Chapter 4.Congruence of Line Segments,Angles,and Triangles

1 Chapter 4. Congruence of Line Segments,Angles, and Triangles2467. :and Transitive property : 2EF 5 DBand Prove:EF 2EF 5DB1. Halves of equalquantities are Transitive property :CE 5CF,CD 52CE,CB 52 CFProve:CD 52CE1. 5CF2. Substitution 52CF4. Transitive property :RT 5RS,,Prove:RD 5RS2. Substitution Substitution :AD 5 BEand BC 5CD Prove:AC 5BE1. 5CD2. Subtraction 1CD, 4. Partition Substitution >EFEF>CDAB>CDAB>EFEF>CDAB>CD4-1 Postulates of Lines, Line Segments,and Angles (pages 139 140)Writing About Eand F must name the same There is only one positive real number that is the length of a given line segment. A different number is the length of a different line Skills3. :AB5 ADand DC5 5AD1.

2 Transitive propertyof :and Transitive propertyof : m/1 1m/2 590, m/A 5m/2 Prove: m/1 1 m/1 1m/2 5901. m/A 5m/22. m/1 1m/A590 3. : m/A 5m/B, m/1 5m/B,m/2 5m/AProve: m/1 5m/2 m/1 5m/B 1. m/A 5m/B2. m/1 5m/A3. Transitive propertyof m/2 5m/A4. m/1 5m/25. Transitive propertyof >BDBD>CDAD>CDAD>BDBD>CDAD>CD12. :/CDB >/CBDand /ADB >/ABDP rove:/CDA > >/CBD1. >/ABD2. Addition postulate.>/CBD1 Partition postulate.>/CDB1/BDA,/CBA>/CBD1 >/CBA5. 545 Applying Skills16. will :A triangle is :The measures of the sides are triangle is equilateral then all of its sides arecongruent. Congruent sides have will :Points Eand F are distinct and twolines intersect at :The lines do not intersect at lines can intersect at only one they intersect at both E and F, then Eand Fmust name the same point.

3 But we are given that Eand F are distinct. This is a con-tradiction, so the lines do not intersect at will :A line through a vertex of a triangle isperpendicular to the opposite :The line separates the triangle into tworight line through a vertex of a triangle that isperpendicular to the opposite side separates itinto two Triangles . Perpendicular linesintersect to form right angles, so each of thetriangles has a right angle. A triangle with aright angle is a right will :Two points on a circle are endpoints ofa line :The length of the line segment isshorter than the length of the portionof the circle with the same shortest distance between two points isthe line segment between them. Hence, itslength is shorter than the length of the Through two distinct points, one and onlyone line can be Using Postulates and Definitions inProofs (pages 142 144)Writing About symbol refers to a line segment withendpoints A and C with point B lying betweenthem.

4 The symbol refers to a line segment with endpoints A and B with point C lyingbetween them. These symbols could not refer tosegments of the same Since m/ABD590, then:m/ABD 1m/DBC5180 or m/DBC590m/DBC 1m/CBE 5180 or m/CBE 590m/ABE1m/CBE5180 or m/ABE590 Developing Skills3. :,bisects , and bisects . Definition ofcongruent ,3.. the midpoint4. Definition of of ,Eis of .5.,5. Definition of Halves of equalquantities are Definition ofcongruent >CECE512 CBAD512 ABCBABCBFEABFDAB>CBAD>CECBFEABFDAB>CBACB ABC4. :bisects /DCB,bisects /DAB,and /DCB> :/CAB > >/DAB1. m/DCB2. Definition of 5m/DABcongruent /DCB, 3. /DAB4. m/CAB4. Definition of angle5, m/CAB5. Halves of equal 5m/DCAquantities >/DCA6. Definition ofcongruent : ,2.

5 Partition :is a segment and AB : 5CD1. 5AB 1BC,BD 5BC 1CD2. Partition 5BC 1CD3. 5BD4. :is a segment,B is the midpointof , and Cis the midpoint of .Prove: is the midpoint 1..ACBDACABCDABCDAE>BDAE>BE 1 DEBD>BE 1 EDAE>AD 1 DEAD>BEAE>BDAD>BE12m/DCB12m/DABAChCAhACh CAh248 StatementsReasonsCis the midpoint of . Definition of 5BC,3. Definition of BC 5 CDcongruent 5BC 5CD4. Transitive propertyof : Pand Tare distinct points,Pis themidpoint of .Prove: Tis not the midpoint of . states that a line segment hasonly one is the midpoint of ,and T does not name P. Therefore,Tis not : Symmetric Subtraction ,4. Partition :,Eis the midpoint of ,and Fis the midpoint of . 5BC2. Definition ofcongruent is the midpoint 3.

6 Fis the midpointof .4.,4. Definition of Halves of equalquantities are Definition ofcongruent >FCFC512 BCAE512 ADBCADAD>BCAE>FCBCADAD>BCDE>BFBE>BF1 FEDF>DE1EF>BE 2 FEDF 2 EFEF>FEDF>BEDE>BFDF>BERSRSRSBC>CDAB>BC,B D11. :, and and bisect eachother at 5DB 2. Definition ofcongruent bisect 3. other at the midpoint4. Definition of of and of . ,5. Definition of 5EB6. Halves of equalquantities are Definition ofcongruent :bisects /CDA,/3 >/1,/4 >/2 Prove:/3 > /CDA. 1. >/22. Definition of >/1,3. > >/44. Skills13. will distance from Ato is AB. Thedistance from a point to a segment is thelength of a perpendicular from the point to thesegment. In this case, it is will :and bisect each other at N,, and bisect 1.

7 Other at the midpoint 2. Definition of of and of . 'LMLMRSBDDRhDRhAE>EBEB512 DBAE512 ACDBACDBACAC>DBAE>EBDBACAC> 52RN,3. Definition of LM Doubles of equalquantities are 5LM distance from Lto is LN. The distance from a point to a segment is the length of aperpendicular from the point to the this case, it is LN5 Proving Theorems About Angles (pages 152 154)Writing About and /BEDare supplements ofcongruent angles /BEC and /AED. If twoangles are congruent, then their supplements The converse is If two angles aresupplementary, then the angles form a linearpair. supplementary angles do not need to m/ACD1. >/DCA,2. >/DCB3. m/B 5m/DCB,3. Definition of m/A>m/DCBcongruent m/B1m/A5904. Substitution and /Bare 5.

8 Definition of /BFG1. Definition of a form a linear and /ADCform a linear /BFG2. If two angles form aare pair, then they/ADE and /ADCare also >/BFG3. >/BFG4. If two angles arecongruent, then theirsupplements (48) a right 1. a right 3. Definition of >/CEB4. If two angles are rightangles, then they /BCD1. right m/ABC590 and 2. Definition of right m/EBA1m/EBC3. Partition m/EBA1m/EBC4. Substitution /EBC5. Definition of are >/ECB6. >/ECD7. If two angles arecongruent, then theircomplements /DBF1. right m/ABC590 and 2. Definition of right m/ABC1m/DBC3. Partition m/ABC1m/DBC4. /DBC5. Definition of are /DBCand >/DBC6. Reflexive property ' >/CBF7. If two angles arecongruent, then theircomplements 1.

9 /DGE2. Definition of are vertical >/DGE3. Vertical angles m/BHG4. >/CGH5. Definition ofcongruent >/DGE6. Transitive property /DBC1. Definition of a linearform a linear /ECBform a linear /DBC2. If two angles form aare pair, then they/ACBand /ECBare >ACB3. >/ECB4. If two angles arecongruent, theirsupplements 1. at >/CEB2. If two angles intersect to formcongruent adjacentangles, then they a right 1. m/ABC5902. Definition of m/DBA1m/CBD3. Partition 'CEDgCEDgAEBgDCgEFg(Cont.)StatementsReas ons4. m/DBA1m/CBD4. Substitution /CBD5. Definition of are and /DBA6. >/CBD7. If two angles arecomplements of thesame angle, then theyare 570, m/DEB 5110, m/AEC , m/AED5 5m/DEA575,m/AEC5 5m/DEA545,m/BED 5m/AEC 5135 16.

10 520,x 550, m/LPS 5130,m/MPS two angles are straight angles, then they bothmeasure 180 degrees. Hence, they both have thesame measure. By definition of congruent angles,the two angles are two angles,/1 and /2, are both supplementsof the same angle,/3, then m/1 1m/3 5180and m/2 1m/3 5180. By the substitutionpostulate, m/1 1m/3 5m/2 1m/3. By thesubtraction postulate, m/1 5m/2 so /1 > two angles,/1 and /2, are congruent and /3 and /4 are their respective supplements, thenm/1 1m/3 5180 and m/2 1m/4 the substitution postulate, m/1 1m/3 5m/2 1m/4. Since m/1 5m/2, by thesubtraction postulate, m/3 5m/4. Therefore,/3 > 4-4 Congruent Polygons andCorresponding Parts (page 157)Writing About One pair of congruent corresponding sides isnot sufficient to prove two Triangles >nSTRimplies that /R>/S,/S>/T,/T>/R,,, and.