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Holt McDougal Geometry
Triangle Congruence: CPCTCTriangle Congruence: CPCTC
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
Triangle Congruence: CPCTC
Warm-up
Identify the postulate or theorem that proves the triangles congruent.
ASAHL
SAS or SSS
Holt McDougal Geometry
Triangle Congruence: CPCTC
Warm-up
4. Given: PN bisects MO, PN MO
Prove: ∆MNP ∆ONP
1. Given2. Def. of bisect3. Reflex. Prop. of 4. Given5. Def. of 6. Rt. Thm.7. SAS
1. PN bisects MO2. MN ON3. PN PN4. PN MO 5. PNM and PNO are rt. s6. PNM PNO
7. ∆MNP ∆ONP
Reasons Statements
Holt McDougal Geometry
Triangle Congruence: CPCTC
Use CPCTC to prove parts of triangles are congruent.
Objective
Holt McDougal Geometry
Triangle Congruence: CPCTC
CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.
Holt McDougal Geometry
Triangle Congruence: CPCTC
SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
Remember!
Holt McDougal Geometry
Triangle Congruence: CPCTC
Example 2: Proving Corresponding Parts Congruent
Prove: XYW ZYW
Given: YW bisects XZ, XY YZ.
Z
Holt McDougal Geometry
Triangle Congruence: CPCTC
Example 2 Continued
WY
ZW
Holt McDougal Geometry
Triangle Congruence: CPCTC
Check It Out! Example 2
Prove: PQ PS
Given: PR bisects QPS and QRS.
Holt McDougal Geometry
Triangle Congruence: CPCTC
Check It Out! Example 2 Continued
PR bisects QPS
and QRS
QRP SRP
QPR SPR
Given Def. of bisector
RP PR
Reflex. Prop. of
∆PQR ∆PSR
PQ PS
ASA
CPCTC
Holt McDougal Geometry
Triangle Congruence: CPCTC
Work backward when planning a proof. To show that ED || GF, look for a pair of angles that are congruent.
Then look for triangles that contain these angles.
Helpful Hint
Holt McDougal Geometry
Triangle Congruence: CPCTC
Example 3: Using CPCTC in a Proof
Prove: MN || OP
Given: NO || MP, N P
Holt McDougal Geometry
Triangle Congruence: CPCTC
5. CPCTC5. NMO POM
6. Conv. Of Alt. Int. s Thm.
4. AAS4. ∆MNO ∆OPM
3. Reflex. Prop. of
2. Alt. Int. s Thm.2. NOM PMO
1. Given
ReasonsStatements
3. MO MO
6. MN || OP
1. N P; NO || MP
Example 3 Continued
Holt McDougal Geometry
Triangle Congruence: CPCTC
Check It Out! Example 3
Prove: KL || MN
Given: J is the midpoint of KM and NL.
Holt McDougal Geometry
Triangle Congruence: CPCTC
Check It Out! Example 3 Continued
5. CPCTC5. LKJ NMJ
6. Conv. Of Alt. Int. s Thm.
4. SAS4. ∆KJL ∆MJN
3. Vert. s Thm.3. KJL MJN
2. Def. of mdpt.
1. Given
ReasonsStatements
6. KL || MN
1. J is the midpoint of KM and NL.
2. KJ MJ, NJ LJ
Holt McDougal Geometry
Triangle Congruence: CPCTC
Lesson Quiz: Part I
1. Given: Isosceles ∆PQR, base QR, PA PB
Prove: AR BQ
Holt McDougal Geometry
Triangle Congruence: CPCTC
4. Reflex. Prop. of 4. P P
5. SAS5. ∆QPB ∆RPA
6. CPCTC6. AR = BQ
3. Given3. PA = PB
2. Def. of Isosc. ∆2. PQ = PR
1. Isosc. ∆PQR, base QR
Statements
1. Given
Reasons
Lesson Quiz: Part I Continued
Holt McDougal Geometry
Triangle Congruence: CPCTC
Lesson Quiz: Part II
2. Given: X is the midpoint of AC . 1 2
Prove: X is the midpoint of BD.
Holt McDougal Geometry
Triangle Congruence: CPCTC
Lesson Quiz: Part II Continued
5. CPCTC
4. ASA4. ∆AXD ∆CXB
6. Def. of mdpt.6. X is mdpt. of BD.
3. Vert. s Thm.3. AXD CXB
2. Def. of midpt.2. AX CX
1. Given1. X is mdpt. of AC. 1 2
ReasonsStatements
5. DX BX