Ruler Postulate Distance A – B Segment Addition .
DAY23GOAL:Use Definitions, Postulates, Properties and Theoremsto prove other shortcuts for measuring.GEOMETRIC POSTULATES;Basic rules for measuringRuler PostulateSegment Addition PostulateProtractor PostulateAngle Addition PostulateDistance a – b If B is between A and C then AB BC ACAngle Measure a – b If B is in the interior of AOC, then m AOB m BOC m AOCALGEBRAIC PROPERTIESCONGRUENCE PROPERTIESAddition, Subtraction, Multiplication, Division, Reflexive, Symmetric,Transitive.Reflexive, Symmetric, TransitiveDEFINITIONSStraight Angle Definition:Linear Pair Definition:Perpendicular PairSupplementary Angles Definition:Complementary Angles Definition:Vertical AnglesUSED IN PROOFSAn angle is a straight angle iff its measure is 180Two angles form a linear pair iff non-shared sides form astraight angle.Two angles form a perpendicular pair iff non-shared sidesform a right angle.The measure of two angles adds up to 180 iff the anglesare supplementary.The measure of two angles adds up to 90 iff the anglesare complementary.Two angles are vertical angles iff their sides form twopairs of opposite rays.
THEOREMS:Have to be proven before being usedTO1.2.3.PROVE THEOREMSStart with the given information.Use a Definition to explain the given information.Reason back from what you would like to prove using Definitions, Properties, Postulates andTheorems that have already been proven.4. Conclude with what needs to be proven.LINEAR PAIR THEOREM:If two angles form a linear pair, then the angles aresupplementary.GIVEN: 1 and 2 form a linear pairPROVE: 1 and 2 are supplementarySTATEMENTREASONS1. 1 and 2 form a linear pair.Given2. TAM is a straight angle.Linear Pair Definition(Two angles form a linear pair iff their non-sharedsides form a straight angle)!3. m TAM 180Straight Angle Definition(An angle is a straight angle iff its measure is180)4.m 1 m 2 m TAMAngle Addition Postulate!5. m 1 m 2 1806.Substitution 1 and 2 are supplementary anglesSupplementary Angle Definition(The measure of two angles is 180 iff the anglesare supplementary)GivenQEDDefinition of SupplementaryThe measure of two angles sums to 180 iff the angles are supplementary.Substitution Property of EqualityIf mÐ 1 m Ð 2 mÐ TAM and mÐTAM 180 Then mÐ1 m Ð2 180 Definition of Linear PairTwo angles form a linear pair
f two angles sums180 re supplementary.m BAT 180 .TO PROVE THEOREMSStart with the given information.Use a Definition to explain the given information.Reason back from what you would like to prove using Definitions, Properties, Postulates andTheorems that have already been proven.f Straight Anglea straight angle, 1.asure is 180 .ition Postulate 2.es are adjacent,3.f their individualthe measure of they their non-shared4.idesConclude with what needs to be provenAT form linearpairIf two angles form a linear pair, then the angles aresupplementary.AT 180 on Property ofualityVERTICAL ANGLES THEOREM:If two angles are vertical angles, then the angles havec d then a b dequal measures.Vertical Angles Theoremform a linear pairIf two angles are vertical angles 2m a linear pair 1 3Thentheyhaveequalmeasures,m 3 180 180 GIVEN: 1 and 2 are vertical anglesair Theorem 1 haveand equal2 are verticalAngles 1Givenand 2measuresform a linearPROVE:pair,plementary.Prove that 1 and 2 have equal measures.GivenLINEAR PAIR THEOREM: BAT m CAT180 - m 3180 - m STATEMENT3Statements1. 1 and 2 are vertical angles1 m 2ve equal measures)of Linear Pairm a linear pair iff 2.sides form a straightare supplementarye supplementaryoperty of Equalitymber is subtracted3.es of an equation,uation is equivalentvertical anglesf Supplementarytwo angles sums to4.180 e supplementary.roperty of Equality1) for (180 - m 3) - m 3) 1 and 3 form a linear pair 2 and 3 form a linear pairREASONSReasonsGivenLinear Pair Definition. form a linear pair iff their non(Two anglesshared sides form a straight angle) 1 and 3 are supplementary 2 and 3 are supplementaryLinear Pair Theorem(If two angles form a linear pair then they aresupplementary)m 1 m 3 180!m 2 m 3 180!5. m 1 180 m 3!Supplementary Angle Definition(The measure of two angles is 180 iff the anglesare supplementary)Subtraction Prop. of Equalitym 2 180 m 3!6. m 1 m 2Substitution
2.4.Proof:ReasonsStatementsLINEAR PAIR THEOREM:If two angles form a linear pair, then the angles arel. Angle Addition Postulatel. mll m/-3 180;supplementarymL2*m/-3-1802. Substitution Prop.mL32. m/-l mL3 - mL2VERTICAL ANGLES THEOREM:If two angles are vertical angles, they have equal measuresm/-33. Reflexive Prop.m13 3.4. Subtraction Prop. of 4. mll- mL2Class DiscussionExampleIGOMILOG- 1.?'-!2.3In the diagram,/-4:/-5.Name two other angles congruent to 15'JJSolutionL8: /-5 4 5 5L4:congruentL5, 17:7 : otherL4 andSinceIn the diagram,. NameLtwoanglesto L5.35,andandwLUOY indthem/-YOW angle.3.Findofthevalueof XExs.Find11-14m/-YOW Y13.15.nIVOWnIVOWFind the measures15.16.m/-SOUm/-SOUand a supplement of LA.of a complement16.17.wLTOUwLTOU17.l.mLA 1018.mLZOTmLZOT18.2.mLA-753.mLA wo right angles.6. Name two adjacent complementary angles.angles that are not adjacent.7. Name two complementary(4x 8)"(4x 8)"8. a. Name a supplement of /- MLQ., b. Name another pair of supplementary angles.19.19.a.b.c.d.e.m/-QIR m/-PIQ m/-VIT -WIVIQ wLSIT:?/x'-nExs.S-EffixOPB9. In the diagram, assume that m/-CDB:90. Name:/\a. TWo congruent supplementary anglesan obtuse angle is ?c. a right angle is ?arenotcongruentthatanglesb. Two supplementaryle is ?supplements.and/-2/-2are/-l22,/-lcomplementary anglessupplements.c.areTwoand22,22 d. A straighthave a complement?GG hitofofnLDFBassume10. In the diagram,- 90 and FE bi- esa.a.lflfm/-l-27,find m13-27, m13each resflrndFindthemeasuresm/-3IAFD.mll sectsofofl-2them/-3b.b.IfIfmll x,x,flrndExs. 9, l0c. upplements be congruent?a. ec.c.IfIftwo?,/ l"your conclusion.Points, Lines, Planes, and evaluevalueofofr BareItItLAangles.angles.23.mLA-2x,mLB-x-15BB 23.mLA-2x,mLB-x-15nd a supplementl.mLBmLB-- 2x2x-- 161616,mLBm/-A xx** ndthethevaluevalueofofyyandcomplementary, findarecomplementary,andlDlDareLCandof LB.lflfLC- iz.sangles.angles.4. m/B - 3x25.mLCmLC 3/ 5,mLD 2y25. 2y 3/ 5,mLD 226.m/-Cm/-C /26.-3y 2 / -8,mLD-8,mLD-3yd complementary. Find their measures.findthetheequationtotofindd supplementary. cribed.measuresofofthethetwomeasuresry angles.entary angles.'hat may or may 0.TheThedifferencedifferencebetween ThreeThreetimes32.of the angle./ 3l
(Def.ofofll nZ.Given3.3.Given4.4.t 5.Def.ofllHnes(Seesteps2 2and,and,4.)4.)Corresponding Angles, Alternate Interior Angles,AlternateExteriorAngles,Same Side Interior Angles, Same Side Exterior lyonlytotocoplanaronlylines. line and two coplanar sversalisisa agram,diagram,I isI isa atransversaltransversalofofn es:anglesExterioranglesl, tionsrelativeing positions relative sisa .c.c.Nameangles.//Chapter5656 Classify2 2 pair of angles as alternate interior angles, same'sideChaptereachinterior angles, corresponding angles, or none of these.2.4.6.8.1217L7Lll3.5.7.9.and L4and Ll5and LIOand 114L617and /-101landand /-12Ll4 and Ll5Lll10.Name alternateexterioralternate exterior angles, yorr10. Althoughwe have notdefinedangles.two pairs of them.whattheyexteriorare. Nameguess11. canNamesamesideangles.11. Name two pairs of angles we would call same-side exteriorangles.t2. Suppose one pair of alternate interior angles are congruent(say, /-2: /-7). Explain why the other pair of alternate13.interior angles must also be congruent.Suppose a pair of same-side interior angles (say, /-2 arrdL3) are supplementary. What must be true of any pair ofcorresponding angles?eac!-pair of lines as intersectrlQ parallel, or ske[c. AB and IDb. AB and FKa. AE and Eie ef. CN and FGe. EF and NMd. EF and IH14. Classify--.4,-. !15. Name six lines parallel rcdi.
nt.CORRESPONDINGANGLES POSTULATEExs. 1-11IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSALLineso 15. THEN CORRESPONDING ANGLES ARE CONGRUENTandangles?a protractor in the lastrsversals,numberedGiven: Parallelcut by a TransversalConclude: Alternate Interior Angles are Congruentthat correspondingangles Linesare conhe didnotinHowever,our previouspostulatesandtheo 1 2If k l, then(Corresponding Angles Postulate)m. we Lineswill accept it as a postulate.llelallelcorresversals, and a protractor in the lastd that corresponding angles are conin ourthenpreviouspostulatesangresand theorsal,correspondingareem. we will accept it as a postulate.ALTERNATE INTERIOR ANGLE THEOREMersal, thenIFcorrespondingTWOtheorems.PARALLELLINES ARE CUT BY A TRANSVERSALangres areove the followingnTHEN ALTERNATE INTERIOR ANGLES ARE CONGRUENTparallel lines are cut by a transverhen alt. int. A- are :.Given: Parallel Lines cut by a TransversalConclude: Alternate Interior Angles are Congruentarc:.al,Athenalternate interior angles areIfk l,theorems.then 1 2(Alternate Interior Angle Theorem)rovefollowingsitive theProp.of zal, then alternate interior angles are.ALTERNATE EXTERIOR ANGLE THEOREM/-1.ReasonsIF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL/-1. l. GivenTHEN ALTERNATE INTERIOR ANGLES ARE CONGRUENT2. Yert. A- arc:3.Given:If two parallelare cut by a transReasonsParallellinesLinescut by a TransversalConclude: Alternate Exterior Angles are Congruentversal, then corr. A- arco.4.TransitivePropertyl.Given2. Yert. A- arc:SAME SIDE INTERIOR ANGLE THEOREM3. If two parallel lines are cut by a transIF TWO PARALLEL LINESARE CUT BY A TRANSVERSALversal, thencorr.A- arcl, then same-sideinterioranglesareo.SAME SIDE INTERIOR ANGLES ARE CONGRUENT4.THENTransitive PropertyGiven: Parallel Lines cut by a Transversalal, then same-side interior angles areIf k l, then 1 4Conclude: Same Side Interior Angles areSupplementary(Same Side Interior Angle Theorem)SAME SIDE EXTERIOR ANGLE THEOREMIF TWO PARALLEL LINES ARE CUT BY A TRANSVERSALTHEN SAME SIDE INTERIOR ANGLES ARE CONGRUENTGiven: Parallel Lines cut by a TransversalConclude: Same Side Exterior Angles areSupplementary
CW#23If l p cut by transversal t, which angle pairs are Corresponding Angles, Alternate Interior Angles,Alternate Exterior Angles, Same Side Interior Angles, Same Side Exterior Angles?Which angle pairs are congruent?Which angle pairs are supplementary?Classroom Exercises1. What do the arrowheads in the diagram tell you?2. a. How are liiies k and I rclated?b. How are lines k and p related? Why?State the postulate or theorem that justifies each statement.3. 1l:/-55. mL4 m167. m/-4 - mL5-1809.kIpll. lf mll :130,12.4. 13: /-66. mL4 - m188. /-6: L710. L3 is supplementary to 15.Exs. 1-11what are the measures of the other numbered angles?/{an tried to prove Postulate l0as shown below. However, he did nothaveavalidproof.Explain12. Find the values of x and whyy. not.If two parallel lines are cut by a transversal, then corresponding angles are congruent.Given: k ll/; transversal I cuts k andProve: 1l: 12/.Proof:Statementst ll t;/1.LJ:Reasonst is a transversal./1LL"V(3v 5,"/-l:3.4.Ll:/-312l. Given2. It 2 parallel lines are cut by a transversal, then alt. int. A- are :.3. Vert. A arc:.4. Transitive Prop. of zWritten ExercisesA l, If q llD, name all angles that must be congruent to /-1.2. If cll d, name all angles that must be congruent to /-1.Assume that allnanA cll a.3. Name all angles congruent to 14.4. Name all angles supplementary to /-4.5. If m/- 16 50, then mLl4 - ? and mL2 6. Ifm/-9 x,thenm/-12- ? andm/ 7- ?62/Chapter 2R?
BHW23:CATPIL 03Linear Pair Theorem ProofIf two angles form a linear pair,Then the angles are supplementary.Given: ÐCAB and ÐBAT form a linear pair.& ÐCAT forms a straight angleProve: ÐCAB and ÐBAT are supplementary.Ð CAB andÐBAT aresupplementaryGivenÐ CAT is a straight angle.StatementsReasonsDefinition of SupplementaryThe measure of two angles sumsto 180 iff the angles are supplementary.mÐCAB m ÐBAT 180 .Definition of Straight AngleIf an angle is a straight angle,then its measure is 180 .Angle Addition PostulateIf two angles are adjacent,then the sum of their individualmeasures equals the measure ofthe angle formed by their nonshared sidesÐCAB and ÐBAT form linear pairGivenmÐCAB m ÐBAT mÐ CATmÐ CAT 180 Substitution Property of EqualityIf a b c and c d then a b dPIL04Ð1 and Ð 3 form a linear pairÐ2 and Ð 3 form a linear pairmÐ 1 m Ð 3 180 mÐ 2 m Ð 3 180 Linear Pair TheoremIf two angles form a linear pair,then they are supplementary.mÐ1 180 - m Ð 3mÐ 2 180 - m Ð 3Vertical Angles TheoremIf two angles are vertical anglesThen they have equal measures,Ð2Ð1 Ð3Given Ð 1 and Ð 2 are vertical AnglesProve that Ð 1 and Ð 2 have equal measures.StatementsReasonsGivenmÐ 1 mÐ 2(Ð 1 and Ð 2 have equal measures)Definition of Linear PairTwo angles form a linear pair ifftheir non-shared sides form a straightangleÐ 1 and Ð 3 are supplementaryÐ 2 and Ð 3 are supplementarySubtraction Property of EqualityIf the same number is subtractedfrom both sides of an equation,Then the new equation is equivalentto the original.Ð1 and Ð 2 are vertical anglesDefinition of SupplementaryThe measure of two angles sums to180 iff the angles are supplementary.
are congruent.Given: k ll/; transversal I cuts k andProve: 1l: 12Substitution Property of EqualityProof:Substitute (mÐ 1)for (180 - m Ð 3)in mÐ 2 (180 - m Ð 3)StatementsReasonsHW#23t ll t; t is a transversal./1.-/.l. Given/12. It 2 parallel lines are cut by a transversal,Correspondingthen alt. int. A-Angles,are :. Alternate Interior Angles,If c d cut by transversal a, which angle pairs are3. ExteriorAlternateSide Exterior Angles?/-3 Angles, Same Side Interior Angles,3. Vert. SameA arc:.LJ:LL/-l:4. Ll:124. Transitive Prop. ofWhich angle pairs are congruent? zWhich angle pairs are supplementary?ShowWrittenyour work.ExercisesA l, If q llD, name all angles that must be congruent to /-1.2. If cll d, name all angles that must be congruent to /-1.Assume that allnanA cll a.3. Name all angles congruent to 14.4. Name all angles supplementary to /-4.5. If m/- 16 50, then mLl4 - ? and mL2 6. Ifm/-9 x,thenm/-12- ? andm/ 7- ?262Find/ Chapterofthe valuesr?and y.ry"A13. Write the reasons.Given: k ll/Prove: L6 is supp."V(3v 5,"13.to L7.Proof:StatementsReasonsproofs in two-column folzr.l.2.3.?4. I5.?*ll r: ft ll/2. m16 to L7.m/-8e: Z I is supplementaryt.-3. m/-8 m/-7 1804. m/-6 m/-7 - 180/-6AC.is supp. to /-7.h 18 ll OC and5.eOll3 on page 60. Then write a proof in?1
3. Straight Angle Definition (An angle is a straight angle iff its measure is 180) 4. Angle Addition Postulate 5. Substitution 6. and are supplementary angles Supplementary Angle Definition (The measure of two angles is 180 iff the angles are supplementary) 1 2 Substitution Given Definition of Linear Pair Two angles form a linear pair
PARALLEL LINE POSTULATE: Through a point not a given line, there is a line parallel to the given line RULER POSTULATE BC 1. Any two points can have coordinates 0 and 1 2. Distance a - b SEGMENT ADDITION POSTULE: If B is in between A and C, then AB BC AC CONGRUENT SEGMENTS: Segments that have same shape and equal lengths.
Euclidean Geometry 300 BCE - Euclid’s Elements Five Postulates. 5th Postulate - not needed in rst 28 propositions. Proclus (410-485) Equivalent postulate. Giralomo Saccheri (1667-1733) Assume 5th postulate false and get contradiction. Used assumption - lines in nite. Led to contradiction of P1, almost P2. Postulate 1 Postulate 2 r Postulate 3
Postulate 3 (The Unique Line Postulate). Given any two distinct points, there is a unique line that contains both of them. Postulate 4 (The Distance Postulate). For every pair of points A and B, the distance from A to B is a nonnegative real number determined by A and B. Pos
Apply the protractor postulate and angle addition postulate to calculate angle measures. Apply the ruler postulate and segment addition postulate to calculate the lengths of line segments. Identify a midpoint or bisector of a line segment or angles. Introduction to Proof Comple
POSTULATE 1 Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between pointsA andB, written asAB, is the absolute value of the difference of the coordi
Geometry Cheat Sheet Chapter 1 Postulate 1-6 Segment Addition Postulate - If three points A, B, and C are collinear and B is between A and C, then AB BC AC. Postulate 1-7 Angle Addition Postulate - If point
Feb 05, 2010 · Euclidean Parallel Postulate. A geometry based on the Common Notions, the first four Postulates and the Euclidean Parallel Postulate will thus be called Euclidean (plane) geometry. In the next chapter Hyperbolic (plane) geometry will be developed substituting Alternative B for the Euclidean Parallel Postulate (see text following Axiom 1.2.2).
postulate is false. If a proof in Euclidean geometry could be found that proved the parallel postulate from the others, then the same proof could be applied to the hyperbolic plane to show that the parallel postulate is true, a contradiction. The existence of the hyperbolic plane shows that the Fifth postulate cannot be proven from the others.
