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August 28, 2014 August 28, 2014 August 28, 2014 Name:_______________________! B A E ! Given: ABCD is a rectangle. Prove: AC = BD. ABCD is a parallelogram. Prove: AE = EC and D C BE = ED. ! Statement ABCD is a parallelogram given AB || DC, AD || BC definition of a parallelogram angle EAB congr. to angle ECD, angle ABE congr. to angle EDC AIAs congruent triangle AEB congr. to triangle CED ASA postulate AE congr. to CE, DE congr. to BE CPCTC ! ! ! ! ! ! ! ! ! ! ! ! ! ! Reason C D Statement Reason A Given: ABCD is a rhombus. B A Date: _____________________ Prove: AC BD. ! B E D C ! Statement Reason ABCD is a rectangle given ABCD is a rhombus given angles BAD, ADC, DCB, and CBA are right definition of a rectangle AD congr. BC congr. CD congr. BA definition of a rhombus AD congr. to BC opposite sides of a parallelogram are congruent (this is the first theorem we proved on parallelograms) DE congr. EB, AE congr. EC diagonals of a parallelogram bisect one another (see leftmost proof on this page) angle ADC congr. to angle BCD all right angles are congruent triangle ABE congr. triangle CBE SSS postulate DC congr. to DC reflexive property of congruence angle AEB congr. angle CEB CPCTC triangle ADC congr. to triangle CBD SAS postulate m(angle AEB) + m(angle CEB) = 180º linear pair postulate AC congr. to DB CPCTC m(angle AEB) = m(angle CEB) definition of congruence m(angle AEB) + m(angle AEB) = 180ª substitution 2m(angle AEB) = 180ª combining like terms m(angle AEB) = 90ª division property of equality AC perp. to BD definition of perpendicular lines ! ! ! ! ! ! ! ! ! ~ (x +would 3y) ABCD still be a parallelogram? Explain. Q(-2, A-4)D Q(3, 6) R(2, -4) R(- 1, - 1) B C _____________________ ! S(2, 3) Date: S(3, - a) y-1 Name:_______________________! Write a two-column or a paragraph proof usingWrite each amethod. two-column or a paragraph proof. 15. Given: ~MJK’~ ~KLM a. By Theorem 6.6: If both 14. Given: AB -~ CD, BC ~pairs AF of opposite sides of a quadrilateral are congruent, then !-AFD ~ !-ADF the quadrilateral is a parallelogram. Prove: 6.10: ABCD isIfa one parallelogram. b. By Theorem pair of opposite Prove: MJKL is a parallelogram, K Write a two-column or a paragraph proof. 15 Given: ~ ~,ONPHHK 17. Given:ARQP Parallelogram ~ ~JOI of ~QQ. R GHOI is the midpoint Prove:MRON H1JKisisa aparallelogram. rhombus. Prove: sides of a quadrilateral are congruent and parF C D II M L ! ! Geometry Statement Reason Chapter 6 Resource Book triangle MJK congr. triangle KLM given angle JKM congr. angle LMK, angle JMK congr. angle LKM CPCTC JK || ML, JM || KL AIAs congruent implies lines are parallel MJKL is a parallelogram ! ! ! definition of a parallelogram A ! Cop~,right © McDougal Littel~ Inc. All nghts reserved, Statement ! given triangle AFD is isosceles base angles of isosceles triangle are congruent AD congr. AF definition of isosceles triangle BC congr. AD ABCD is a parallelogram ! M Copyright © McDougal Littell h~c, NI rights reserved. Reason AB congr. CD, BC congr. AF; angle AFD congr. angle ADF transitive property of congruence opposite sides of a parallelogram are congruent (this is the first theorem we proved on parallelograms) R H Q allel, then the quadrilateral is a parallelogram. Statement N Geometry Chapter 6 Resource Book Reason HIJK is a parallelogram; triangle HOI congr. triangle JOI ! given Copyright@ McDougal Li~eII Inc. HIrights congr. KJ, HK congr. opposite sides of a All reserve& IJ parallelogram are congruent (this is the first theorem we proved on parallelograms) HI congr. JI CPCTC HI congr. HK transitive property HI congr. HK congr. JI congr. JK restated for clarity HIJK is a rhombus definition of a rhombus ! ! 18. Given Prov 14. P(7,-1) 15. P(-4, 0) Q(3, 6) R(- 1, - 1) S(3, - a) 16. P(1, 1) Q(3, 7) Q(-a, 4) R(6, 4) S(- 1, - 3) R(-5, 1) Name:_______________________! S(- 2, - ~) ragraph proof. 18. Given: Rectangle RECT Prove: GART ~- ~ACE HK R ! Statement Reason Geometry Chapter 6 Resource Book ! ! ! ! ! ! Date: _____________________