Proving Triangles
Congruent
Triangle Congruency Short-Cuts
If you can prove one of the following short
cuts, you have two congruent triangles
1.
2.
3.
4.
5.
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)
HL (hypotenuse-leg) right triangles only!
Built – In Information in
Triangles
Identify the ‘built-in’ part
Shared side
SSS
Parallel lines
-> AIA
Vertical angles
SAS
Shared side
SAS
SOME REASONS For Indirect
Information
•
•
•
•
•
•
•
Def of midpoint
Def of a bisector
Vert angles are congruent
Def of perpendicular bisector
Reflexive property (shared side)
Parallel lines ….. alt int angles
Property of Perpendicular Lines
This is called a common side.
It is a side for both triangles.
We’ll use the reflexive property.
HL ( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.
HL
ASA
Name That Postulate
(when possible)
Reflexive
Property
SAS
Vertical
Angles
SAS
Vertical
Angles
SAS
Reflexive
Property
SSA
Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
For AAS:
A  F
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 4
G
K
I
H
J
ΔGIH  ΔJIK by AAS
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 5
B
A
C
D
E
ΔABC  ΔEDC by ASA
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 6
E
A
C
B
D
ΔACB  ΔECD by SAS
Determine if whether each pair of triangles is congruent by
SSS, SAS, ASA, or AAS. If it is not possible to prove
that they are congruent, write not possible.
Ex 7
J
M
K
L
ΔJMK  ΔLKM by SAS or ASA
Determine if whether each pair of triangles is
congruent by SSS, SAS, ASA, or AAS. If it is
not possible to prove that they are congruent,
write not possible.
J
T
Ex 8
K
L
V
Not possible
U
Problem #4
AAS
A
Given: A  C
BE  BD
Prove: ABE  CBD
C
B
E
Statements
D
Reasons
Given
Vertical Angles Thm
Given
4.
ABE 
CBD
AAS Postulate
37
Problem #5 AHL
Given ABC, ADC right s,
AB  AD
Prove:
B
3. AC  AC
ABC  ADC
C
Statements
1. ABC, ADC right s
AB  AD
D
Reasons
Given
Given
Reflexive Property
4.
ABC 
ADC
HL Postulate
38
Congruence Proofs
1. Mark the Given.
2. Mark …
Reflexive Sides or Angles / Vertical Angles
Also: mark info implied by given info.
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?
39
Given implies Congruent
Parts
midpoint


parallel
segment bisector
segments

angles
segments
angle bisector

angles
perpendicular

angles
40
Example Problem
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
A
B
C
D
41
Step 1: Mark the Given
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
… and
what it
implies
A
B
C
D
42
Step 2: Mark .
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
•Reflexive Sides
..
•Vertical Angles
A
B
C
D
… if they exist.
43
Step 3: Choose a Method
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
SSS
SAS
ASA
AAS
HL
A
B
C
D
44
Step 4: List the Parts
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
STATEMENTS
S
AB  AD
A
S
BAC  DAC
A
B
C
D
REASONS
AC  AC
… in the order of the Method
45
Step 5: Fill in the Reasons
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
STATEMENTS
A
B
C
REASONS
S
AB  AD
Given
A
S
BAC  DAC
Def. of Bisector
Reflexive (prop.)
AC  AC
D
(Why did you mark those parts?)
46
Step 6: Is there more?
Given: AC bisects BAD
AB  AD
Prove: ABC  ADC
STATEMENTS
S 1. AB  AD
1.
2. AC bisects BAD 2.
A 3. BAC  DAC 3.
4.
S 4. AC  AC
5. ABC  ADC 5.
A
B
C
D
REASONS
Given
Given
Def. of Bisector
Reflexive (prop.)
SAS (pos.)
47
Congruent Triangles Proofs
1. Mark the Given and what it implies.
2. Mark … Reflexive Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons …
why you marked the parts.
6. Is there more?
53
Using CPCTC in Proofs
• According to the definition of congruence, if two
triangles are congruent, their corresponding parts
(sides and angles) are also congruent.
• This means that two sides or angles that are not
marked as congruent can be proven to be congruent
if they are part of two congruent triangles.
• This reasoning, when used to prove congruence, is
abbreviated CPCTC, which stands for
Corresponding Parts of Congruent Triangles are
Congruent.
54
Corresponding Parts of
Congruent Triangles
• For example, can you prove that sides AD and BC are
congruent in the figure at right?
• The sides will be congruent if triangle ADM is congruent
to triangle BCM.
– Angles A and B are congruent because they are marked.
– Sides MA and MB are congruent because they are marked.
– Angles 1 and 2 are congruent because they are vertical
angles.
– So triangle ADM is congruent to triangle BCM by ASA.
• This means sides AD and BC are congruent by CPCTC.
55
Corresponding Parts of
Congruent Triangles
• A two column proof that sides
AD and BC are congruent in the
figure
at right is
shown below:
Statement
Reason
MA @ MB
Given
ÐA @ ÐB
Given
Ð1 @ Ð2
DADM @
DBCM
Vertical angles
AD @ BC
CPCTC
ASA
56
Corresponding Parts of
Congruent Triangles
• A two column proof that sides
AD and BC are congruent in the
figure
at right isReason
shown below:
Statement
MA @ MB
Given
ÐA @ ÐB
Given
Ð1 @ Ð2
DADM @
DBCM
Vertical angles
AD @ BC
CPCTC
ASA
57
Corresponding Parts of
Congruent Triangles
• Sometimes it is necessary to add an
auxiliary line in order to complete a
proof
Statement
Reason
• For example, to prove ÐR @ ÐO in
@ FO
Given
thisFRpicture
RU @ OU
Given
UF @ UF
reflexive prop.
DFRU @ DFOU SSS
ÐR @ ÐO
CPCTC
58
Corresponding Parts of
Congruent Triangles
• Sometimes it is necessary to add an
auxiliary line in order to complete a
proof
Statement
Reason
• For example, to prove ÐR @ ÐO in
@ FO
Given
thisFRpicture
RU @ OU
Given
UF @ UF
Same segment
DFRU @ DFOU SSS
ÐR @ ÐO
CPCTC
59