Essential Question What are some properties of trapezoids and kites? Recall the types of quadrilaterals shown below.

Transcription

1 7.5 Properties of Trapezoids and ites ssential Question What are some properties of trapezoids and kites? ecall the types of quadrilaterals shown below. Trapezoid Isosceles Trapezoid ite PV I OVI PO To be proficient in math, you need to draw diagrams of important features and relationships, and search for regularity or trends. aking a onjecture about Trapezoids Work with a partner. Use dynamic geometry software. a. onstruct a trapezoid whose ample base angles are congruent. xplain your process. b. Is the trapezoid isosceles? ustify your answer. c. epeat parts (a) and (b) for several other trapezoids. Write a conjecture based on your results. iscovering a Property of ites Work with a partner. Use dynamic geometry software. a. onstruct a kite. xplain ample your process. b. easure the angles of the kite. What do you observe? c. epeat parts (a) and (b) for several other kites. Write a conjecture based on your results. ommunicate Your nswer 3. What are some properties of trapezoids and kites? 4. Is the trapezoid at the left isosceles? xplain quadrilateral has angle measures of 70, 70, 110, and 110. Is the quadrilateral a kite? xplain. ection 7.5 Properties of Trapezoids and ites 441

2 7.5 esson What You Will earn ore Vocabulary trapezoid, p. 442 bases, p. 442 base angles, p. 442 legs, p. 442 isosceles trapezoid, p. 442 midsegment of a trapezoid, p. 444 kite, p. 445 Previous diagonal parallelogram Use properties of trapezoids. Use the Trapezoid idsegment Theorem to find distances. Use properties of kites. Identify quadrilaterals. Using Properties of Trapezoids trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. ase angles of a trapezoid are two consecutive angles whose common side is a base. trapezoid has two pairs of base angles. or example, in trapezoid, and are one pair of base angles, and and are the second pair. The nonparallel sides are the legs of the trapezoid. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. leg base angles base base base angles leg Isosceles trapezoid Identifying a Trapezoid in the oordinate Plane how that OT is a trapezoid. Then decide whether it is isosceles. OUTIO (0, 3) tep 1 ompare the slopes of opposite sides. T(4, 2) slope of = = 1 2 slope of OT = = 2 4 = 1 2 O(0, 0) The slopes of and OT are the same, so OT. slope of T = = 2 = 1 slope of O = = 3 Undefined 0 The slopes of T and O are not the same, so T is not parallel to O. 2 y (2, 4) x ecause OT has exactly one pair of parallel sides, it is a trapezoid. tep 2 ompare the lengths of legs O and T. O = 3 0 = 3 T = (2 4) 2 + (4 2) 2 = 8 = 2 2 ecause O T, legs O and T are not congruent. o, OT is not an isosceles trapezoid. onitoring Progress elp in nglish and panish at igideasath.com 1. The points ( 5, 6), (4, 9), (4, 4), and ( 2, 2) form the vertices of a quadrilateral. how that is a trapezoid. Then decide whether it is isosceles. 442 hapter 7 Quadrilaterals and Other Polygons

3 Theorems Isosceles Trapezoid ase ngles Theorem If a trapezoid is isosceles, then each pair of base angles is congruent. If trapezoid is isosceles, then and. Proof x. 39, p. 449 Isosceles Trapezoid ase ngles onverse If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. If (or if ), then trapezoid is isosceles. Proof x. 40, p. 449 Isosceles Trapezoid iagonals Theorem trapezoid is isosceles if and only if its diagonals are congruent. Trapezoid is isosceles if and only if. Proof x. 51, p. 450 Using Properties of Isosceles Trapezoids The stone above the arch in the diagram is an isosceles trapezoid. ind m, m, and m. OUTIO tep 1 ind m. is an isosceles trapezoid. o, and are congruent base angles, and m = m = 85. tep 2 ind m. ecause and are consecutive interior angles formed by intersecting two parallel lines, they are supplementary. o, m = = 95. tep 3 ind m. ecause and are a pair of base angles, they are congruent, and m = m = 95. o, m = 85, m = 95, and m = onitoring Progress In xercises 2 and 3, use trapezoid. elp in nglish and panish at igideasath.com 2. If =, is trapezoid isosceles? xplain. 3. If m = 70 and m = 110, is trapezoid isosceles? xplain. ection 7.5 Properties of Trapezoids and ites 443

4 I The midsegment of a trapezoid is sometimes called the median of the trapezoid. Using the Trapezoid idsegment Theorem ecall that a midsegment of a triangle is a segment that connects the midpoints of two sides of the triangle. The midsegment of a trapezoid is the segment that connects the midpoints of its legs. The theorem below is similar to the Triangle idsegment Theorem. Theorem Trapezoid idsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one-half the sum of the lengths of the bases. If is the midsegment of trapezoid, then,, and = 1 ( + ). 2 Proof x. 49, p. 450 midsegment Using the idsegment of a Trapezoid In the diagram, is the midsegment of trapezoid PQ. P 12 in. ind. Q OUTIO = 1 (PQ + ) Trapezoid idsegment Theorem 2 = 1 ( ) ubstitute 12 for PQ and 28 for. 2 = 20 implify. The length of is 20 inches. 28 in. Using a idsegment in the oordinate Plane ind the length of midsegment YZ in y trapezoid TUV. 12 T(8, 10) Y (0, 6) OUTIO tep 1 ind the lengths of V and TU. V = (0 2) 2 + (6 2) 2 = 20 = 2 5 TU = (8 12) 2 + (10 2) 2 = 80 = V(2, 2) Z U(12, 2) x tep 2 ultiply the sum of V and TU by 1 2. YZ = 1 ( ) = 1 ( 6 5 ) = o, the length of YZ is 3 5 units. 4 onitoring Progress elp in nglish and panish at igideasath.com 4. In trapezoid, and are right angles, and = 9 centimeters. The length of midsegment P of trapezoid is 12 centimeters. ketch trapezoid and its midsegment. ind. xplain your reasoning. 5. xplain another method you can use to find the length of YZ in xample hapter 7 Quadrilaterals and Other Polygons

5 Using Properties of ites kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. Theorems TUY TIP The congruent angles of a kite are formed by the noncongruent adjacent sides. ite iagonals Theorem If a quadrilateral is a kite, then its diagonals are perpendicular. If quadrilateral is a kite, then. Proof p. 445 ite Opposite ngles Theorem If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If quadrilateral is a kite and, then and. Proof x. 47, p. 450 ite iagonals Theorem iven is a kite,, and. Prove TTT 1. is a kite with and. 2. and lie on the bisector of. 3. is the bisector of. 4. O 1. iven 2. onverse of the isector Theorem 3. Through any two points, there exists exactly one line. 4. efinition of bisector inding ngle easures in a ite ind m in the kite shown. 115 OUTIO y the ite Opposite ngles Theorem, has exactly one pair of congruent opposite angles. ecause, and must be congruent. o, m = m. Write and solve an equation to find m. 73 m + m = 360 m + m = 360 orollary to the Polygon Interior ngles Theorem ubstitute m for m. 2m = 360 ombine like terms. m = 86 olve for m. ection 7.5 Properties of Trapezoids and ites 445

6 onitoring Progress elp in nglish and panish at igideasath.com 6. In a kite, the measures of the angles are 3x, 75, 90, and 120. ind the value of x. What are the measures of the angles that are congruent? Identifying pecial Quadrilaterals The diagram shows relationships among the special quadrilaterals you have studied in this chapter. ach shape in the diagram has the properties of the shapes linked above it. or example, a rhombus has the properties of a parallelogram and a quadrilateral. quadrilateral parallelogram trapezoid kite rectangle rhombus isosceles trapezoid square Identifying a Quadrilateral I I In xample 6, looks like a square. ut you must rely only on marked information when you interpret a diagram. What is the most specific name for quadrilateral? OUTIO The diagram shows and. o, the diagonals bisect each other. y the Parallelogram iagonals onverse, is a parallelogram. ectangles, rhombuses, and squares are also parallelograms. owever, there is no information given about the side lengths or angle measures of. o, you cannot determine whether it is a rectangle, a rhombus, or a square. o, the most specific name for is a parallelogram. onitoring Progress elp in nglish and panish at igideasath.com 7. Quadrilateral has at least one pair of opposite sides congruent. What types of quadrilaterals meet this condition? ive the most specific name for the quadrilateral. xplain your reasoning U T 9. V W Y X hapter 7 Quadrilaterals and Other Polygons

7 7.5 xercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck 1. WITI escribe the differences between a trapezoid and a kite. 2. IT WO, QUTIO Which is different? ind both answers. Is there enough information to prove that trapezoid is isosceles? Is there enough information to prove that? Is there enough information to prove that the non-parallel sides of trapezoid are congruent? Is there enough information to prove that the legs of trapezoid are congruent? onitoring Progress and odeling with athematics In xercises 3 6, show that the quadrilateral with the given vertices is a trapezoid. Then decide whether it is isosceles. (ee xample 1.) 3. W(1, 4), X(1, 8), Y( 3, 9), Z( 3, 3) In xercises 11 and 12, find ( 3, 3), ( 1, 1), (1, 4), ( 3, 0) 5. ( 2, 0), (0, 4), P(5, 4), Q(8, 0) 6. (1, 9), (4, 2), (5, 2), (8, 9) In xercises 7 and 8, find the measure of each angle in the isosceles trapezoid. (ee xample 2.) Q T In xercises 13 and 14, find the length of the midsegment of the trapezoid with the given vertices. (ee xample 4.) 13. (2, 0), (8, 4), (12, 2), (0, 10) 14. ( 2, 4), T( 2, 4), U(3, 2), V(13, 10) In xercises 15 18, find m. (ee xample 5.) In xercises 9 and 10, find the length of the midsegment of the trapezoid. (ee xample 3.) ection 7.5 Properties of Trapezoids and ites 447

8 19. O YI escribe and correct the error in TTI OTIO In xercises 27 and 28, find the value of x. finding. 3x x x 2 = = 14 8 = OI WIT TTI In the diagram, P = 8 inches, and = 20 inches. What is the diameter of the bottom layer of the cake? 20. O YI escribe and correct the error in finding m. P Q PO OVI You and a friend are building a 50 kite. You need a stick to place from X to W and a stick to place from W to Z to finish constructing the frame. You want the kite to have the geometric shape of a kite. ow long does each stick need to be? xplain your reasoning. Opposite angles of a kite are congruent, so m = 50. Y W W Z Z X Q P 29 in. 18 in. In xercises 21 24, give the most specific name for the quadrilateral. xplain your reasoning. (ee xample 6.) X Y OI In xercises 31 34, determine which pairs of segments or angles must be congruent so that you can prove that is the indicated quadrilateral. xplain your reasoning. (There may be more than one right answer.) 31. isosceles trapezoid information is given in the diagram to classify the quadrilateral by the indicated name. xplain. 26. square hapter 7 int_math2_pe_0705.indd square parallelogram 110 OI In xercises 25 and 26, tell whether enough 25. rhombus 32. kite Quadrilaterals and Other Polygons 5/4/16 8:26

9 35. POO Write a proof. iven, is a midsegment of. Prove Quadrilateral is an isosceles trapezoid. 36. POO Write a proof. iven is a kite., Prove 37. TT OI Point U lies on the perpendicular bisector of T. escribe the set of points for which TU is a kite. 38. OI etermine whether the points (4, 5), ( 3, 3), ( 6, 13), and (6, 2) are the vertices of a kite. xplain your reasoning, POVI TO In xercises 39 and 40, use the diagram to prove the given theorem. In the diagram, is drawn parallel to. 39. Isosceles Trapezoid ase ngles Theorem iven is an isosceles trapezoid. Prove, 40. Isosceles Trapezoid ase ngles onverse iven is a trapezoid., Prove is an isosceles trapezoid. V U T 41. I UT Your cousin claims there is enough information to prove that is an isosceles trapezoid. Is your cousin correct? xplain. 42. TTI OTIO The bases of a trapezoid lie on the lines y = 2x + 7 and y = 2x 5. Write the equation of the line that contains the midsegment of the trapezoid. 43. OTUTIO and bisect each other. a. onstruct quadrilateral so that and are congruent, but not perpendicular. lassify the quadrilateral. ustify your answer. b. onstruct quadrilateral so that and are perpendicular, but not congruent. lassify the quadrilateral. ustify your answer. 44. POO Write a proof. iven QT is an isosceles trapezoid. Prove TQ T Q T 45. OI WIT TTI plastic spiderweb is made in the shape of a regular dodecagon (12-sided polygon). PQ, and X is equidistant from the vertices of the dodecagon. a. re you given enough information to prove that PQ is an P isosceles trapezoid? Q X b. What is the measure of each interior angle of PQ? 46. TTI TO PIIO In trapezoid PQ, PQ and is the midsegment of PQ. If = 5 PQ, what is the ratio of to? 3 : 5 5 : 3 1 : 2 3 : 1 ection 7.5 Properties of Trapezoids and ites 449

10 47. POVI TO Use the plan for proof below to write a paragraph proof of the ite Opposite ngles Theorem. iven is a kite., Prove, Plan for Proof irst show that. Then use an indirect argument to show that. 50. TOUT POVOI Is a valid congruence theorem for kites? ustify your answer. 51. POVI TO To prove the biconditional statement in the Isosceles Trapezoid iagonals Theorem, you must prove both parts separately. a. Prove part of the Isosceles Trapezoid iagonals Theorem. iven is an isosceles trapezoid., Prove 48. OW O YOU IT? One of the earliest shapes used for cut diamonds is called the table cut, as shown in the figure. ach face of a cut gem is called a facet. a., and and are not parallel. What shape is the facet labeled? b., and and are congruent but not parallel. What shape is the facet labeled? 49. POVI TO In the diagram below, is the midsegment of, and is the midsegment of. Use the diagram to prove the Trapezoid idsegment Theorem. b. Write the other part of the Isosceles Trapezoid iagonals Theorem as a conditional. Then prove the statement is true. 52. POO What special type of quadrilateral is? Write a proof to show that your answer is correct. iven In the three-dimensional figure,.,,, and are the midpoints of,,, and, respectively. Prove is a. aintaining athematical Proficiency eviewing what you learned in previous grades and lessons raph PQ with vertices P( 3, 2), Q(2, 3), and (4, 2) and its image after the translation. (kills eview andbook) 53. (x, y) (x + 5, y + 8) 54. (x, y) (x + 6, y 3) 55. (x, y) (x 4, y 7) Tell whether the two figures are similar. xplain your reasoning. (kills eview andbook) hapter 7 Quadrilaterals and Other Polygons