid EFH. olution E 8 F G Use the formula for the midsegment of a trapezoid. 0 H G 1 (EF H) Formula for midsegment of a trapezoid 1 (8 0) ubstitute 8 for EF and 0 for H. 1 (8) 14 dd. ultiply. WE The length of the midsegment G &** is 14. ids t f Tra id Find the length of the midsegment &** of the trapezoid. 4. 8 5. 6. 4 10 6 14 18 6.5 Trapezoids
6.5 Exercises Guided ractice Vocabulary heck 1. ame the bases of trapezoid.. ame the legs of trapezoid. kill heck ecide whether the quadrilateral is a trapezoid, an isosceles trapezoid, or neither.. 4. 5. Find the length of the midsegment. 6. 7 7. 8. 19 7 11 15 ractice and pplications Extra ractice ee p. 686. arts of a Trapezoid atch the parts of trapezoid Q with the correct description. 9. Q &* and &*. legs 10. aq and a. base angles 11. a and aq. opposite angles 1. &**. bases 1. Q&* and &* E. midsegment Homework H lp Example 1: Exs. 14 19 Example : Exs. 0 6 Finding ngle easures JKL is an isosceles trapezoid. Find the missing angle measures. 14. K L 15. K L 16. L 60 45 J K J J 18 4 hapter 6 Quadrilaterals
Finding ngle easure QT is a trapezoid. Find the missing angle measures. 17. 18. 19. 1 78 110 T 78 T 150 T Finding idsegments Find the length of the midsegment &** of the trapezoid. 0. 9 1. 14. 15 9 7 16 Using lgebra Find the value of x.. x 4. 16 5. 9 10 7 x ake esign 1 6. ake esign The top layer of the cake in the diagram at the right has a diameter of 10 inches. The bottom layer has a diameter of inches. What is the diameter of the middle layer? x 10 in. in. KE EIGE form and sculpt shapes and figures onto cakes by using tools such as icing bags, handmade paper cones, or cutters. oordinate Geometry The vertices of a trapezoid are (, 6), (8, 6), (8, ), and (4, ). 7. lot the vertices on a coordinate plane. onnect them to form trapezoid. 8. ame the bases of trapezoid. 9. ame the legs of trapezoid. 0. Find the coordinates of the midpoint of each leg. Then plot these points on the coordinate plane you drew in Exercise 7. What is the line segment that connects these two points called? 6.5 Trapezoids 5
tudent Help I LZOE.O HOEWOK HEL Extra help with problem solving in Exs. 1 is at classzone.com Visualize It! In Exercises 1, use the figures shown below. The figure on the left is a trapezoid with midsegment of length m. The figure on the right is formed by cutting the trapezoid along its midsegment and rearranging the two pieces. 1 4 9 5 m 4 m m 9 5 1 1. Which theorem or postulate from hapter can you use to show that a1 ca and a ca4 in the figure on the left?. What kind of quadrilateral is on the right? Explain your answer.. hallenge How does the diagram help you see that the length of the midsegment is half the sum of the lengths of the bases? tandardized Test ractice 4. ultiple hoice In the trapezoid at the right, what is the value of x? 1 15 J 1 15 17 8 K x L 5. ultiple hoice Which of the following must a trapezoid have? F G H J congruent bases diagonals that bisect each other exactly one pair of parallel sides a pair of congruent opposite angles ixed eview Logical easoning Tell whether the quadrilateral is a parallelogram. Explain your reasoning. (Lesson 6.) 6. 7. 8. 115 65 lgebra kills ultiplying ultiply. Write the answer in simplest form. (kills eview, p. 659) 9. 1 0 40. 5 1 4 41. 16 1 8 4. 60 4 4. 7 44. 7 8 14 45. 5 6 1 46. 4 1 7 16 6 hapter 6 Quadrilaterals