Regents Geometry – Mr. Mezzano
Chapter 4.6 Practice
C.P.C.T.C. Proofs
1)
Name: ________________________
Given: AD  DC, AC  BD
Prove: ABD  CBD
Statement
Reason
1. AD  DC, AC  BD
2. BDA and BDC are right angles
3. BDA  BDC
4. BD  BD
5. ∆BDA  ∆BDC
6. ABD  CBD
1. Given
2. Definition of Perpendicular Lines
3. Right Angles Congruence Theorem
4. Reflexive Property of Congruence
5. SAS Congruence Postulate
6. C.P.C.T.C.
2)
Given: SR  RP, SRT  RPQ , ST || RQ
Prove: ST  RQ
Statement
Reason
1. SR  RP, SRT  RPQ , ST || RQ
2. RST  PRQ
3. ∆RST  ∆PRQ
4. ST  RQ
1. Given
2. Corresponding Angles Postulate
3. ASA Congruence Postulate
4. C.P.C.T.C.
3)
Given: AB  DC, AD  BC
Prove: A  C
Statement
Reason
1. AB  DC, AD  BC
1. Given
2. BD  BD
3. ∆ABD  ∆CDB
4. A  C
2. Reflexive Property of Congruence
3. SSS Congruence Postulate
4. C.P.C.T.C.
4)
Given: YX  WX
ZX bisects YXW
Prove: YZ  WZ
Statement
Reason
1. YX  WX
ZX bisects YXW
2. YXZ  WXZ
3. XZ  XZ
4. ∆YXZ  ∆WXZ
5. YZ  WZ
1. Given
5)
2. Definition of Angle Bisectors
3. Reflexive Property of Congruence
4. SAS Congruence Postulate
5. C.P.C.T.C.
Given: AC  DC, A  D
Prove: B  E
Statement
Reason
1. AC  DC, A  D
2. ACB  DCE
3. ∆ACB  ∆DCE
4. B  E
1. Given
2. Vertical Angles Congruence Theorem
3. ASA Congruence Postulate
4. C.P.C.T.C.
6)
Given: AB  BE , ADB  ECB
Prove: DB  CB
Statement
Reason
1. AB  BE , ADB  ECB
2. ABD  EBC
3. ∆ABD  ∆EBC
4. DB  CB
1. Given
2. Vertical Angles Congruence Theorem
3. AAS Congruence Theorem
4. C.P.C.T.C.
7)
Given: MQ  NT , MQ || NT
Prove: MN  TQ
Statement
Reason
1. MQ  NT , MQ || NT
2. NQM  QNT
1. Given
2. Alternate Interior Angles Theorem
3. NQ  NQ
4. ∆NQM  ∆QNT
5. MN  TQ
3. Reflexive Property of Congruence
4. SAS Congruence Postulate
5. C.P.C.T.C.
8)
Given: O is the midpoint of NP
N  P
Prove: O is the midpoint of SR
Statement
Reason
1. O is the midpoint of NP
N  P
2. NO  OP
3. NOS  POR
4. ∆NOS  ∆POR
5. SO  OR
6. O is the midpoint of SR
1. Given
9)
2. Definition of Midpoint
3. Vertical Angles Congruence Theorem
4. ASA Congruence Postulate
5. C.P.C.T.C.
6. Definition of Midpoint
Given: AB  CD
DAB and BCD are right angles
Prove: ADB  CBD
Statement
Reason
1. AB  CD
DAB and BCD are right angles
2. ∆DAB and ∆BCD are right triangles
3. BD  BD
4. ∆DAB  ∆BCD
5. ADB  CBD
1. Given
2. Definition of Right Triangles
3. Reflexive Property of Congruence
4. HL Congruence Theorem
5. C.P.C.T.C.