E D

A C

F

B

B

RC K

A

QuadrilateralReview—ProofSOLUTIONSAsalways,pleasenotethatthereareavarietyofwaystoapproacheachoftheseproofs—thesesolutionsshowjustonepossiblewaytoproveeach.1. Given:ACDFisaparallelogram ∠𝐴𝐹𝐵 ≅ ∠𝐸𝐶𝐷 Prove:FBCEisaparallelogramHint:Thinkaboutthewayswecanprovesomethingisaparallelogram.Whichoneseemslikesomethingwecanprove?

Youcanalsotakeasimilarapproachandprovebothtrianglescongruentandthenworkyourwaytogettingbothpairsofoppositeanglesoftheinteriorquadrilateralcongruentandthusitisaparallelogram.

2.Given:∆𝐶𝐴𝑅isisosceleswithbase 𝐶𝑅 𝐴𝐶 ≅ 𝐵𝐾 ∠𝐶 ≅ ∠𝐾 Prove:BARKisapparallelogramHints: Again,whatarethewayswecanprovesomethingisaparallelogram? Whatdoweknowaboutisoscelestriangles?

1.ACDFisaparallelogram Given2.∠𝐴 ≅ ∠𝐷 Oppositeanglesofaparallelogramarecongruent3.𝐴𝐹 ≅ 𝐷𝐶 Oppositesidesofaparallelogramarecongruent4. ∠𝐴𝐹𝐵 ≅ ∠𝐸𝐶𝐷 Given5.∆𝐴𝐹𝐵 ≅ ∆𝐷𝐶𝐸 ASA6.𝐹𝐵 ≅ 𝐸𝐶 CPCTC

7.𝐹𝐷 ≅ 𝐴𝐶 Oppositesidesofaparallelogramarecongruent

8.𝐹𝐷 ≅ 𝐹𝐸 + 𝐸𝐷and𝐴𝐶 ≅ 𝐴𝐵 + 𝐵𝐶 Whole=sumofpartsorsegmentaddition9.𝐹𝐸 + 𝐸𝐷 = 𝐴𝐵 + 𝐵𝐶 Substitutionproperty

10.𝐹𝐸 ≅ 𝐵𝐶 Subtractionpropertyofequality11.FBCEisaparallelogram Aquadrilateralwithtwopairsofopposite

congruentsides(steps6and9)isaparallelogram

S

TW

RN

X

P

V

3. Given:NRTWisaparallelogram 𝑁𝑋 ≅ 𝑇𝑆 𝑊𝑉 ≅ 𝑃𝑅 Prove:XPSVisaparallelogramHints:Howdoyouprovesomethingisaparallelogram?

1.∆𝐶𝐴𝑅isisosceleswithbase𝐶𝑅 Given2.𝐴𝐶 ≅ 𝐵𝐾 Given

3.𝐴𝐶 ≅ 𝐴𝑅 Definitionofisoscelestriangle

4. 𝐵𝐾 ≅ 𝐴𝑅 Transitiveproperty5.∠𝐴𝐶𝑅 ≅ ∠𝐴𝑅𝐶 ITT6.∠𝐴𝐶𝑅 ≅ ∠𝐵𝐾𝑅 Given7.∠𝐴𝑅𝐶 ≅ ∠𝐵𝐾𝑅 TransitiveProperty8.𝐵𝐾 ∥ 𝐴𝑅 Converseofcorrespondinganglestheorem9.BARKisaparallelogram Iftwooppositesidesofaquadrilateralare

congruent(step4)andparallel(step8),thenthequadrilateralisaparallelogram

1.NRTWisaparallelogram Given2.𝑁𝑊 ≅ 𝑇𝑅and𝑊𝑇 ≅ 𝑁𝑅 Oppositesidesofaparallelogramarecongruent

3.𝑁𝑋 ≅ 𝑇𝑆 Given

4.𝑁𝑊 ≅ 𝑁𝑋 + 𝑋𝑊and𝑇𝑅 ≅ 𝑇𝑆 + 𝑅𝑆 Whole=sumofpartsorsegmentaddition5.𝑁𝑋 + 𝑋𝑊 = 𝑇𝑆 + 𝑅𝑆 Substitutionproperty6.𝑋𝑊 ≅ 𝑅𝑆 Subtractionpropertyof=7.𝑊𝑇 ≅𝑊𝑉 + 𝑉𝑇and𝑁𝑅 ≅ 𝑁𝑃 + 𝑃𝑅 Whole=sumofparts8.𝑊𝑉 + 𝑉𝑇 = 𝑁𝑃 + 𝑃𝑅 Substitutionproperty9.𝑊𝑉 ≅ 𝑃𝑅 Given10.𝑁𝑃 ≅ 𝑉𝑇 Subtractionpropertyof=11.∠𝑁 ≅ ∠𝑇 and ∠𝑊 ≅ ∠𝑅 Oppositeanglesofaparallelogramarecongruent12.∆𝑋𝑊𝑉 ≅ ∆𝑆𝑅𝑃 SAS(steps5,9,7)13.𝑋𝑉 ≅ 𝑆𝑃 CPCTC14.∆𝑋𝑁𝑃 ≅ ∆𝑆𝑇𝑉 SAS(steps4,9,8)15.𝑋𝑃 ≅ 𝑉𝑆 CPCTC16.XPSVisaparallelogram Ifaquadrilateralhastwopairsofoppositecongruentsides

(steps11and13),thenitisaparallelogram.

D

B C

A

JH

M

G K

O

4. Given:ABCDisaparallelogram 𝐵𝐷 bisects ∠𝐴𝐷𝐶 and ∠𝐴𝐵𝐶 Prove:ABCDisarhombus Hints:whatmakesaparallelogramarhombus?

5. Given:GJMOisaparallelogram 𝑂𝐻 ⊥ 𝐺𝐾 𝑀𝐾isanaltitudeof ∆𝑀𝐾𝐽 Prove:OHKMisarectangle

1.ABCDisaparallelogram Given2.𝐵𝐷 bisects ∠𝐴𝐷𝐶 and ∠𝐴𝐵𝐶 Given

3.∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷 and ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵 Definitionofbisector4.∠𝐴𝐵𝐶 ≅ ∠𝐴𝐷𝐶 Oppositeanglesofaparallelogramarecongruent5.∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷 ≅ ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵 Halvesofcongruentanglesarecongruent6.𝐵𝐷 ≅ 𝐵𝐷 Reflexiveproperty7.∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷 ASA8.𝐴𝐵 ≅ 𝐵𝐶and𝐴𝐷 ≅ 𝐷𝐶 CPCTC

9.𝐵𝐶 ≅ 𝐷𝐶and𝐴𝐵 ≅ 𝐴𝐷 ConverseofIsoscelestriangletheorem10.𝐴𝐷 ≅ 𝐴𝐵 ≅ 𝐵𝐶 ≅ 𝐷𝐶 Transitiveproperty11.ABCDisarhombus Definitionofrhombus

1.𝑂𝐻 ⊥ 𝐺𝐾 Given2.∠𝑂𝐻𝐽 = 90 Definitionofperpendicular3.𝑀𝐾isanaltitudeof ∆𝑀𝐾𝐽 Given4.∠𝑀𝐾𝐽 = 90 Definitionofaltitude5.∠𝑂𝐻𝐽 and ∠𝑀𝐾𝐽aresupplementary Theirsumis180°5.𝑂𝐻 ∥ 𝑀𝐾 Converseofconsecutiveanglestheorem

6.GJMOisaparallelogram Given7.𝑂𝑀 ∥ 𝐺𝐾 Definitionofparallelogram8.OHKMisaparallelogram Definitionofparallelogram(steps5and8)9.OHKMisarectangle Aparallelogramwith1rightangleisarectangle

6. Given:ABCDisrectangle 𝐴𝐸 ≅ 𝐵𝐹Prove:DEFCisanisoscelestrapezoid Hint:whattwothingsmustyouprovetoshowthatitisanisoscelestrapezoid?

7. Given:𝐼𝐷bisects𝑅𝐵 𝐵𝐼 ≅ 𝐼𝑅Prove:𝐵𝐼𝑅𝐷isakite

1.ABCDisarectangle Given

2.𝐴𝐸 ≅ 𝐵𝐹 Given

3.𝑚∠𝐴 = 𝑚∠𝐵 = 90 Definitionofrectangle4.𝐴𝐷 ≅ 𝐵𝐶 Oppositesidesofarectangleareparallel5.∆𝐴𝐸𝐷 ≅ ∆𝐵𝐹𝐶 SAS6.𝐸𝐷 ≅ 𝐹𝐶 CPCTC7.𝐸𝐹 ∥ 𝐷𝐶 Oppositesidesofarectangleareparallel(becausea

rectangleisaparallelogram)8.DEFCisanisoscelestrapezoid Definitionofisoscelestrapezoid(twoparallelbases(step8)

andtwocongruentlegs(step7))

1.𝐼𝐷bisects𝑅𝐵 Given

2.𝐵𝐾 ≅ 𝐾𝑅 Definitionofbisector3.𝐵𝐼 ≅ 𝐼𝑅 Given4.𝐼𝐾 ≅ 𝐼𝐾 ReflexiveProperty5.∆𝐵𝐼𝐾 ≅ ∆𝑅𝐼𝐾 SSS6.∠𝐵𝐾𝐼 ≅ ∠𝑅𝐾𝐼 CPCTC7.∠𝐵𝐾𝐼 + ∠𝑅𝐾𝐼 = 180 Defofsupplementaryangles8.∠𝐵𝐾𝐼 ≅ ∠𝑅𝐾𝐼 = 90 Congruentsupplementaryanglesarerightangles9.∠𝑅𝐾𝐷 ≅ ∠𝐵𝐾𝐷 Transitiveproperty10.𝐷𝐾 ≅ 𝐷𝐾 Reflexiveproperty11.∆𝐵𝐷𝐾 ≅ ∆𝑅𝐷𝐾 SAS(steps2,9,10)12.𝐵𝐷 ≅ 𝑅𝐷 CPCTC13.𝐵𝐼𝑅𝐷isakite Definitionofkite(steps3and12)

F B

D C

A E