Proof. Proof of Theorem 6-7. Given: AC and BD bisect each other at E. Prove: ABCD is a parallelogram.

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1 6- hat You ll Learn o determine whether a quadrilateral is a parallelogram... nd hy o use a parallel rule to plot a ship s course, as in ample roving hat a Quadrilateral s a arallelogram heck kills You ll eed GO for Help Use the figure at the right.. Find the coordinates of the midpoints of and. hat is the relationship between and?. Find the slopes of and.. How do they compare? ee back of book.. re and parallel? plain. 4. hat type of figure is? parallelogram Lessons -8 and -7 y O lan Objectives o determine whether a quadrilateral is a parallelogram amples Finding Values for arallelograms s the Quadrilateral a arallelogram? Real-orld onnection s the Quadrilateral a arallelogram? Key oncepts heorem 6-5 heorems 6-5 and 6-6 are converses of heorems 6- and 6-, respectively, from the previous lesson. hey provide two ways to conclude that a quadrilateral is a parallelogram. f both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ath ackground he conditions necessary for a quadrilateral to be a parallelogram are also sufficient, as proved in this lesson. his allows using the biconditional if and only if to combine and catalogue the theorems and their converses in these lessons. ore ath ackground: p. 04 Real-orld onnection he frame remains a parallelogram as it is raised and lowered, and the backboard stays vertical. roof of heorem 6-5 Key oncepts heorem 6-6 Given: X > ZY and XY > Z rove: XYZ is a parallelogram. roof: raw diagonal XZ. ince opposite sides of XYZ are congruent, #XZ > #YZX by. Using, &XZ > &YZX, so X 6 ZY. lso, &ZX > &YXZ, so Z 6 XY. XYZ is a parallelogram by definition. You will complete a proof of heorem 6-6 in ercise. f both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. X Lesson 6- roving hat a Quadrilateral s a arallelogram Y Z Lesson lanning and ee p. 04 for a list of the resources that support this lesson. oweroint ell Ringer ractice heck kills You ll eed For intervention, direct students to: Finding the idpoint Lesson -8: ample tra kills, ord roblems, roof ractice, h. lope lgebra Review, p. 65: ample hecking for arallel Lines Lesson -7: ample tra kills, ord roblems, roof ractice, h. pecial eeds L ut out the bottom of a cardboard bo. easure the opposite sides of the bo to show they are congruent. hen change the angles between the adjacent edges of the bo to demonstrate heorem 6-8. learning style: visual elow Level L For ample, have students make a parallel ruler using straws and connectors such as brads. learning style: tactile

2 . each Guided nstruction uditory Learners efore students actually read the flow proof for heorem 6-7, have them focus on the diagram of and suggest a lan for roof. tudents who suggest the same basic ideas found in the proof will profit from the logical sequencing of their ideas in the proof. ath ip heorem 6-8 is the only theorem in this lesson that is not the converse of a theorem from Lesson 6-. Key oncepts heorem 6-7 roof heorem 6-7 is the converse of heorem 6- of the previous lesson. f the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. roof of heorem 6-7 Given: and bisect each other at. rove: is a parallelogram. Vertical ' are. and bisect each other at. Given ef. of seg. bisector Vertical ' are. XL fter students find and y, have them find the length of each segment to check their work. sk: How can you tell that your answers are correct? he diagonals bisect each other. lternative ethod ncourage students to suggest other ways to prove heorem 6-5. For eample, instead of using again, for &ZX &YXZ and XY 6 Z, use heorem 6-8. oint out to students that two of the bonuses of learning geometry are becoming more creative mathematically and seeing many ways to prove relationships. Real-orld onnection Frank Lloyd right, a famous architect, used parallelograms in designs of many houses, such as the Kraus House in Kirkwood, issouri. f alt. int. ', then lines are. is a parallelogram. ef. of parallelogram f alt. int. ', then lines are. heorem 6-8 suggests that if you keep two objects of the same length parallel, such as cross-country skis, then the quadrilateral determined by their endpoints must be a parallelogram. You will prove heorem 6-8 in ercise. Key oncepts heorem 6-8 f one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. You can use algebra and heorems 6-7 and 6-8 to find values for which quadrilaterals are parallelograms. hapter 6 Quadrilaterals dvanced Learners L4 Have students find countereamples if the word both is deleted from heorems 6-5 and 6-6. learning style: verbal nglish Language Learners LL Have students make a table listing the five ways to prove two quadrilaterals congruent: four theorems and the definition of a parallelogram. hen discuss why both is such a critical term in some theorems. learning style: visual

3 4 5 est-aking ip Read test questions carefully. Here the value of y is choice, but you are asked for the value of. Quick heck For: arallelogram ctivity Use: nteractive etbook, 6- XL Finding Values for arallelograms ultiple hoice For what value of must L be a parallelogram? 9 7 iagonals of a parallelogram bisect each other, so y 7 5 y and 5 y. y - 7 = y + rite the equation with one variable. y - 7 = y = 9 ollect the variables on one side. olve. ubstitute 9 for y in the second equation. Find. = y = 9 ubstitute 9 for y. = olve. For =, L is a parallelogram. nswer is correct. Find the values of a and c for which QR must be a parallelogram. 70, You can conclude that a quadrilateral is a parallelogram if both pairs of opposite sides are parallel. heorems 6-5 through 6-8 provide four shortcuts to prove that a quadrilateral is a parallelogram. XL s the Quadrilateral a arallelogram? an you prove the quadrilateral is a parallelogram from what is given? plain. a. Given: & > &, b. Given: L > LO, > O rove: & > &, & > & rove: is a parallelogram. L Q (a 40) y 7 y R a c - c + rove: LO is a parallelogram. L y lternative ethod fter students read the eplanation for part a, ask: hat is another way you could prove the quadrilateral is a parallelogram? wo sets of alternate interior angles are congruent, so both pairs of opposite sides are parallel. XL XL iversity any students are unfamiliar with navigation and plotting a ship s course. sk for volunteers who can eplain what a ship s compass looks like and how it is used. oweroint dditional amples Find values of and y for which must be a parallelogram. y 80 G 0 4 8, y 89 etermine whether the quadrilateral must be a parallelogram. plain. a y 9 7 a. Yes; Q and R are congruent and parallel, so QR is a parallelogram. Quick heck b. o; it is possible that the diagonals do not bisect each other, so FG would not be a parallelogram. O Yes, both pairs of opposite o, the given information angles are congruent. is a is not enough to prove parallelogram by heorem 6-6. LO is a parallelogram. an you prove the quadrilateral is a parallelogram? plain. a b. ee left. a. Given: Q > R, Q 6 R b. Given: H > GH, H > FH rove: QR is a parallelogram. rove: FG is a parallelogram. Q R H F G b. o; you do not know whether both pairs of opposite sides are congruent Yes; both pairs of opposite angles are congruent. Lesson 6- roving hat a Quadrilateral s a arallelogram

4 oweroint dditional amples he captain of a fishing boat plots a course toward a school of bluefish. One side of a parallel rule connects the boat with the school of bluefish. he other side makes a 6 angle north of due east on the chart s compass. plain how the captain knows in which direction to sail to reach the bluefish. ecause the parallel rule forms a parallelogram, the captain should sail 6 north of due east. aily otetaking Guide 6- L aily otetaking Guide 6- dapted nstruction L Real-orld onnection areers marine navigator has great responsibility for the ship, its crew, its cargo, its mission, and the surrounding natural marine environment. XL Real-orld onnection avigation parallel rule is a navigation tool that is used to plot ship routes on charts. t is made of two rulers connected with congruent crossbars, such that = and =. You place one ruler on the line connecting the ship s present position to its destination. hen you move the other ruler onto the chart s compass to find the direction of the route. plain why this instrument works. losure Using the theorems you have learned in hapter 6, write two different biconditionals about parallelograms. ample: quadrilateral is a parallelogram if and only if both pairs of opposite angles are congruent. quadrilateral is a parallelogram if and only if its diagonals bisect each other. Quick heck. Once in place, both rulers show the direction and remain n. Keep the second ruler in place and move the first ruler to get the compass reading. 4 Quick heck XR ample (page ) ample (page ) 4 hapter 6 Quadrilaterals he crossbars and the sections of the rulers are congruent no matter how they are positioned. o, is always a parallelogram. ince is a parallelogram, the rulers are parallel. herefore, the direction the ship should travel is the same as the direction shown on the chart s compass. ritical hinking uppose the ruler connecting the ship s position to its destination gets in the way of reading the compass. How can you get the desired reading? ee margin. For more eercises, see tra kill, ord roblem, and roof ractice. ractice and roblem olving GO ractice by ample for Help 7. Yes; both pairs of opp. sides are O. 8. o; the quad. could be a kite. 9. Yes; both pairs of opp. ' are O. lgebra Find the values of and y for which must be a parallelogram y y, y 4.6, y (4 - ) ( + 8) ee margin, pp etermine whether the quadrilateral must be a parallelogram. plain y y + +.

5 GO ample (page 4) pply Your kills nline Homework Help Visit: Hchool.com eb ode: aue Fishing Quadrilaterals are formed on on the side of this fishing tackle bo by the adjustable shelves and connecting pieces. plain why the quadrilaterals remain parallelograms no matter what position the shelves are in. ee margin.. ombine each of heorems 6-, 6-, and 6- with heorems 6-5, 6-6, and 6-7, respectively, into biconditional statements. ee back of book.. eveloping roof omplete the proof of heorem 6-6. Given: & > & and & > & rove: is a parallelogram. tatements Reasons. + y + + y = 60. he sum of the measures of the angles of a quadrilateral = 60.. ( + y) = 60 a. 9 istr. rop.. + y = 80 b. 9 iv. rop. of q. 4. & and & are supplementary. 4. efinition of supplementary & and & are supplementary. f same-side int.? are supp., the lines are n. c. 969, 969 ee below. d is a parallelogram. e. 9 ef. of ~ c. 6, roof. rove heorem Given: 6 YX and > YX rove: XY is a parallelogram. X (Hint: raw one or both diagonals. Find congruent triangles. Use.) Y ee back of book. lgebra Find the values of the variables for which must be a parallelogram y ( 0) G (8 5) 6 y 4 5y 5, y 5 y 9 6., y 7. (4a - ) c - 5 c + (a + 5) c 8, a 4 k 9, m.4 8. ultiple hoice From which given information can R you conclude that R is a parallelogram? R 6, R > R >, R > R 6, > R RZ > Z, Z > Z m k Open-nded ketch two noncongruent parallelograms and FGH such that > G and > FH. ee margin. m y.6k Z. ractice ssignment Guide -5 hallenge 6-8 est rep 9- ied Review -9 Homework Quick heck o check students understanding of key skills and concepts, go over ercises, 7, 6,, 7. ercises 6 Have students identify the theorems they use to establish that the quadrilateral is a parallelogram. rror revention! ercise 7 Remind students that parallelograms with more precise names are still parallelograms. sk: re both pairs of opposite sides congruent? yes hat figure has this description? parallelogram ercise 8 Have students copy the figure and write the givens for each answer choice to see if the given information is sufficient. G nrichment Guided roblem olving Reteaching dapted ractice ractice ame lass ate ractice 6- roving hat a Quadrilateral s a arallelogram tate whether the information given about quadrilateral is sufficient to determine that it is a parallelogram... X X, X X X., 4. X X, 5. X X, X X 6., 6 lgebra Find the values of and y for which the figure must be a parallelogram y 6 9 y 4 6 () (y 8) (5y ) L4 L L L L 0. t remains a ~ because the shelves and connecting pieces remain n. Lesson 6- roving hat a Quadrilateral s a arallelogram 5 F earson ducation, nc. ll rights reserved. lgebra Find the value of. hen tell whether the figure must be a parallelogram. plain your answer ( 0) ( 5) ( 5) 6 () ecide whether the quadrilateral is a parallelogram. plain your answer (4 0) ( 5) () 9. nswers may vary. ample: H G 5

6 4. ssess & Reteach oweroint Lesson Quiz roof 0. Given: &K > &L. Given: nr > nr rove: &LK > & rove: R is a parallelogram. rove: KL is a parallelogram. 0. ee back of book. K Find the values of the variables for which GH must be a parallelogram.. G 6y H 0y L R oordinate Geometry Given points,, and in the coordinate plane as shown, find the fourth point described below.. point so that is a parallelogram (4, 0). point so that is a parallelogram (6, 6) 4. point F so that F is a parallelogram (, 4) y 4 O 4. 6, y 0.75 G a (4b 8) H a 4, b 6 etermine whether the quadrilateral must be a parallelogram. plain.. hallenge 5. riting ummarize the ways to show that a quadrilateral is a parallelogram. ee margin. 6. robability f two opposite angles of a quadrilateral measure 0 and the measures of the other angles are multiples of 0, what is the probability that the quadrilateral is a parallelogram? 6 roof 7. n the figure at the right, point is constructed by drawing two arcs. One has center and radius. he other has center and radius. rove that is a median of #. ee margin. 8. oordinate Geometry he diagonals of quadrilateral FGH intersect at (-, 4). wo vertices of FGH are (, 7) and F(-, 5). hat must be the coordinates of G and H to ensure that FGH is a parallelogram? G( 4, ), H(, ) est rep 4. Yes; the diagonals bisect each other. ultiple hoice 9. n ~, what is m&? n ~, what is m&? F F. 8 G. 90 H cm 00 cm 5. Yes; one pair of opposite sides is both congruent and parallel. hort Response tended Response. Given: #R > #, 6 rove: is a parallelogram. ee margin, p. 7.. a. rite an equation and solve for. b. s F 6? plain. c. s F a parallelogram? plain. ee back of book. R (7 - ) 6 (5-7) F o; both pairs of opposite sides are not congruent. 5. You can show a quad. is a ~ if both pairs of Opp. sides are n or O, if both pairs of opp. ' are O, if diagonals bisect each other, if all consecutive ' are suppl., or if one pair of opp. sides are both n and O. 6 6 hapter 6 Quadrilaterals 7. nswers may vary. ample:. O, O (Given). is a ~. (f opp. sides are O, then it is a ~.). is the midpoint of. (he diagonals of a ~ bisect each other.) 4. is a median. (ef. of a median). [] tatements Reasons. kr Ok (Given). O (). n (Given) 4. is a ~. (f opp. sides of a quad. are both n and

7 GO ied Review for Help Lesson 6- lgebra Find the value of each variable in each parallelogram. 5. ee margin.. 4. m - 4h 5. 6e ( c a + 5 ) (8 + 5) f - 8 h k m lternative ssessment Give pairs of students a set of two straws of unequal lengths with which to construct the diagonals of a parallelogram. Have them use the theorems in this lesson to eplain why the construction works. Lesson 4-7 Lesson - 6. plain how you can use overlapping congruent triangles to prove >. ee margin. rite the two conditional statements that make up each biconditional. 7. he diagonals of a quadrilateral bisect each other if and only if the quadrilateral is a parallelogram. ee margin. 8. wo lines are parallel if and only if the two lines and a transversal form corresponding angles that are congruent. ee margin. 9. wo nonvertical lines are perpendicular if and only if the product of their slopes is -. ee margin. est rep For additional practice with a variety of test item formats: tandardized est rep, p. 6 est-aking trategies, p. 56 est-aking trategies with ransparencies heckpoint Quiz Lessons 6- through 6-. ml 59, ml, ml 59. ml 4, ml 6, ml 6. ml 06, ml 74, ml 6 lesson quiz, Hchool.com, eb ode: aua-060 O, then the quad. is a ~.) [] proof missing steps. a 8, h 0, k 0 4. m 9.5, 5 5. e, f, c 04 Find the measures of the numbered angles for each parallelogram.... lassify each quadrilateral in as many ways as possible trapezoid, isosc. trapezoid rectangle, ~ rectangle, ~ lgebra Find the values of the variables for which is a parallelogram. 7. ( 0) 8. (y 0) 4 ( 0), y 45, y 60 * ) * ) 4 9. n the figure at the right, 6 6 Find hat is the most precise name for a quadrilateral with vertices (, 5), (-, 4), (, -5), and (7, 4)? kite * F ) Lesson 6- roving hat a Quadrilateral s a arallelogram 7 6. t is given that O and l Ol. y the Refleive rop. of O O, thus k Ok by, so O by. 0 y 0. ee left. 48 F 6 7. f a quad. is a ~, then the diagonals bisect each other; if the diagonals of a quad. bisect each other, then it is a ~. Use this heckpoint Quiz to check students understanding of the skills and concepts of Lessons 6- through 6-. Grab & Go heckpoint Quiz 8. f two lines and a transversal form O corr. ', then the two lines are n; if two lines are n, then a transversal forms O corr. '. 9. f the prod. of the slopes of two nonvertical lines is, then they are #; if two nonvertical lines are #, then the prod. of their slopes is. 7