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Geometry
5.7 Using Congruent Triangles
Essential Question
How can you use indirect triangle to find an
indirect measurement?
January 8, 2016
5.7 Using Congruent Triangles
Goals
Use congruent triangles to solve problems.
 Use CPCTC.

January 8, 2016
5.7 Using Congruent Triangles
How do we prove triangles
congruent?
SSS
 SAS
 ASA
 AAS
 HL(Rt.

January 8, 2016
)
5.7 Using Congruent Triangles
We’ve been using:
If corresponding parts of two triangles
are congruent, then the triangles are
congruent.
January 8, 2016
5.7 Using Congruent Triangles
Now we add:
If two triangles are congruent, then
their corresponding parts are
congruent.
Corresponding Parts of Congruent Triangles are Congruent.
January 8, 2016
5.7 Using Congruent Triangles
The Basic Idea:
Given
Information
•SSS
•SAS
•ASA
•AAS
Prove
Triangles
Congruent
CPCTC
January 8, 2016
5.7 Using Congruent Triangles
Shows
Corresponding
Parts
Congruent
Example 1
B
A
K
C
J
Is ABC  JKL? YES
What’s the reason? SAS
January 8, 2016
5.7 Using Congruent Triangles
L
Example 1 continued
B
K
ABC  JKL
A
C
J
What other angles are congruent?
B  K and C  L
What other side is congruent?
BC  KL
January 8, 2016
5.7 Using Congruent Triangles
L
Example 1 continued
B
K
ABC  JKL
A
C
J
What other angles are congruent?
B  K and C  L
What other side is congruent?
BC  KL
January 8, 2016
5.7 Using Congruent Triangles
L
Example 2
Explain how you can
use the given
information to prove
the hang glider parts
are congruent.
January 8, 2016
The triangles are congruent by ______.
AAS
This means parts like QT and ST are
CPCTC
congruent because ________.
5.7 Using Congruent Triangles
Example 3
SSS
CPCTC
January 8, 2016
5.7 Using Congruent Triangles
Example 4
Surveyors use this strategy
to measure difficult
distances. Explain how you
can use the given
information to find the
distance across the river.
ASA
CPCTC
January 8, 2016
5.7 Using Congruent Triangles
Proofs
Show that two triangles are congruent.
 Then show corresponding parts are
congruent. CPCTC

January 8, 2016
5.7 Using Congruent Triangles
Example 5
Given: HJ || LK and JK || HL
Prove: H  K
Plan: Show JHL  LKJ by
ASA, then use CPCTC.
J
H
Statements
Reasons
1. HJL  KLJ (Alt Int s)
L
K
2. LJ  LJ
3.HLJ  KJL
4. JHL  LKJ
5. H  K
QED
January 8, 2016
5.7 Using Congruent Triangles
(Reflexive)
(Alt Int s)
(ASA)
(CPCTC)
Example 6
Given: MS || TR and MS  TR
Prove: A is the midpoint of MT.
M
Since MS || TR, M  T
(Alt. Int. s)
SAM  RAT (Vert. s)
R
MS  TR (Given)
A
SAM  RAT (AAS)
S
T
Midpoint Definition
Plan: Show the triangles are
MA  AT (CPCTC)
congruent using AAS, then
If M is the midpoint of
MA =AT. By definition, A is the
AB, then
AM MT.
MB.
midpoint
of segment
A is the midpoint of MT
(Def. midpoint)
January 8, 2016
5.7 Using Congruent Triangles
Example 7
Given: MP bisects LMN
and LM  NM
L
LM  NM
PM  PM
PMN  PML
LP  NP
QED
M
January 8, 2016
(Given)
NMP  LMP (def.  bis)
Prove: LP  NP
P
N
MP bis. LMN
5.7 Using Congruent Triangles
(Given)
(Ref)
(SAS)
(CPCTC)
Your Turn
Try the next two proofs on your own.
January 8, 2016
5.7 Using Congruent Triangles
Your Turn Proof 1
Given: AB  DC, AD  BC
Prove: A  C
A
B
Statements
Reasons
1. AB  DC
1. Given
2. AD  BC
2. Given
3. BD  BD
3. Reflexive
4. ABD  CDB 4. SSS
D
C
January 8, 2016
5. A  C
5.7 Using Congruent Triangles
5. CPCTC
Your Turn Proof 2
A
E
C
B
D
1. AC  DC
2. A  D
(given)
(given)
3. ACB  DCE (vert s)
4. ACB  DCE (ASA)
(CPCTC)
5. B  E
January 8, 2016
5.7 Using Congruent Triangles
Proofs
Ask: to show angles or segments
congruent, what triangles must be
congruent?
 Then, how do you prove triangles
congruent? (SSS, SAS, ASA, AAS)
 Prove triangles congruent, then use
CPCTC.

January 8, 2016
5.7 Using Congruent Triangles
Assignment
January 8, 2016
5.7 Using Congruent Triangles