Recognize and apply properties of rectangles. Determine whether parallelograms are rectangles. are rectangles used in tennis?

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1 ectangles Vocabulary rectangle ecognize and apply properties of rectangles. etermine whether parallelograms are rectangles. are rectangles used in tennis? any sports are played on fields marked by parallel lines. tennis court has parallel lines at half-court for each player. arallel lines divide the court for singles and doubles play. The service box is marked by perpendicular lines. Study Tip ectangles and arallelograms rectangle is a parallelogram, but a parallelogram is not necessarily a rectangle. OETIES O ETGLES rectangle is a quadrilateral with four right angles. Since both pairs of opposite angles are congruent, it follows that it is a special type of parallelogram. Thus, a rectangle has all the properties of a parallelogram. ecause the right angles make a rectangle a rigid figure, the diagonals are also congruent. Theorem 8.13 If a parallelogram is a rectangle, then the diagonals are congruent. bbreviation: If is rectangle, diag. are. You will prove Theorem 8.13 in Exercise 40. If a quadrilateral is a rectangle, then the following properties are true. Words rectangle is a quadrilateral with four right angles. ectangle roperties Examples 1. Opposite sides are congruent and parallel. 2. Opposite angles are congruent. 3. onsecutive angles m m 180 are supplementary. m m 180 m m 180 m m iagonals are congruent and bisect each other. and bisect each other. 5. ll four angles are m m right angles. m m hapter 8 uadrilaterals

2 Example 1 iagonals of a ectangle LGE uadrilateral O is a rectangle. If O 6x 14 and 9x 5, find x. The diagonals of a rectangle are congruent, so O. O O 6x 14 9x 5 iagonals of a rectangle are. efinition of congruent segments Substitution 14 3x 5 Subtract 6x from each side. 9 3x Subtract 5 from each side. 3 x ivide each side by 3. O ectangles can be constructed using perpendicular lines. Study Tip Look ack To review constructing perpendicular lines through a point, see Lesson Example 1 ectangle Use a straightedge to draw line. Label a point on. lace the point at and locate point on. ow construct lines perpendicular to through and through. Label them m and n. 2 lace the compass point at and mark off a segment on m. Using the same compass setting, place the compass at and mark a segment on n. Label these points and S. raw S. 3 Locate the compass setting that represents and compare to the setting for S. The measures should be the same. m n m n S m n S Example 2 LGE ngles of a ectangle uadrilateral is a rectangle. a. ind x. is a right angle, so m 90. (9x 20) (4x 5) (y 2 1) (4y 4) m m m ngle ddition Theorem 4x 5 9x Substitution 13x Simplify. 13x 65 Subtract 25 from each side. x 5 ivide each side by Lesson 8-4 ectangles 425

3 b. ind y. Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent. m m y 2 1 4y 4 y 2 4y 5 0 lternate Interior ngles Theorem efinition of angles Substitution (y 5)(y 1) 0 actor. y 5 0 y 1 0 Subtract 4y and 4 from each side. y 5 y 1 isregard y 1 because it yields angle measures of 0. OVE THT LLELOGS E ETGLES The converse of Theorem 8.13 is also true. Theorem 8.14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. bbreviation: If diagonals of are, is a rectangle. You will prove Theorem 8.14 in Exercise 41. Example 3 iagonals of a arallelogram WIOWS Trent is building a tree house for his younger brother. He has measured the window opening to be sure that the opposite sides are congruent. He measures the diagonals to make sure that they are congruent. This is called squaring the frame. How does he know that the corners are 90 angles? irst draw a diagram and label the vertices. We know that WX ZY, XY WZ, and WY XZ. Z Y ecause WX ZY and XY WZ, WXYZ is a parallelogram. Windows It is important to square the window frame because over time the opening may have become out-of-square. If the window is not properly situated in the framed opening, air and moisture can leak through cracks. Source: guide/measurement XZ and WY are diagonals and they are congruent. parallelogram with congruent diagonals is a rectangle. So, the corners are 90 angles. Example 4 ectangle on a oordinate lane OOITE GEOETY uadrilateral GH has vertices (4, 1), G(2, 5), H(4, 2), and (2, 2). etermine whether GH is a rectangle. ethod 1: Use the Slope ormula, m y 2 y1, x2 x1 to see if consecutive sides are perpendicular. W O y X x H slope of 2 2 ( 1) or 1 ( 4) 2 G 426 hapter 8 uadrilaterals Emma Lee/Life ile/hotoisc

4 slope of GH 2 (5) or 1 4 (2) 2 slope of G 5 ( 1) or 2 2 ( 4) slope of H 2 (2) or ecause GH and G H, quadrilateral GH is a parallelogram. The product of the slopes of consecutive sides is 1. This means that G, H, H GH, and G GH. The perpendicular segments create four right angles. Therefore, by definition GH is a rectangle. ethod 2: Use the istance ormula, d (x x 2 1 ) 2 (y 2, y 1 ) 2 to determine whether opposite sides are congruent. irst, we must show that quadrilateral GH is a parallelogram. (4 2) 2 (1 2) GH (2 4) 2 [5 (2)] G [4 (2)] 1 [5)] ( H (2 ) 4 [2)] ( Since each pair of opposite sides of the quadrilateral have the same measure, they are congruent. uadrilateral GH is a parallelogram. H (4 4) 2 [1 (2)] G (2 2) 2 (5 2) The length of each diagonal is 65. Since the diagonals are congruent, GH is a rectangle by Theorem oncept heck 1. How can you determine whether a parallelogram is a rectangle? 2. OE EE raw two congruent right triangles with a common hypotenuse. o the legs form a rectangle? 3. I THE EO ckenna and onsuelo are defining a rectangle for an assignment. ckenna rectangle is a parallelogram with one right angle. onsuelo rectangle has a pair of parallel opposite sides and a right angle. Who is correct? Explain. Lesson 8-4 ectangles 427

5 Guided ractice 4. LGE is a rectangle. 5. LGE is a rectangle. If 30 x and 4x 60, If 2x 10 and 2x 30, find x. find. LGE uadrilateral ST is a rectangle. ind each measure or value. 6. x 7. ms T (3x 11) (x 2 1) S 8. OOITE GEOETY uadrilateral EGH has vertices E(4, 3), (3, 1), G(2, 3), and H(5, 1). etermine whether EGH is a rectangle. pplication 9. IG rs. Walker has a rectangular picture that is 12 inches by 48 inches. ecause this is not a standard size, a special frame must be built. What can the framer do to guarantee that the frame is a rectangle? ustify your reasoning. ractice and pply or Exercises 10 15, , 35, See Examples Extra ractice See page LGE uadrilateral K is a rectangle. K 10. If = 5x 3 and 4x 6, find K. 11. If 2x 3 and K 5x 9, find. 12. If 8x 14 and K x 2 1, find K. 13. If m 2x 3 and mk x 5, find x. 14. If mk x 2 4 and mk x 30, find mk. 15. If mk 2x 2 2 and mk 14x, find x. WXYZ is a rectangle. ind each measure if m1 30. W X 16. m1 17. m2 18. m m4 20. m5 21. m m7 23. m8 24. m Z Y 25. TIOS contractor has been hired to pour a rectangular concrete patio. How can he be sure that the frame in which to pour the concrete is rectangular? 26. TELEVISIO Television screens are measured on the diagonal. What is the measure of the diagonal of this screen? 21 in. 36 in. 428 hapter 8 uadrilaterals Zenith Electronics orp.//wide World hotos

6 OOITE GEOETY etermine whether GH is a rectangle given each set of vertices. ustify your answer. 27. (9, 1), (9, 5), G(6, 5), H(6, 1) 28. (6, 2), (8, 1), G(10, 6), H(12, 3) 29. (4, 3), (5, 8), G(6, 9), H(7, 2) OOITE GEOETY The vertices of WXYZ are W(2, 4), X(2, 0), Y(1, 7), and Z(9, 3). 30. ind WY and XZ. 31. ind the coordinates of the midpoints of WY and XZ. 32. Is WXYZ a rectangle? Explain. Golden ectangles The arthenon in ancient Greece is an example of how the golden rectangle was applied to architecture. The ratio of the length to the height is the golden ratio. Source: OOITE GEOETY The vertices of parallelogram are (4, 4), (2, 1), (0, 3), and (6, 0). 33. etermine whether is a rectangle. 34. If is a rectangle and E,, G, and H are midpoints of its sides, what can you conclude about EGH? 35. IITUE GOL The windmill section of a miniature golf course will be a rectangle 10 feet long and 6 feet wide. Suppose the contractor placed stakes and strings to mark the boundaries with the corners at,, and. The contractor measured and and found that. escribe where to move the stakes L and K to make a rectangle. Explain. GOLE ETGLES or Exercises 36 and 37, use the following information. any artists have used golden rectangles in their work. In a golden rectangle, the ratio of the length to the width is about This ratio is known as the golden ratio. 36. rectangle has dimensions of feet and feet. etermine if the rectangle is a golden rectangle. Then find the length of the diagonal. 37. ESEH Use the Internet or other sources to find examples of golden rectangles. 38. What are the minimal requirements to justify that a parallelogram is a rectangle? 39. raw a counterexample to the statement If the diagonals are congruent, the quadrilateral is a rectangle. OO Write a two-column proof. 40. Theorem Theorem Given: ST is a rectangle. 43. Given: E and E are rectangles. VT GKH HK rove: VS G and HK intersect at L. rove: GHK is a parallelogram. S V T K E L L 10 ft E G H G 6 ft K H 44. ITIL THIKIG Using four of the twelve points as corners, how many rectangles can be drawn? Lesson 8-4 ectangles 429 Izzet Keribar/Lonely lanet Images

7 SHEIL GEOETY The figure shows a Saccheri quadrilateral on a sphere. ote that it has four sides with T T, T, and T. 45. Is T parallel to? Explain. 46. How does compare to T? 47. an a rectangle exist in spherical geometry? Explain. T Standardized Test ractice 48. WITIG I TH nswer the question that was posed at the beginning of the lesson. How are rectangles used in tennis? Include the following in your answer: the number of rectangles on one side of a tennis court, and a method to ensure the lines on the court are parallel 49. In the figure, E. If 6, what is? E 6 x x ote: igure not drawn to scale 50. LGE rectangular playground is surrounded by an 80-foot long fence. One side of the playground is 10 feet longer than the other. Which of the following equations could be used to find s, the shorter side of the playground? 10s s 80 4s s(s 10) 80 2(s 10) 2s 80 aintain Your Skills ixed eview 51. TEXTILE TS The avajo people are well known for their skill in weaving. The design at the right, known as the Eye-azzler, became popular with avajo weavers in the 1880s. How many parallelograms, not including rectangles, are in the pattern? (Lesson 8-3) or Exercises 52 57, use. ind each measure or value. (Lesson 8-2) 52. m 53. m 54. m 55. m 56. y 57. x 5x 3y Getting eady for the ext Lesson 430 hapter 8 uadrilaterals ind the measure of the altitude drawn to the hypotenuse. (Lesson 7-1) T O S EEUISITE SKILL ind the distance between each pair of points. (To review the istance ormula, see Lesson 1-4.) 61. (1, 2), (3, 1) 62. (5, 9), (5, 12) 63. (1, 4), (22, 24)