Activity Assess 8-2 CRITIQUE & EXPLAIN Manuel draws a diagram of kite PQRS Kites and ⟷ with ​​ QS​​ as the line of symmetry over Trapezoids a design of a kite-shaped key fob. He Q makes a list of conclusions based on the diagram. PearsonRealize.com P R • PR QS I CAN… use triangle • QP ~ QR congruence to understand = kites and trapezoids. • SP =~ SR • PR bisects QS. S VOCABULARY • ∆PQR is an equilateral • midsegment of a trapezoid triangle. • ∆PSR is an isosceles triangle. A. Which of Manuel’s conclusions do you agree with? Which do you disagree with? Explain. B. Use Structure What other conclusions are supported by the diagram? ESSENTIAL QUESTION How are diagonals and angle measures related in kites and trapezoids? CONCEPTUAL UNDERSTANDING EXAMPLE 1 Investigate the Diagonals of a Kite How are the diagonals of a kite related? A A kite has two pairs of congruent adjacent sides. X STUDY TIP D B Remember that you must show that both B and D are on the perpendicular bisector in order to show that one diagonal is C the perpendicular bisector of the other. It is not sufficient to Point B is equidistant from the endpoints of ¯​​ AC​​, as is D, show that only one is on the so they lie on the perpendicular bisector of ¯​​ AC​​. perpendicular bisector. The diagonals of a kite are perpendicular to each other. Exactly one diagonal bisects the other. Try It! 1. a. What is the measure of ∠AXB? b. If ​AX = 3.8​, what is AC? c. If ​BD = 10​, does BX​ = 5​? Explain. LESSON 8-2 Kites and Trapezoids 365 Activity Assess THEOREM 8-3 The diagonals of a kite are If... X perpendicular. E W Y Z PROOF: SEE EXERCISE 12. Then... ​​WY‾ ​ ⟂ ​XZ‾ ​ EXAMPLE 2 Use the Diagonals of a Kite Quadrilateral PQRS is a kite with diagonals ​​QS‾ ​​ and ​​PR‾ ​​. R 3 1 Q S 2 T 35 P A. What is ​m∠1​? The diagonals of a kite are perpendicular, so ​m∠1 = 90​. COMMON ERROR B. What is ​m∠2​? You may incorrectly assume angles are congruent just from their The sum of the angles of △​ PQT​ is 180. appearance. Always check that you ​m 2 35 90 180​ can prove congruence first. ∠ + + = ​m∠2 = 55​ C. What is m∠3? Since △​ PQR​ is an isosceles triangle, ​m∠3 ≅ m∠QPT.​ So, ​m∠3 = 35.​ Try It! 2. Quadrilateral WXYZ is a kite. X a. What is ​m∠1​? 2 32 b. What is ​m∠2​? 1 W Y Z 366 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess APPLICATION EXAMPLE 3 Explore Parts of an Isosceles Trapezoid Kiyo is designing a trapezoid-shaped roof. In order for the roof to be symmetric, the overlapping triangles △​ DAB​ B C and △​ ADC​ must be congruent. Will the roof be symmetric? A D Step 1 Show △​ ABE ≅ △DCF​. _ _ _ ​​BE ​ ≅_ ​CF ​​, because ​​BE ​​ and ​​CF ​​ are altitudes. GENERALIZE B C How do the lengths of the The roof is an isosceles diagonals and the way they trapezoid, since​ ​DC‾ ​ ≅ ​AB‾ ​ . intersect relate to the sides of a quadrilateral? A E F D ​△ABE​ and △​ DCF​ are right triangles, so they are congruent by the HL Theorem. Step 2 Show △​ DAB ≅ △ADC​. By CPCTC, ∠​ DAB ≅ ∠ADC​. By the Reflexive Property of Congruency, ​​AD‾ ​ ≅ ​DA‾ ​ . So, △​ DAB ≅ △ADC​ by SAS. The overlapping triangles are congruent, so the roof is symmetric. Try It! 3. a. Given isosceles trapezoid PQRS, Q R what are ​m∠P , m∠Q,​ 135 and ​m∠S​? P S b. Given ​​ST‾ ​ ∥ ​RU‾ ​,​ what is the S T measure of ∠​ TUR​? 47 R U LESSON 8-2 Kites and Trapezoids 367 Activity Assess THEOREM 8-4 In an isosceles trapezoid, each pair If... B C of base angles is congruent. A D PROOF: SEE EXERCISE 13. Then... ​∠BAD ≅ ∠CDA, ∠ABC ≅ ∠DCB​ THEOREM 8-5 The diagonals of an isosceles If... B C trapezoid are congruent. A D PROOF: SEE EXERCISE 18. Then... ​​AC‾ ​ ≅ ​DB‾ ​ EXAMPLE 4 Solve Problems Involving Isosceles Trapezoids All horizontal beams of the high-voltage transmission tower are parallel to the ground. 10 ft 1 10 ft 2 The top section is an The center section isosceles trapezoid. is an isosceles 14 ft 14 ft trapezoid. A. If ​m∠1 = 138​, what is ​m∠2​? The sum of the interior angle measures of a quadrilateral is 360. ​m∠1 + m∠1 + m∠2 + m∠2 = 360​ ​138 + 138 + 2(m∠2) = 360​ ​276 + 2(m∠2) = 360​ The base angles ​2(m∠2) = 84​ are congruent. MAKE SENSE AND PERSEVERE ​m∠2 = 42​ What other strategy might you use to solve this problem? The measure of ∠2 is ​42​. CONTINUED ON THE NEXT PAGE 368 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Activity Assess EXAMPLE 4 CONTINUED B. One cross support in the center of the tower measures ​4c 3,​ and the GENERALIZE + Why might this strategy work for other measures ​6c − 5​. What is the length of each cross support? isosceles trapezoids but not for The cross supports are diagonals of trapezoids with noncongruent an isosceles trapezoid, so they are legs? congruent. Step 1 Find the value of c. 14 ft 14 ft ​4c 3 6c 5​ + = − 4c + 3 6c – 5 ​8 = 2c ​ ​4 = c​ Step 2 Find the lengths of the diagonals. ​4c + 3 = 4(4) + 3​ ​ = 19 ​ ​6c − 5 = 6(4) − 5​ =​ 19​ Each cross support measures 19 ft in length. Try It! 4. Given isosceles trapezoid MNOP where the given expressions represent the measures of the diagonals, what is the value of a? N O 2a – 1 a + 13 P M THEOREM 8-6 Trapezoid Midsegment Theorem In a trapezoid, the midsegment If... B C is parallel to the bases, and the length of the midsegment is half X Y the sum of the lengths of the bases. A D PROOF: SEE LESSON 11-2. Then... ​​XY‾ ​ ∥ ​AD‾ ​, ​XY‾ ​ ∥ ​BC‾ ​,​ 1 and ​XY ​ __ ​ (AD BC)​ = 2 + LESSON 8-2 Kites and Trapezoids 369 Activity Assess APPLICATION EXAMPLE 5 Apply the Trapezoid Midsegment Theorem Paxton makes trapezoidal handbags for her friends. She stiches decorative trim along the top, middle, and bottom on both sides of the handbags. How much trim does she need for three handbags? Explain. 6 in. 2 in. 2 in. 2 in. 2 in. 9 in. Formulate The top and bottom sides of the handbag are the bases of a trapezoid. The left and right sides are the legs. Since the middle segment divides both legs in half, it is the midsegment of the trapezoid. The midsegment of a trapezoid is the segment that connects the midpoints of the legs. Let x represent the length of the midsegment in inches. Compute Step 1 Find the value of x. 1 Apply the Trapezoid Midsegment Theorem ​x = ​ __ ​(6 + 9)​ 2 with the base lengths 6 and 9. ​x = 7.5​ The length of the midsegment is 7.5 in. Step 2 Find the amount of trim that she needs. First, find the amount for one side. ​6 + 9 + 7.5 = 22.5​ Then, multiply by 2 for the number of sides per handbag and by 3 for the number of handbags. ​22.5 ∙ 2 ∙ 3 = 135​ Interpret Paxton needs 135 inches of trim. Try It! 5. Given trapezoid JKLM, what is KL? K L 25 m X Y J 4 M 5 XY 370 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com Concept Summary Assess CONCEPT SUMMARY Kites and Trapezoids WORDS Kites Trapezoids A kite is a quadrilateral with two A trapezoid is a quadrilateral with exactly one pair pairs of adjacent sides congruent of parallel sides. The length of the midsegment and no pairs of opposite sides is the average of the lengths of the two bases. congruent. Exactly one diagonal A trapezoid with congruent legs is an isosceles is a perpendicular bisector of trapezoid that has congruent base angles and the other. congruent diagonals. DIAGRAMS Quadrilateral ABCD is a kite. ​Quadrilateral RSTU is an isosceles trapezoid. B S T E A B A C R U D ​SU‾ ​ ≅ TR‾​ ​​ ​​AB‾ ​​ ∥ ​​ST‾ ​​ ∥ ​​RU‾ ​​ ​AC‾ ​ ⟂ BD‾​ ​ 1 ​AB = ​ __ ​(ST + RU)​ m∠S = m∠T BE = ED 2 m∠R = m∠U Do You UNDERSTAND? Do You KNOW HOW? 1. ESSENTIAL QUESTION How are For Exercises 5–7, use kite X diagonals and angle measures related in WXYZ to find the measures. kites and trapezoids? 5. m∠XQY Q W Y 56 12 mm 2. Error Analysis What is Reagan’s error? 6. m∠YZQ 7. WY Q R Z P S For Exercises 8–10, use trapezoid DEFG ✗ with ​EG = 21​ ft and ​m∠DGF = 77​ to find By Theorem 8-5, PR =~ QS each measure. 8. ED E F 9. DF 8 ft 3. Vocabulary If ​​XY‾ ​​ is the midsegment of a trapezoid, what must be true about point X 10. m∠DEF and point Y? 77° 4. Construct Arguments Emaan says every D G kite is composed of 4 right triangles. Is he 11. What is the length 54 cm correct? Explain. B C of ​​PQ‾ ​​? P Q A D 33 cm LESSON 8-2 Kites and Trapezoids 371 Scan for Practice Tutorial PRACTICE & PROBLEM SOLVING Multimedia Additional Exercises Available Online UNDERSTAND PRACTICE 12. Construct Arguments Write a two-column 16. Given kite ABCD, in which B proof to show that the diagonals of a kite ​AN = 4.6 m​, what is ​AC​? are perpendicular. SEE EXAMPLE 1 N A C X 4.6 m W Y Z D 13.