1. Proving Triangles Congruent

2. Triangle Congruency Short-Cuts
If you can prove one of the following short cuts, you have two congruent triangles
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)
HL (hypotenuse-leg) right triangles only!

3. 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?

4. Built – In Information in Triangles

5. Identify the ‘built-in’ part

6. SAS SAS SSS Shared side Vertical angles Parallel lines -> AIA

7. 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

8. Given implies Congruent Parts
segments
midpoint
angles
parallel
segments
segment bisector
angles
angle bisector
angles
perpendicular

9. This is called a common side. It is a side for both triangles.
We’ll use the reflexive property.

10. HL ( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.
ASA HL

12. Statements Reasons Given Given Reflexive Property HL Postulate
Given ABC, ADC right s,
Prove:
Statements
Reasons
Given
1. ABC, ADC right s
Given
Reflexive Property
HL Postulate

14. Name That Postulate SAS SAS SSA SAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
SAS

15. 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 I H J K
ΔGIH  ΔJIK by AAS

16. 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. B A C E D
Ex 5
ΔABC  ΔEDC by ASA

17. 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

18. 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 K L M
ΔJMK  ΔLKM by SAS or ASA

19. 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 L K V U
Not possible

20. Problem #4 AAS Statements Reasons Given Given AAS Postulate
Vertical Angles Thm
Given
AAS Postulate

21. Example Problem

22. Step 1: Mark the Given
… and what it implies

23. Step 2: Mark . . .
Reflexive Sides
Vertical Angles
… if they exist.

24. Step 3: Choose a Method
SSS
SAS
ASA
AAS HL

25. Step 4: List the Parts S A … in the order of the Method STATEMENTS
REASONS S A
… in the order of the Method

26. Step 5: Fill in the Reasons
STATEMENTS
REASONS S A S
(Why did you mark those parts?)

27. Step 6: Is there more?
STATEMENTS
REASONS S 1. 2. 3. 4. 5. A S

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.