1. “Triangle Congruence Theorems”
Geometry
“Triangle Congruence Theorems”

2. The 4 Triangle Congruence Theorems
Don’t be an ASS!
Unless you’re right...
RHS

3. Theorem
If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent.
Think about it… they have to add up to 180°

4. The first step...
If two triangles have two pairs of angles congruent, then their third pair of angles is congruent.
85°
30°
85°
30°
But do the two triangles have to be congruent?

5. Example Draw two non-congruent triangles with angles of 30 and 90.
30°
30°
Why aren’t these triangles congruent?
This leads us to our first theorem of congruent triangles…

6. ASA (Angle, Side, Angle)
If two angles and the included side of one triangle are congruent to two angles and the included side of the other, then the triangles are congruent. A C B X X F E D X
AD
AB  DE
BE
Then the 2 triangles are congruent.

7. AAS (Angle, Angle, Side) Special case of ASAB
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. X X F E D X
The third pair of angles must be congruent, so it’s considered ASA
C  F, A  D, AB  DE, then the 2 triangles are congruent.

8. SAS (Side, Angle, Side)
If in two triangles, two sides and the contained angle of one are congruent to two sides and the contained angle of the other, then the triangles are congruent. A C B F E D
CA  FD
A  D
AB  DE
… then the 2 triangles are congruent.

9. SSS (Side, Side, Side) A C B
In two triangles, if 3 sides of one are congruent to three sides of the other then the triangles are congruent. F E D
CA  FD
AB  DE
CB  FE
Then the triangles are congruent.

10. RHS (Right Angle, Hypotenuse, Side)C B
If both triangles have a right angle, both hypotenuses are congruent, and another pair of sides are congruent, then the triangles are congruent. F E D
A  D = 90°
CB  FE
AB  DE
… then the triangles are congruent.

11. Example A C B
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D E
Yes, by SAS F

12. Example
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B
No, ASS does not guarantee congruence
No, G I H F E D
The angle must be between the congruent sides

13. Example D A C B
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?
Yes, by SSS
ABC  ?
DBC

14. Summary: The four congruence theorems:
ASA - Pairs of congruent sides contained between two congruent angles (AAS)
SAS - Pairs of congruent angles contained between two congruent sides
SSS - Three pairs of congruent sides
RHS - ASS condition where matching angles are 90°