2. Triangle Congruency Short-Cuts
If you can prove one of the following short
cuts, you have two congruent triangles
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) right triangles only!
6. 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
7. This is called a common side.
It is a side for both triangles.
We’ll use the reflexive property.
8. HL( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.
HL ASA
9. Name That Postulate
(when possible)
SAS
SAS
SAS
Reflexive
Property
Vertical
Angles
Vertical
Angles
Reflexive
Property SSA
10. Let’s Practice
Indicate the additional information needed
to enable us to apply the specified
congruence postulate.
For ASA:
For SAS:
B D
For AAS: A F
AC FE
11. 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.
ΔGIH ΔJIK by AAS
G
I
H J
K
Ex 4
12. ΔABC ΔEDC by ASA
B A
C
E
D
Ex 5
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.
13. ΔACB ΔECD by SAS
B
A
C
E
D
Ex 6
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.
14. ΔJMK ΔLKM by SAS or ASA
J K
L
M
Ex 7
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.
15. Not possible
K
J
L
T
U
Ex 8
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.
V
17. Problem #5
3. AC AC
Statements Reasons
C
B D
AHL
Given
Given
Reflexive Property
HL Postulate
4. ABC ADC
1. ABC, ADC right s
AB AD
Given ABC, ADC right s,
Prove:
AB AD
ABC ADC
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18. 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
23. Step 3: Choose a Method
SSS
SAS
ASA
AAS
HL
C
B D
A
Given: AC bisects BAD
AB AD
Prove: ABC ADC
44
24. Step 4: List the Parts
STATEMENTS REASONS
… in the order of the Method
C
B D
A
Given: AC bisects BAD
AB AD
Prove: ABC ADC
BAC DAC
AB AD
AC AC
S
A
S
45
25. Step 5: Fill in the Reasons
(Why did you mark those parts?)
STATEMENTS REASONS
C
B D
A
Given: AC bisects BAD
AB AD
Prove: ABC ADC
BAC DAC
AB AD
AC AC
Given
Def. of Bisector
Reflexive (prop.)
S
A
S
46
26. S
A
S
Step 6: Is there more?
STATEMENTS REASONS
C
B D
A
Given: AC bisects BAD
AB AD
Prove: ABC ADC
BAC DAC
AB AD
AC AC
Given
AC bisects BAD Given
Def. of Bisector
Reflexive (prop.)
ABC ADC SAS (pos.)
1.
2.
3.
4.
5.
1.
2.
3.
4.
5. 47
27. 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
28. 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
29. 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
30. 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
A B Given
1 2 Vertical angles
ADM BCM ASA
AD BC CPCTC
56
31. 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
A B Given
1 2 Vertical angles
ADM BCM ASA
AD BC CPCTC
57
32. Corresponding Parts of
Congruent Triangles
• Sometimes it is necessary to add an
auxiliary line in order to complete a
proof
• For example, to prove ÐR @ ÐO in
this picture
Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF reflexive prop.
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC
58
33. Corresponding Parts of
Congruent Triangles
• Sometimes it is necessary to add an
auxiliary line in order to complete a
proof
• For example, to prove ÐR @ ÐO in
this picture
Statement Reason
FR @ FO Given
RU @ OU Given
UF @ UF Same segment
DFRU @ DFOU SSS
ÐR @ ÐO CPCTC
59
