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Congruent Triangles
Polygons MNOL and ZYXW are congruent
∆ABC and ∆DEF are congruent
Rectangles ABCD and EFGH are not congruent
A
B
D
C
E
F
H
G
Y
Z
∆ZXY and ∆JLP
are not
congruent
L
X
J
P
4-1 Congruent Figures
Objective:
To recognize congruent figures and their
corresponding parts
Vocabulary/ Key Concept
• Congruent polygonstwo polygons are congruent if
their corresponding sides and
angles are congruent
Naming Congruent Figures
Ang Legs Triangle: Construct two triangles with the following sides-1 red, 1
blue, 1 yellow
∆ABC and ∆DEF
óA
óB
óC
Warm Up: WXYZ  JKLM. List 4 pairs
of congruent sides and angles.
•
•
•
•
WX  JK
XY  KL
YZ  LM
ZW MJ
•
•
•
•
W  J
K  X
Y  L
Z  M
Each pair of polygons are congruent.
Find the measure of each numbered
angle
• M1 = 110
• m 2 = 120
• M3 = 90
• m 4 = 135
We know:
óB
óF
óA
óE
Then we can conclude:
óC
óD
Key Concept: If two angles in a triangle are
congruent to two angles in another triangle, then
the third angles are congruent.
WARNING: This is only true for ANGLES not side lengths!
Concept Check!
How do we know
if two triangles
are congruent?
Objective:
To prove two triangles are
congruent using SSS and SAS
Postulates
Key Concepts
• SSS – Side-side-side corresponding
congruence.
If three sides of one triangle are congruent
to three sides of another triangle, then the
two triangles are congruent
(all corresponding sides are equal)
Example 1: State if the two triangles are congruent. If they are, write a
congruence statement and state how you know they are congruent.
Student
Slide #1
Example 2: State if the two triangles are congruent. If they are, write a
congruence statement and state how you know they are congruent.
Student
Slide #2
Key Concepts
• SAS – Side-Angle-Side corresponding
Congruence.
ANGLE MUST
BE IN BETWEEN
THE TWO SIDES
(INCLUDED
ANGLE)
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of
another triangle, then the two triangles are congruent
Example 1: State if the two triangles are congruent. If they are, write a
congruence statement and state how you know they are congruent.
Student
Slide #3
Example 2: Can you use SAS to prove these two triangles are congruent?
If no, what information would you need in order to use SAS to prove these
triangles are congruent?
Student
Slide #4
Determine if you can use SSS or SAS to
prove two triangles are congruent.
Write the congruence statement.
AB  CB --CONGRUENCE MARKING
BD  BD – REFLEXIVE PROPERTY OF
CONGRUENCE
ABD  CBD –CONGRUENCE
MARKING
 ABD   CBD by SAS
Example:
If we know:
óB
óE
What other information must we know in order to prove
∆ABC
∆DEF using SAS?
WARM UP (will be collected):
a) Name the three pairs of corresponding sides
b) Name the three pairs of corresponding angles
c) Do we have enough information to conclude that
the two triangles are congruent? Explain your
reasoning.
*CORRESPONDING
DOES NOT MEAN
THEY ARE
CONGRUENT!
WUP#1: Determine if you can use SSS or
SAS to prove two triangles are congruent.
Write the congruence statement.
What do you know?
NP QP -- CONGRUENT MARKS
NR QR -- CONGRUENT MARKS
RP RP -- REFLEXIVE PROPERTY OF 
 PRN   PRQ by
SSS
WUP #2: What one piece of additional
information must we know in order to prove
the triangles are congruent using SAS. Explain
your reasoning and then write a congruence
statement.
Explanation:
Statement:
Objective:
To prove two triangles are
congruent using ASA, AAS,
and HL Postulates
Key Concepts
• ASA – Two angles and an included side.
SIDE IS IN BETWEEN
THE ANGLES
If two angles and the included side of one triangle are
congruent to two angles and the included side of
another triangle, then the two triangles are congruent.
Key Concepts
• AAS – Two angles and a non-included side.
If two angles and the non-included side of a
triangle are congruent to two angles and the nonincluded side of another triangle, then the two
triangles are congruent.
Determine if you can use ASA or AAS
to prove two triangles are congruent.
Write the congruence statement.
Determine if you can use ASA or AAS to
prove two triangles are congruent and
explain your reasoning. Then write the
congruence statement.
Explain:
Determine if you can use ASA or AAS to prove
two triangles are congruent and explain your
reasoning. Then write the congruence
statement.
Explain:
TRY ONE
Congruence that works: 
SSS
SAS
AAS
ASA
Congruence that does not work: 
ASS
SSA
AAA
*Remember, we
don’t swear in
math (not even
backwards). And
no screaming!
What did you learn today?
• What are the five ways
(one for right triangles)
to prove triangles are
congruent?
So what do we know about the parts of congruent triangles?
Congruent Parts of Congruent Triangles are Congruent
Hence,
*Remember, you can only use CPCTC,
AFTER you have proven two triangles to
be congruent!
CPCTC Song (sung to the tune of “YMCA” by the Village People)
Author of lyrics: Eagler
Young man, there's no need to feel down
I said, young man, pick yourself off the ground
I said, young man, 'cause there's a new thing I've found
There's no need to be unhappy
Young man, there's this thing you can do
I said, young man, it's so easy to prove
You can use it, and I'm sure you will see
Many ways to show congruency
It's fun to solve it with C-P-C-T-C
It's fun to solve it with C-P-C-T-C
Barely takes any time, uses only one line
It's the easiest thing you'll find
It's fun to solve it with C-P-C-T-C
It's fun to solve it with C-P-C-T-C
If you don't have a clue, it's so simple to do
Write five letters and you'll be through
Example 1:
Complete the 2 column proof:
Given:
,
óABE
óDEB
Prove:
Statements
Reasons
Write a Proof
Statement
1. FJ  GH JFH  GHF
2. HF  FH
3.  JFH  GHF
4. FG  JH
Reasons
1. Given
2. Reflexive property of
congruence
3. SAS
4. CPCTC
TRY ONE: Write a Proof
Given: óBAC  óCDE, AC  CD
Prove: óB  óE
Statement
1. AC  CD, óBAC 
óCDE
2.  ACB  ECD
3.  DEC   ABC
4.  B  E
Reasons
1. Given
2. Vertical angles
3. ASA
4. CPCTC
What did you learn today?
• Write down one thing
you understand
• Write down one thing
you still find confusing