definition of a parallelogram opposite sides of parallelogram are congruent opposite angles of a parallelogram are congruent diagonals of a parallelogram bisect each other alternate interior angles theorem consecutive interior angles theorem alternate interior angles theorem converse consecutive interior angles theorem converse definition of a rhombus

A B C D Given:ABCD is a parallelogram 1/7 Prove:AB = CD AD = CB statementreason ABCD is a parallelogramgiven Draw ACthrough any two points there exists exactly one line <1 <2 <3 <4

A B C D Given:ABCD is a parallelogram Prove:AB = CD AD = CB statementreason ABCD is a parallelogramgiven Draw ACthrough any two points there exists exactly one line BC // DAdefinition of a parallelogram AB // CDdefinition of a parallelogram <1 = <2alternate interior angles theorem <3 = <4alternate interior angles theorem AC = ACreflexive property ∆ABC = ∆CDAASA AB = CDCPCTC AD = CBCPCTC <1 <2 <3 <4

A B C D Given:ABCD is a parallelogram 2/7 Prove:<B = <D hint: opp sides of parallelogram are cong statementreason ABCD is a parallelogramgiven Draw ACthrough any two points there exists exactly one line <1 <2 <3 <4

A B C D Given:ABCD is a parallelogram 2/5 Prove:<B = <D hint: opp sides of parallel are cong statementreason ABCD is a parallelogramgiven Draw ACthrough any two points there exists exactly one line AB = CDopposite sides of parallelogram are congruent BC = DAopposite sides of parallelogram are congruent AC = ACreflexive property ∆ABC = ∆CDASSS <B = <DCPCTC <1 <2 <3 <4

A B C D Given:ABCD is a parallelogram 3/7 Prove:AE = CE BE = DE statementreason ABCD is a parallelogramgiven E <2 <3 <4 <1

A B C D Given:ABCD is a parallelogram 5/5 Prove:AE = CE BE = DE statementreason ABCD is a parallelogramgiven <1 = <2alternate interior angles theorem <3 = <4alternate interior angles theorem BC = DAopposite sides of parallelogram are congruent ∆BCE = ∆DAEASA AE = CECPCTC BE = DECPCTC E <2 <3 <4 <1

A B C D Given:BC // DA BC = DA Prove:ABCD is a parallelogram statementreason BC // DAgiven BC = DAgiven <2 <3 <4 <1 4/7

A B C D Given:BC // DA BC = DA Prove:ABCD is a parallelogram statementreason BC // DAgiven BC = DAgiven <1 = <3alternate interior angles theorem AC = ACreflexive property ∆ABC = ∆CDASAS <2 = <4CPCTC AB // CDalternate interior angles converse ABCD is a parallelogramdefinition of a parallelogram <2 <3 <4 <1

A B C D Given:ABCD is a parallelogram AC  BD Prove:AB = AD statementreason ABCD is a parallelogramgiven AC  BDgiven E Note: figure not drawn to scale 5/7

A B C D Given:ABCD is a parallelogram AC  BD Prove:AB = AD statementreason ABCD is a parallelogramgiven AC  BDgiven BE = DEdiagonals of a parallelogram bisect <AEB = 90definition of  lines <AED = 90definition of  lines <AEB = <AEDsubstitution AE = AEreflexive property ∆AEB = ∆AEDSAS AB = ADCPCTC E Note: figure not drawn to scale

A B C D Given:ABCD is a rhombus Prove:<AEB = 90 statementreason ABCD is a rhombusgiven E Note: figure not drawn to scale 6/7

A B C D Given:ABCD is a rhombus Prove:<AEB = 90 statementreason ABCD is a rhombusgiven AB = BCdefinition of a rhombus AE = CEdiagonals of a rhombus bisect each other BE = BEreflexive property ∆ABE = ∆CBESSS <AEB = <CEBCPCTC <AEB + <CEB = 180linear pair postulate <AEB + <AEB = 180substitution 2(<AEB) = 180combine like terms <AEB = 90division property E Note: figure not drawn to scale

BC D Given:ABCD is a parallelogram AC = BD Prove:<A = 90 statementreason ABCD is a parallelogramgiven AC = BDgiven A 7/7

BC D Given:ABCD is a parallelogram AC = BD Prove:<A = 90 statementreason ABCD is a parallelogramgiven AC = BDgiven AD = BCopposite sides of a parallelogram are equal AB = ABreflexive property ∆ABD = ∆BACSSS <A = <BCPCTC BC // DAdefinition of a parallelogram <A + <B = 180same side interior angles theorem <A + <A = 180substitution 2(<A) = 180combine like terms <A = 90division property A