4)Prove𝐴𝐵 ≅ 𝐵𝐶

Statements Reasons (Comments)𝐴𝐵=x+4𝐵𝐶=2x𝐴𝐶=16

Given Great

𝐴𝐵+𝐵𝐶=𝐴𝐶 SegmentAddition Youhadthecorrectidea.ButyouhavetoaddtheSEGMENTS,nottheirmeasurementsfirst

x+4+2x=16 Substitution NOWyouhavethecorrectstatement.WecannowsaythisbecauseweSUBSTITUTEthemeasurementsinfortheirrespectivesegments

3x=12 AdditionPOE Youhavetoshoweachalgebrastep

x=4 DivisionPOE GOOD!(4)+4=82(4)=8

Substitution GOOD!

𝐴𝐵 = 8    8 = 𝐵𝐶 Transitive GOOD!𝐴𝐵 ≅ 𝐵𝐶 Iftwosegmentshavethe

samemeasure,thentheyarecongruent(bydefinition)

Youhaveshownbothsegments=8,butyouhavebeenaskedtoprovethetwosegmentsareCONGRUENT.Soyourlaststatementshouldbeexactlythat.

#9Prove∠𝐵𝐶𝐷  ≅ ∠𝐴Statement Reason Comment∠𝐴Isarightangle𝐶𝐸 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐵𝐶𝐷∠𝐵𝐶𝐷 = 45°

Given GOOD!

∠𝐵𝐶𝐷 ≅ ∠𝐷𝐶𝐸 Whenaraybisectsanangle,thetworesultinganglesarecongruent(bydefinition)

GOOD!

∠𝐷𝐶𝐸 = 45° Iftwoanglesarecongruentthentheyhavethesamemeasure(bydefinition)

Weneedtoestablishthatthemeasureofbothanglesare45°sothatwecanaddthemtogetasumof90°

∠𝐵𝐶𝐷 + ∠𝐷𝐶𝐸 = ∠𝐵𝐶𝐸 Angleaddition Thefinalfocusisonthecongruenceof∠𝐵𝐶𝐸,butallwehavetalkedaboutsofaristhesmallerangles.Sowehavetotransitionfromthesmalleranglestothebiggerangle.

45°+ 45° = ∠𝐵𝐶𝐸 Substitution NOWwecansubstitutethevaluesofthesmalleranglesintothepreviouslyestablishedrelationship

90° = ∠𝐵𝐶𝐸 Addition Wehavetomaketheconnectionthat∠𝐵𝐶𝐸iscongruenttoarightangleA.Toapproachthis,wemustmaketheconnectionwith90°

∠𝐴 =  90° Ifanangleisarightanglethenitmeasures90°(bydefinition)

Toaccommodateyourlaststatementreason,Iaddedinthat∠𝐴was90°.Languageisveryimportant.Ifyouaregoingtoclaimcongruence,youmustestablishbothhavethesamemeasure(90°)ORbotharerightangles.Itisnotenoughtoestablishoneisarightangle,andtheotheris90°.

∠𝐵𝐶𝐷  ≅ ∠𝐴 Iftwoangleshavethesamemeasure,thentheyarecongruent(bydefinition)

GOOD!

#10:Prove∠𝐶𝐸𝐴 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒Statements Reasons CommentsACbisects∠𝐵𝐴𝐷AEbisects∠𝐷𝐴𝐹∠𝐵𝐴𝐹 = 180°

Given

∠𝐵𝐴𝐶  ≅ ∠𝐶𝐴𝐷∠𝐷𝐴𝐸  ≅ ∠𝐸𝐴𝐹

When a ray bisects and angle,the resulting two angles arecongruent(bydefinition)

YouhavetoestablishcongruencebeforeyoucanlabelthemBOTHwith“x”

∠𝐵𝐴𝐶 + ∠𝐶𝐴𝐷 + ∠𝐷𝐴𝐸+ ∠𝐸𝐴𝐹= ∠𝐵𝐴𝐹

Angleaddition Youmustestablishthepieceswehavebeendiscussingadduptothewhole,ifyouplantodiscussthewholeinthenextstatement

∠𝐵𝐴𝐹 𝑖𝑠 𝑎 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 Assumedfromdiagram Ifyouwanttomoveintothefactthatanangleis180°, youhavetoexplainHOWyouknowitis180°.Youknowitis180°becauseitisastraightline.HOWdoyouknowitisastraightline?Youareallowedtolookatthediagramandassumeit.

∠𝐵𝐴𝐹 = 180° If an angle is a straight angle,thenithasameasureof180(bydefinition)

NOWyoucansayit’s180°.

∠𝐵𝐴𝐶 ≅ ∠𝐶𝐴𝐷 = 𝑥∠𝐷𝐴𝐸  ≅ ∠𝐸𝐴𝐹 = 𝑦

Labelingdiagram Ifyouwanttousexandyinyourproof,youhavetoestablishwhatyouarelabeling

𝑥 + 𝑥 + 𝑦 + 𝑦 = 180° Substitution(4times) Nowthatyourvariablesareestablished,youmaysubstitutethemintousethem.

2𝑥 + 2𝑦 = 180° Addition Youmustdescribeeachalgebraicstep.

𝑥 + 𝑦 = 90° DivisionPOE Greatjobrecognizingthis!∠𝐶𝐴𝐸 = ∠𝐶𝐴𝐷 + ∠𝐸𝐴𝐷 AngleAddition Youmustestablishthepieces

wehavebeendiscussingadduptothewhole,(sincethewholeisthefocusoftheproof)

∠𝐶𝐴𝐸 = 𝑥 + 𝑦 Substitution ∠𝐶𝐴𝐸 = 90° TransitivePOE

∠𝐶𝐸𝐴 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 If an angle has a measure of90°, then the angle is a rightangle(bydefinition)

Itisnotenoughtoestablish∠𝐶𝐴𝐸 = 90°. Youwereaskedtoprove“rightangle”not90°.Soyoucantstopat90°.

#11a)Prove𝑚∠1 = 𝑚∠𝐽 +𝑚∠𝐻.Statements Reasons Comments𝑚∠𝐽 +𝑚∠𝐻 +𝑚∠𝐽𝐾𝐻

= 180°Given

∠𝐽𝐾𝐻 + ∠1 = ∠𝐽𝐾𝑀 AngleAddition Wehavetomaketheconnectionbetweenthepartsandthewhole.

∠𝐽𝐾𝑀 𝑖𝑠 𝑎 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 Assumedfromdiagram Wecan’tclaimthemeasureis180untilweestablishtheangleisstraight.

∠𝐽𝐾𝑀 = 180° Ifanangleisastraightangle,thentheanglemeasures180°

∠𝐽𝐾𝐻 + ∠1 = 180° TransitivePOE Careful!Youclaimed,“assume”here.Butyoucan’ttechnicallyassume180.YouCANassumestraightlinesandangles,whichiswhywehadtoestablishstraightangleforstep3.

∠𝑱𝑲𝑯 + ∠1 = ∠𝐽 + ∠𝐻+ ∠𝑱𝑲𝑯

TransitivePOE GOOD!

𝑚∠1 = 𝑚∠𝐽 +𝑚∠𝐻 SubtractionPOE Whenyousubtractcongruentpartsfrombothsidesoftheequalsign,youmaintainequality!