Trapezoids. are the bases. TP. / are the legs.

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1 8 5 What You ll Learn You ll learn to identify and use the properties of trapezoids and isosceles trapezoids. rapezoids any state flags use geometric shapes in their designs. an you find a quadrilateral in the aryland state flag that has exactly one pair of parallel sides? Why It s Important rt rapezoids are used in perspective drawings. See Example. aryland state flag trapezoid is a quadrilateral with exactly one pair of parallel sides. he parallel sides are called bases. he nonparallel sides are called legs. base angles Study trapezoid RP. R P R and P are the bases. R P / P and R are the legs. R leg leg P base base angles al Wo rld Re Each trapezoid has two pairs of base angles. In trapezoid RP, and R are one pair of base angles; P and are the other pair. base Example rt Link rtists use perspective to give the illusion of depth to their drawings. In perspective drawings, vertical lines remain parallel, but horizontal lines gradually come together at a point. In trapezoid ZOI, name the bases, the legs, and the base angles. Vanishing point O Z Eye level I ases Z and O I are parallel segments. Legs Z O and I are nonparallel segments. ase ngles Z and are one pair of base angles; O and I are the other pair. Lesson 8 5 rapezoids 333

2 he median of a trapezoid is the segment that joins the midpoints of its legs. In the figure, N is the median. E median nother name for the median of a trapezoid is the midsegment of the trapezoid. N G F Words: he median of a trapezoid is parallel to the bases, and the length of the median equals one-half the sum of the lengths of the bases. heorem 8 3 odel: N Example 2 Find the length of median N in trapezoid if 2 and 8. N 2 ( ) 2 (2 8) 2 (30) or 5 Symbols: N, N N 2 ( ) heorem 8 3 Replace with 2 and with 8. Simplify. N he length of the median of trapezoid is 5 units. Your urn a. Find the length of median N in trapezoid if 20 and 6. Isosceles riangle: Lesson 6 4 If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. In Lesson 6 4, you learned that the base angles of an isosceles triangle are congruent. here is a similar property for isosceles trapezoids. heorem 8 4 Words: Each pair of base angles in an isosceles trapezoid is congruent. odel: Symbols: W X W X, Z Y Z 334 hapter 8 Quadrilaterals Y

3 Example 3 Find the missing angle measures in isosceles trapezoid RP. R Find mp. P heorem 8 4 mp m efinition of congruent mp 60 Replace m with P RP. Find m. Since RP is a trapezoid, onsecutive Interior ngles: Lesson 4 2 m mp 80 onsecutive interior angles are supplementary. m Replace mp with 60. m Subtract 60 from each side. m 20 Simplify. Find mr. R heorem 8 4 mr m efinition of congruent mr 20 Replace m with 20. Your urn b. he measure of one angle in an isosceles trapezoid is 48. Find the measures of the other three angles. In this chapter, you have studied quadrilaterals, parallelograms, rectangles, rhombi, squares, trapezoids, and isosceles trapezoids. he Venn diagram illustrates how these figures are related. he Venn diagram represents all quadrilaterals. Parallelograms and trapezoids do not share any characteristics except that they are both quadrilaterals. his is shown by the nonoverlapping regions in the Venn diagram. Every isosceles trapezoid is a trapezoid. In the Venn diagram, this is shown by the set of isosceles trapezoids contained in the set of trapezoids. ll rectangles and rhombi are parallelograms. Since a square is both a rectangle and a rhombus, it is shown by overlapping regions. Quadrilaterals Parallelograms Rhombi Rectangles Squares rapezoids Isosceles rapezoids Lesson 8 5 rapezoids 335

4 heck for Understanding ommunicating athematics. raw an isosceles trapezoid and label the legs and the bases. trapezoid bases legs base angles median isosceles trapezoid 2. Explain how the length of the median of a trapezoid is related to the lengths of the bases. 3. opy and complete the following table. Write yes or no to indicate whether each quadrilateral always has the given characteristics. haracteristics Parallelogram Rectangle Rhombus Square rapezoid Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. onsecutive angles are supplementary. iagonals bisect each other. iagonals are congruent. iagonals are perpendicular. Each diagonal bisects two angles. Guided Practice Example Q 4. In trapezoid QRS, name the bases, the legs, and the base angles. R S Example 2 Find the length of the median in each trapezoid ft 6. 5 ft Example 3 7. rapezoid is isosceles. Find the missing angle measures. 3 m 0 m hapter 8 Quadrilaterals

5 Example 3 8. onstruction hip roof slopes at the ends of the building as well as the front and back. he front of this hip roof is in the shape of an isosceles trapezoid. If one angle measures 30, find the measures of the other three angles. Exercises Practice For each trapezoid, name the bases, the legs, and the base angles. 9. Homework Help For Exercises 8 20, 22 9, 29 See Examples 3 2 7, 2, 30 2 Extra Practice See page V H. J G R K S Find the length of the median in each trapezoid in yd 32 m 30 yd 2 in. 64 m mm cm 8 ft 9.6 cm 35 ft 20 mm Find the missing angle measures in each isosceles trapezoid. 8. Z 9. O J K 20. W 20 X 00 I 85 L Z Y Lesson 8 5 rapezoids 337

6 2. Find the length of the shorter base of a trapezoid if the length of the median is 34 meters and the length of the longer base is 49 meters. 22. One base angle of an isosceles trapezoid is 45. Find the measures of the other three angles. etermine whether it is possible for a trapezoid to have the following conditions. Write yes or no. If yes, draw the trapezoid pplications and Problem Solving three congruent sides 24. congruent bases four acute angles 26. two right angles one leg longer than either base two congruent sides, but not isosceles 29. ridges Explain why the figure outlined on the Golden Gate ridge is a trapezoid. 30. lgebra If the sum of the measures of the bases of a trapezoid is 4x, find the measure of the median. 3. ritical hinking sequence of trapezoids is shown. he first three trapezoids in the sequence are formed by 3, 5, and 7 triangles.,,..., a. How many triangles are needed for the 0th trapezoid? b. How many triangles are needed for the nth trapezoid? ixed Review Name all quadrilaterals that have each property. 32. four right angles 33. congruent diagonals 34. lgebra Find the value for x that will make quadrilateral a parallelogram. (Lesson 8 3) 20 4x Standardized est Practice (Lesson 8 4) 2x Extended Response raw and label a figure to illustrate that JN and are medians of JKL and intersect at I. (Lesson 6 ) L 36. ultiple hoice In the figure, 60, 2, and is the. hoose the correct statement. (Lesson 2 5) midpoint of 338 hapter 8 Quadrilaterals here is not enough information.

7 esigner re you creative? o you find yourself sketching designs for new cars or the latest fashion trends? hen you may like a career as a designer. esigners organize and design products that are visually appealing and serve a specific purpose. any designers specialize in a particular area, such as fashion, furniture, automobiles, interior design, and textiles. extile designers design fabric for garments, upholstery, rugs, and other products, using their knowledge of textile materials and geometry. omputers especially intelligent pattern engineering (IPE) systems are widely used in pattern design.. Identify the geometric shapes used in the textiles shown above. 2. esign a pattern of your own for a textile. FS FS bout Fashion esigners Working onditions vary by places of employment overtime work sometimes required to meet deadlines keen competition for most jobs Education a 2- or 4-year degree is usually needed computer-aided design () courses are very useful creativity is crucial Earnings edian Hourly Wage in 200 $0 $20 $30 $40 $50 $60 $70 Source: ureau of Labor Statistics areer ata For the latest information on a career as a designer, visit: hapter 8 ath In the Workplace 339

8 hapter 8 aterials Investigation Kites unlined paper compass kite is more than just a toy to fly on a windy day. In geometry, a kite is a special quadrilateral that has its own properties. straightedge Investigate protractor. Use paper, compass, and straightedge to construct a kite. ruler a. raw a segment about six inches in length. Label the endpoints I and E. ark a point on the segment. he point should not be the midpoint of IE. Label the point X. b. onstruct a line that is perpendicular to IE through X. ark point K about two inches to the left of X on the perpendicular line. hen mark another point,, on the right side X X. of X so that K c. onnect points K, I,, and E to form a quadrilateral. KIE is a kite. Use a ruler to measure the lengths of the sides of KIE. What do you notice? I K X E d. Write a definition for a kite. ompare your definition with others in the class. 340 hapter 8 Quadrilaterals

9 2. Use compass, straightedge, protractor, and ruler to investigate kites. a. Use a protractor to measure the angles of KIE. What do you notice about the measures of opposite and consecutive angles? b. onstruct at least two more kites. Investigate the measures of the sides and angles. c. an a kite be parallelogram? Explain your reasoning. In this extension, you will investigate kites and their relationship to other quadrilaterals. Here are some suggestions.. Rewrite heorems 8 2 through 8 6 and 8 0 through 8 2 so they are true for kites. 2. ake a list of as many properties as possible for kites. 3. uild a kite using the properties you have studied. Presenting Your onclusions Here are some ideas to help you present your conclusions to the class. ake a booklet showing the differences and similarities among the quadrilaterals you have studied. e sure to include kites. ake a video about quadrilaterals. ast your actors as the different quadrilaterals. he script should help viewers understand the properties of quadrilaterals. Investigation For more information on kites, visit: hapter 8 Investigation Go Fly a Kite! 34