definition of a parallelogram opposite sides of parallelogram

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

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

C B <1 <4 <3 <2 Given: ABCD is a parallelogram Prove: AB = CD AD = CB D A statement reason ABCD is a parallelogram Draw AC given through any two points there exists exactly one line definition of a parallelogram alternate interior angles theorem reflexive property ASA CPCTC BC // DA AB // CD <1 = <2 <3 = <4 AC = AC ∆ABC = ∆CDA AB = CD AD = CB

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

C B <1 <4 <3 Given: ABCD is a parallelogram Prove: <B = <D hint: opp sides of parallel are cong <2 D A statement reason ABCD is a parallelogram Draw AC given through any two points there exists exactly one line opposite sides of parallelogram are congruent reflexive property SSS CPCTC AB = CD BC = DA AC = AC ∆ABC = ∆CDA <B = <D 2/5

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

B C <1 <3 E <4 <2 A Given: ABCD is a parallelogram Prove: AE = CE BE = DE D statement reason ABCD is a parallelogram <1 = <2 <3 = <4 BC = DA ∆BCE = ∆DAE AE = CE BE = DE given alternate interior angles theorem opposite sides of parallelogram are congruent ASA CPCTC 5/5

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

C B <3 <2 Given: BC // DA BC = DA Prove: ABCD is a parallelogram <4 <1 D A statement reason BC // DA BC = DA <1 = <3 AC = AC ∆ABC = ∆CDA <2 = <4 AB // CD ABCD is a parallelogram given alternate interior angles theorem reflexive property SAS CPCTC alternate interior angles converse definition of a parallelogram

C B Given: E Prove: ABCD is a parallelogram AC BD AB = AD D A Note: figure not drawn to scale statement reason ABCD is a parallelogram AC BD given 5/7

C B E Given: ABCD is a parallelogram AC BD Prove: AB = AD D A Note: figure not drawn to scale statement reason ABCD is a parallelogram given AC BD given BE = DE diagonals of a parallelogram bisect <AEB = 90 definition of lines <AED = 90 definition of lines <AEB = <AED substitution AE = AE reflexive property ∆AEB = ∆AED SAS AB = AD CPCTC

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

C B E Given: ABCD is a rhombus Prove: <AEB = 90 D A Note: figure not drawn to scale statement reason ABCD is a rhombus AB = BC AE = CE BE = BE ∆ABE = ∆CBE <AEB = <CEB <AEB + <CEB = 180 <AEB + <AEB = 180 2(<AEB) = 180 <AEB = 90 given definition of a rhombus diagonals of a rhombus bisect each other reflexive property SSS CPCTC linear pair postulate substitution combine like terms division property

B C Given: Prove: A ABCD is a parallelogram AC = BD <A = 90 D statement reason ABCD is a parallelogram AC = BD given 7/7

B C A Given: ABCD is a parallelogram AC = BD Prove: <A = 90 D statement reason ABCD is a parallelogram AC = BD AD = BC AB = AB ∆ABD = ∆BAC <A = <B BC // DA <A + <B = 180 <A + <A = 180 2(<A) = 180 <A = 90 given opposite sides of a parallelogram are equal reflexive property SSS CPCTC definition of a parallelogram same side interior angles theorem substitution combine like terms division property