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NOTE: Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS, and HL.
- SSS (side, side, side)
If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.
In the two triangles above, 
.
That is because 
Proving triangles congruent by the side-side-side postulate.
Example 1.
Given: 
, 
is a diagonal.
Prove that 
Example 2.
Given: 
Prove that 
- SAS (side, angle, side)
In this case, two triangles are congruent if two sides and the included angle in a given triangle are equal to the corresponding two sides and the included angle in another triangle.
Example 3.
Given: 
Prove that 
Example 4.
Given: 
Prove that 
3. ASA (angle, side, angle)
The ASA postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Example 5.
Given: 
Prove that 
Example 6.
Given: 
Prove that 
4. AAS (angle, angle, side)
This postulate says that iftwo angles and a non-included sideof one triangle are equal totwo angles and a non-included sideof another triangle, then the triangles are congruent.
Example 7.
Given: 
Prove that
Example 8.
Given: 
Prove that 
5. HL (hypotenuse, leg)
The HL Theorem states: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Example 9.
Given: 
Prove that 
Example 10.
Given: 
Prove that 
Practice Questions 1
Solutions
- Congruent. SAS.
- Not Congruent
- Congruent. SSS
- Congruent. SSS