6-3 Conditions for ||’ograms Geometry

Thm 6-3-1 B A > ABCD is a parallelogram > C D If one pair of opposite sides of a quadrilateral are  and || then the quadrilateral is a ||’ogram. B A > ABCD is a parallelogram > C D

Thm 6-3-2 A B __ ABCD is a parallelogram __ __ C D __ If both pairs of opposite sides of a quadrilateral are @, then the quadrilateral is a parallelogram. A B __ ABCD is a parallelogram __ __ C D __

Thm 6-3-3 B A (( ) ABCD is a parallelogram )) ( D C If both pair of opposite s of a quadrilateral are @, then the quadrilateral is a ||’ogram. B A (( ) ABCD is a parallelogram )) ( D C

Thm 6-3-4 B A (180-x) x ABCD is a parallelogram x C D If an  of a quadrilateral is supplementary to both of its consecutive s, then the quadrilateral is a ||’ogram. B A (180-x) x ABCD is a parallelogram x C D

Thm 6-3-5 B A __ __ __ __ D C ABCD is a parallelogram If the diagonals of a quadrilateral bisects each other, then the quadrilateral is a ||’ogram. B A __ __ __ __ D C ABCD is a parallelogram

Ex. 1a.) Determine whether the quadrilateral is a ||’ogram and explain why or why not. __ Yes it’s a ||’ogram because both pair of opposite sides are @. __ __ __

Ex. 1b ) Determine whether the quadrilateral is a ||’ogram and explain why or why not. Yes because the 2 triangles are  by SAS post, so it’s a parallelogram because both pairs of sides are . B A __ ) __ __ ( D C

Ex. 1c) Determine whether the quadrilateral is a ||’ogram and explain why or why not. Yes because its parallelogram by the def of a parallelogram. > ^ ^ >

Ex. 1d) Determine whether the quadrilateral is a ||’ogram and explain why or why not. No, because consecutive ‘s are not supplementary. 65o 65o 110o

Ex. 2a Show that JKLM is a parallelogram for a=3 & b=9. K 5b+ 6 L 15a-11 10a+4 J 8b-21 M

Ex. 2b.) Show that PQRS is a parallelogram for x=10 & y=6.5. R Q S P

Ex. 3 Prove that the pts. represent the vertices of a ||’ogram J (-6,2) K(-1,3) L(2,-3) M(-3,-4) Graph; then use one of today’s theorems and distance formula and/or slope formula or use def. of a parallelogram and slope formula.

Ex. 3a.) Using thm 6-3-2 and distance formula JK= ML=

Ex. 3a.) using distance formula JM= KL= Both pairs of opp sides are @

Ex. 3b.) using the def of ||’ogram and slope m of JK= m of ML=

Ex. 3b.) m of JM= m of KL= Both pairs of Opposite sides are ll

Ex. 3c.) Using thm 6-3-1 one pr. Of opp. Sides are congruent… JK= ML=

Ex. 3c.) Using Thm. 6-3-1… & parallel m of JK= m of ML= 1 pr of opp. Sides are congruent & parallel

Assignment