Transcript Chapter 8 Section 4 (CPCTC)

DRILL
Prove each pair of triangles are congruent.
Given: AB ║ CD,
BC ║ DA
Given: PQ is congruent
to QR and PS is
congruent to SR.
B
A
Q
C
D
R
P
S
Objectives:
Use congruent triangles to plan and write
proofs.
Use congruent triangles to prove
constructions are valid.
8.4
Using Triangle Congruence
CPCTC
Corresponding parts
When you use a shortcut (SSS, AAS,
SAS, ASA, HL) to show that 2 triangles
are congruent, that means that ALL the
corresponding parts are congruent.
B
That means that EG  CB
A
F
What is AC congruent to?
C
Segment FE
E
G
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding parts of congruent
triangles are congruent.
Corresponding Parts of Congruent
Triangles are Congruent.
CPCTC
You can only use CPCTC in a
proof AFTER you have proved
congruence.
Ex. Proof
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
M
R
A
S
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
R
A
S
Statements:
Reasons:
1. A is the midpoint of
MT, A is the midpoint
of SR.
2. MA ≅ TA, SA ≅ RA
3. MAS ≅ TAR
4. ∆MAS ≅ ∆TAR
5. NS is congruent to TR
1. Given
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
Statements:
1. A is the midpoint of
MT, A is the midpoint
of SR.
2. MA ≅ TA, SA ≅ RA
3. MAS ≅ TAR
4. ∆MAS ≅ ∆TAR
5. NS is congruent to TR
Reasons:
R
A
S
1. Given
2. Definition of a
midpoint
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
R
A
S
Statements:
Reasons:
1. A is the midpoint of
MT, A is the midpoint
of SR.
2. MA ≅ TA, SA ≅ RA
3. MAS ≅ TAR
4. ∆MAS ≅ ∆TAR
5. NS is congruent to TR
1. Given
2. Definition of a
midpoint
3. Vertical Angles
Theorem
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
R
A
S
Statements:
Reasons:
1. A is the midpoint of
MT, A is the midpoint
of SR.
2. MA ≅ TA, SA ≅ RA
3. MAS ≅ TAR
4. ∆MAS ≅ ∆TAR
5. NS is congruent to TR
1. Given
2. Definition of a
midpoint
3. Vertical Angles
Theorem
4. SAS Congruence
Postulate
T
M
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: NS is congruent to TR.
R
A
S
Statements:
Reasons:
1. A is the midpoint of
MT, A is the midpoint
of SR.
2. MA ≅ TA, SA ≅ RA
3. MAS ≅ TAR
4. ∆MAS ≅ ∆TAR
5. NS is congruent to TR
1. Given
2. Definition of a
midpoint
3. Vertical Angles
Theorem
4. SAS Congruence
Postulate
5. CPCTC
T
Ex. 2: Using more than one pair of
triangles.
Given: 1≅2,
3≅4.
Prove ∆BCE≅∆DCE
Prove: segment DC
is congruent to
segment CD
D
C
2
1
4
E
B
3
A
Given: QSRP,
segment PT≅ segment RT 1
Prove:
segment PS≅ segment RS
Q
4
3
P
Statements:
Reasons:
1. QS  RP
2. PT ≅ RT
1. Given
2. Given
T
S
R
Homework
Page 429
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