PLrallelogrums. Tests for. Lesson 6-2. If EF = GH; FG = EH, then Quad. EFGH is a parallelogram. Proving Quadrilaterals as Parallelograms

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1 Lesson 6-2 Tests for PLrallelogrums Proving Quadrilaterals as Parallelograms Theorem L: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If EF = GH; FG = EH, then Quad. EFGH is a parallelogram. G Theorem 2: E If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrtlaterul is a parallelogram. If EF = GH and EF ll HG, then Quad. EFGH is a parallelogram.

2 Theorem 3: Theorem.' If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If /.H = /.F and /.E = /-G, then Quad. EFGH is a parallelogra Theorem 4: If the diagonals of a quadrilateral bisect each other, then the_ quadrilateralis aparallelogram.ll M i.s'thentic{point d li(i ancl I;H then Quad. EFGH is a parallelogram. ',nn. _ /.\n/r ^-r rrn/r _ rrn EM: GM and HM: FM H 5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are ll.lctejinttionl 2. Show that both pairs of opposite sides are =. 3. Show that one pair of opposite sides are both = and ll. 4. Show that both pairs of opposite angles are =. 5. Show that the diagonals bisect each other.

3 Examples... Example 1: Find the value of x and y that ensures the quadrilateral is a parallelogram. 6x: 4x+8 2x:8 x: 4 units 2y : y+2 y -2 unit 4x+8 Examplg 23 Find the value of x and y that ensure the quadrilateral is aparallelogram. Zx+ g: 120 5y + 120: 1g0 (2x + 8)', 2x: 112 5y: " x: 56 units y :12 5y", units

4 lr-l Chapter 6: Quadrilaterals Lesson 6-2: Tests for Parallelograms Classwork Name: Date: Period: Determine if each quadrilateral is a parallelogram. If so, by which theorem(s)? If not, why? l. 4. one?i^y' oqgoj,fie eacs NL \1fl UY1L{AeLtr oj^d' T;!rd!he[, ln ^n,-.p 'thth T)e{r: W' 3Y-tl =i Find the values of x and y that ensure each quadrilateral is a parallelogram. 3x-11 l\b-- 4v-4 140, 'lx v TY=37 UD=25 v 7x=31 x = lb.5 ^ffi 911=LS ua''5 8. ry X = 3x-15 -Zx, -lf _ (*-4 10y+6 t3b+l}u+b= l@ #=q t 4l-+t0t;: 136 loyt,fr Iv=7.51

5 For problems 9-13 use the given information to detennine that quadrilateral ABCD is a parallelogram or not. If you are not given enough information write "can't determine." 9. AB=13, CD=13. Lo*{+ d,elur**e, 10. mzabe = mzedc, AD = BC C^trli+ d+fuyvttyrn 1 l. AE=12,8C=24 ca,rl + d,,glwn^,t W 12. AB=BC, mzbad = mzbcd CA,n'+ d*fwnl " "5t6'ffit'"-Ui.*C eo{h 6t'0 13. BD=2(BE), AC=2(EC) For problems 14, 15 use the given information to determine that the quadrilateral is a parallelogram. If you are not given enough information write "can't determine." 14. Two consecutive pairs of angles are supplementary. 1@ Xt,. 15. Opposite sides are parallel yu, o

6 U-L Notes Chapter 6: Quadrilaterals Lesson 6-2: Tests for Parallelograms Homework zf zu-n l+\l T'*nqu, &,oduri dt *u, qn*'.e'ti- Nar ffnn h-l \$.8'tl P#,.?u\t-:hu." 4"*3 'b*$ ffi,r+ q Determine if each quadrilateral is a parallelogram. If there is not enough intbrmation, write " cara' t determine." 1 4. Y';rpoi, QEr s;&s?*^11d,^y^ Find the values of x and y that ensure each quadrilateral is a parallelogram SxJfl = }x-l 2x=lt_ ne+2x+4 = 180 Il 7+2y"16 2x= to1 faffi1 8. 4y=Zx+\ 2x= -J = Sg+At - 6^-- 3b 6'zl #=ro 3q = 2x+1,,I= rqy1 3-u = b+1?w

7 For problems 9-13 use the given information to determine that quadrilateral ABCD is a parallelogram or not. If you are not given enough information write "can't determine." 9. AD=BC, mzbad+mzabc =1-80" tlc6.,, 6Y\e Vl( opp. 6idet g aj,\d, il. 10. AB = CD, BCIIAD NO ml_bcd = J2", nlcda= 1 1 1" po. BE=ED, mzdab = mzbcd N0. mzbec = 90" NO. For problems 14, 15 use the given information to determine that the quadrilateral is a parallelogram. If you are not given enough information write "can't determine." 14. Diagonals are congruent NO. 15. Opposite angles are congruent.\b- J o

8 Javascript not enabled Nu*", I 6-2: Self Check Quiz Multiple Choice Identify the choice that best completes the strtentent or answers the question. rb;i i For problems 1-3, find the values of x and y that will ensure the quadrilateral shown in each problem is a parallelogram. '7-- L ]1 Jita 28=3x+4 24= 3y X=8 t(*) 4b = zy 40-la =zt ''l Y"I r-ft-f.xx:6,y=9 b. x:8,y-12 --xx: 12,y:4 d. x: 8, y:6 tz=3y X:b t\-_43+z ll-- 4vt ^J u= 5 rc--l J. a. x:6, y=3.h *:8, y:4,{ x:4, Y:3 d. x: 6, y:5 a. b. x:12. y: 15 x:25, y: 15 c. d. x:15, y:25 x:18,

9 l-f-l ' \-/ r5-r 4. Which of the following would not indicate that a quadrilateral is a parallelogram? Fboth pairs of opposite angles are congruent --br opposite sides are parallel and congruent 5. c. vertical angles are congruent k diagonals bisect each other Which statement would necessarily be true if quadrilateral ABCD is a parallelogram? a. Dl :D3 b. AB:BC fn-l ' LJ ---r parallelograrn? 6. Which piece of information would indicate that quadrilateral QRST is a c. d. AC:BD BE: DE f---l t_j A,cp a. t-l,,-1,+ =,qf I. b. g,q t tst II. c. I and II d. none ofthese 7. Which piece of information would indicate that quadrilateral QRST is a parallelogram? a. US :9 -YRT:3, QS :9 c. QS : 18. RT: 6 d. RU:3

10 [ru 8. Which piece of information would indicate that quadrilateral QRST is a L, parallelogram? t-l a. IRUS =IQUT b. IURS =ITQU c. IURS =le'tu d. IRQU =IUST I$-f '.i/ 9. Which piece of information would indicate that quadrilateral QRST is a parallelogram? a- a. RU:UT b. RS: QT c. QT: RQ d. TS: RQ f.j2. I 10. Which piece of information would indicate that quadrilateral QRST is a ' t..) parallelograrn? 2v -1= ll 2x= 2u Y= lo F a. x:8 b. x: 10 c. x:11.5 d. x: 15.5

11 l-a -l I1. Which piece I of infonnation would indicate that quadrilateral QRST is a -l-1 'r \ parallelogram? r-c: Te-f u QT=T5 PE=-1-l F, a. x:8.67 b. x=8 c. x: d. x: Which piece of information would indicate that quadrilateral QRST is a parallelogram? a P a. x:0,y:0 b. x:20,y:30 c. x: 60,y:40 d. x:60,y:20 Pj+ a. x=0.5,y:9 b. x:31.67,y=9.y x: 0.5, y :7.44.il x:31.67,y:7.44 3x-?+ 3v$ = 3+ bx- l8 = 3* R 2KtX =6A 3x = l6d Y= b'o z(r.) = :3 tza,31,1'4"" 13. Which piece of infonnation would indicate that quadrilateral QRST is a paralletograrrr? Qq_1 = 1+ I "' cltl =Sl,J u' tl,) by=b2 x= g.b1 5x- b+x-4=tt0 [rx -10 = l&) b{= 0a X > 31,b1

12 6-2 Warmup,l^,"* 21 Numeric Response 1. in the above diagram, find the values of x and y to ensure the figure is a parallelogram. AB = 4x*10, rln : 46, AD :jy-4,andtc:2y+11 U**tlr==Zt If.----\ v \,, Ir -ffi [x = tt \ z\a= 2o 2.,48=zx+2, ce=.u*13, DE: ly-5,andee: y+15 zxyza-\ 1 2Yrz-- x+13 3l-5 = \+t5 3. m/.d=(2x-1,1)o,ru/.c=117o,m,{.8=f /\ zy+lflo,lp - G* ud 2]=14 U=or 4. I.4p=B,E5=5r-y,AB=4r+yandcD=1e 8,5V1 4x +(sx+),n f4il-g If A+!,1 sx 5. wt.fl =83o,rn/5= (2rJy+3) o,nl/z= 4Oo, m/6 =1iO'- to;'' 8e-- Zo3+b,-{o = K'-zo AA --'-l 5> = lox lox- {o Tv-g = n q'15-? V=ix-g "'qi=27 ffi B ry CD