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6-6 6-6 Exercises Exercises KEYWORD: MG7 6-6 KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide −− −− RS and−− PV; bases:−− legs: PR and VS; −− midsegment: QT Assign Guided Practice exercises as necessary. If you finished Examples 1–2 Basic 13–16, 24, 25, 28, 29, 31 Average 13–16, 24, 25, 28, 29, 31, 37, 38 Advanced 13–16, 24, 25, 28, 29, 31, 37–39 SEE EXAMPLE 1 p. 428 If you finished Examples 1–5 Basic 13–34, 40–42, 45, 47–49, 52–56 Average 13–23, 24–32 even, 33–36, 38–42 even, 43–45, 47–49, 52–56 Advanced 13–23, 28–32 even, 33, 35–42, 44–56 Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In trapezoid PRSV, name the bases, the legs, and the midsegment. , - + / 2. Both a parallelogram and a kite have two pairs of congruent sides. How are the congruent sides of a kite different from the congruent sides of a parallelogram? * 6 3. Crafts The edges of the kite-shaped glass in the sun catcher are sealed with lead strips. JH, KH, and LH are 2.75 inches, and MH is 5.5 inches. How much lead is needed to seal the edges of the sun catcher? If the craftsperson has two 3-foot lengths of lead, how many sun catchers can be sealed? about 20.1 in.; 3 sun catchers SEE EXAMPLE 2 p. 429 Homework Quick Check In kite WXYZ, m∠WXY = 104°, and m∠VYZ = 49°. Find each measure. 8 9 4. m∠VZY 41° Quickly check key concepts. Exercises: 13, 14, 18, 20, 22, 28, 32 6 5. m∠VXW 63° 6. m∠XWZ 54° SEE EXAMPLE 3 p. 430 Answers 2. Possible answer: In a , 2 pairs of opp. sides are . In a kite, exactly 2 distinct pairs of cons. sides are . < 7 8. RW = 17.7, and SV = 23.3. Find TW. 7. Find m∠A. 106° Ç{ - 5.6 / 7 SEE EXAMPLE 4 p. 430 , 9. Find the value of z so that EFGH is isosceles. 2 or -2 6 10. MQ = 7y - 6, and LP = 4y + 11. Find the value of y so that 17 = 5 2 LMPQ is isosceles. _ _ 3 3 * SEE EXAMPLE 5 ÇÊâÊÓ Ê Óä  ʣÓâÊÓ Â 11. Find QR. 14 p. 431 + 12. Find AZ. 9.5 + 8 , ÓÓ * 9 ££° - Ç°£ Îä 432 < Chapter 6 Polygons and Quadrilaterals LESSON 6-6 Practice A 6-6 PRACTICE A Properties of Kites and Trapezoids Fill in the blanks to complete each theorem or definition. diagonals 1. If a quadrilateral is a kite, then its are perpendicular. angles 2. If a quadrilateral is a kite, then exactly one pair of opposite are congruent. 3. A kite is a quadrilateral with exactly two pairs of congruent ge07se_c06_0427_0435.indd 432 consecutive sides . 9/12/05 11:45:42 AM $ 3.25 ABCD is a kite. Use the figure to find each measure in Exercises 4–6. ! # 118° 6 " 4. mD 5. AB 6. CD 118 3.25 6 0 A trapezoid is a quadrilateral with exactly one pair of parallel sides. Name the parts of trapezoid PQRS asked for in Exercises 7–9. 1 3 2 _ _ PQ SR _ ;_ PS ; QR S and R or P and Q 7. both bases 8. both legs 9. one pair of base angles Fill in the blanks to complete each theorem or definition. 10. A trapezoid is isosceles if and only if its diagonals or legs or base s are congruent. 11. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles . 12. If the legs of a trapezoid are an isosceles trapezoid. congruent , then the trapezoid is 13. If a quadrilateral is an isosceles trapezoid, then each pair of KEYWORD: MG7 Resources 432 Chapter 6 base angles is congruent. In an art museum, a statue sits on a pedestal with sides that are isosceles trapezoids. Name the parts of isosceles trapezoid EFGH asked for in Exercises 14 and 15. % & ( ' 14. both pairs of congruent angles HEF GFE; EHG FGH 15. both pairs of congruent segments EH FG ; EG FH _ _ _ _ PRACTICE AND PROBLEM SOLVING 13 14–16 17–18 19–20 21–22 1 2 3 4 5 "" ÇÊ° 13. Design Each square section in the iron railing contains four small kites. The figure shows the dimensions of one kite. What length of iron is needed to outline one small kite? How much iron is needed to outline one complete section, including the square? Independent Practice For See Exercises Example ÇÊ° ÇÊ° In Exercises 27−32, students may not be sure how to begin solving the problem. Point out that they must first determine if the figure is a kite, a trapezoid, or an isosceles trapezoid. Then suggest that they redraw each figure, marking known properties for that type of quadrilateral. £ÇÊ° about 56.6 in.; about 418.3 in. In kite ABCD, m∠DAX = 32°, and m∠XDC = 64°. Find each measure. Extra Practice 14. m∠XDA 58° Skills Practice p. S15 15. m∠ABC 122° Ê,,", ,/ 16. m∠BCD 52° 8 Application Practice p. S33 18. SZ = 62.6, and KZ = 34. Find RJ. 96.6 17. Find m∠Q. 62° - * ££n < + , 19. Algebra Find the value of a so that XYZW is isosceles. Give your answer as a simplified radical. ±4 √ 5 9 < Ó>ÓÊÊÈx®Â 8 7 >ÓÊÊ£x®Â 20. Algebra GJ = 4x - 1, and FH = 9x - 15. Find the value of x so that FGHJ is isosceles. 2.8 21. Find PQ. Î 3.6 * 72.5 + {°Ó 22. Find KR. / , xÓ°x 6 / ÎÓ°x Tell whether each statement is sometimes, always, or never true. 23. The opposite angles of a trapezoid are supplementary. S 24. The opposite angles of a kite are supplementary. S 25. A pair of consecutive angles in a kite are supplementary. N 26. Estimation Hal is building a trapezoid-shaped frame for a flower bed. The lumber costs $1.29 6 ft per foot. Based on Hal’s sketch, estimate the cost of the lumber. (Hint: Find the angle measures 60° in the triangle formed by the dashed line.) 6 ft 20 ft Possible answer: about $60 Find the measure of each numbered angle. 28. m∠1 = 116°; m∠2 = 46° 27. 29. m∠1 = 51°; m∠2 = 16° xÓ 30. 30. m∠1 = 112°; m∠2 = 40° 28. £ m∠1 = 82°; n m∠2 = 128° Ó £ ££È 32. {äâÊÊx®Â £nâ {n Ý Î Ç{Â Ó £ ÎÝ ÇÓ £ Ó 31. Î{Â Ó 29. £ nÓ £ m∠1 = 120° m∠1 = 117° £äâ 6- 6 Properties of Kites and Trapezoids ,%33/. ÈÈ *À>VÌViÊ 0ROPERTIESOF+ITESAND4RAPEZOIDS 6-6 PRACTICE B M!"$ M!"# $ ÈÈ ' ge07se_c06_0427_0435.indd 433 &INDTHEAREAOFN%&' UNIT ( ) % &INDM: 7 : +-AND.-&IND,. + , . 8 9 82° * N XOR 0 ( A ^ (X 2 10)° (X 2 98)° $ 1 ^ ZqORq 3 5SETHEFIGUREFOR%XERCISESAND4HEFIGURESHOWSA ZIGGURAT!ZIGGURATISASTEPPEDFLATTOPPEDPYRAMIDTHAT WASUSEDASATEMPLEBYANCIENTPEOPLESOF-ESOPOTAMIA 4HEDASHEDLINESSHOWTHATAZIGGURATHASSIDES ROUGHLYINTHESHAPEOFATRAPEZOID - TRAPEZOIDMIDSEGMENT M .OPOSSIBLEANSWERTHEREISNOWAYTOUSETHE0YTHAGOREAN4HEOREMTO FINDTHELENGTHOF!%ANDTHUS!#WITHTHEINFORMATIONPROVIDED 0 B1 1 H 3 4 5 2 . 7RITEAPARAGRAPHPROOF 4HEBOTTOMOFTHEZIGGURATISMETERSLONGANDTHETOPOFTHEZIGGURAT ISMETERSLONG&IND-. 0YTHAGOREAN4HEOREM"$ISTHESUMOF"%AND%$4HEAREAIS ?? !# "$ 2 ? %ACHhSTEPvINTHEZIGGURATHASEQUALHEIGHT'IVETHEVOCABULARYTERMFOR-. 5/8/06 1:34:21 PM THELENGTHOF!#AND"%MAYBEFOUNDFROM"!AND!%BYUSINGTHE 0ROVE ? 0154ISARECTANGLE0OSSIBLEANSWER)TISGIVEN ? ? B2 ? 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'IVEN01I3215320432 & " 4 !REA?? !# "$ 5SETHEFIGUREOFTRAPEZOID0123FOR%XERCISE % ' 13Z AND24Z &INDTHE VALUEOFZSOTHAT1234ISISOSCELES (12N 4)° 3 "$AAND!#A&INDTHE VALUEOFASOTHAT!"#$ISISOSCELES 1 (10N 19)° ! % $ 3UPPOSEYOUAREGIVEN"!$!AND"$4ELLWHETHERITISPOSSIBLETOFINDTHEAREAOF !"#$WITHWHATYOUHAVELEARNEDSOFARINTHISGEOMETRYCLASS%XPLAINYOURANSWER 2 &INDTHEVALUEOFXSOTHAT%&'(ISISOSCELES - &INDTHEVALUEOFNSOTHAT0123ISISOSCELES 6-6 PRACTICE C 3UPPOSEYOUAREGIVEN"!!#AND%$4ELLWHETHERITISPOSSIBLETOFINDTHEAREA OF!"#$%XPLAINYOURANSWER9ESPOSSIBLEANSWERTHELENGTHOF !%ISHALF & *À>VÌViÊ 0ROPERTIESOF+ITESAND4RAPEZOIDS 5SETHEFIGUREOFKITE!"#$FOR%XERCISESn 4HEFIGURESHOWSKITE!"#$&INDAFORMULAFORTHE AREAOFAKITEINTERMSOFTHEDIAGONALS!#AND"$ # M$#! ,%33/. " )NKITE!"#$M"!#ANDM"#$ &OR%XERCISESnFINDEACHMEASURE ! 433 . 'IVEN)SOSCELESTRAPEZOID*+, 0ROVEN*.-ISISOSCELES + * , - ? 0OSSIBLEANSWER)TISGIVENTHAT *+,-ISANISOSCELESTRAPEZOIDSO*+ ? ? ? -,AND+,J*-"ASEANGLESINANISOSCELESTRAPEZOIDARECONGRUENTSO *-#ORRESPONDINGANGLESAREEQUALSO.+,*AND.,+ -"YTHE4RANSITIVE0ROPERTYOF#ONGRUENCE.+,.,+.+,AND ? .,+ ? ARETHEBASEANGLESOFO.+,SOITISANISOSCELESTRIANGLE4HUS.+ .,"YTHE3EGMENT!DDITION0OSTULATE*.*+.+AND-.-, .,"YTHE!DDITION0ROPERTYOF%QUALITYANDTHEDEFINITIONOFCONGRUENT ? ? SEGMENTS*.-."ECAUSE*.AND-.HAVETHESAMELENGTHTHEYARE CONGRUENT3OO*.-ISISOSCELES Lesson 6-6 433