Triangle Congruence Conditions
⭐ Higher Tier Content
Two triangles are congruent if they are exactly the same shape and size. This means all corresponding sides and angles are equal. To prove triangles are congruent, you must use a recognised congruence condition and write a clear, formal argument.
General Structure of a Congruence Proof
A formal proof usually follows this pattern.
State the triangles you are comparing.
List the matching information that is given or can be deduced.
Name the congruence condition that applies.
Conclude that the triangles are congruent.
Then you can state that any other corresponding sides or angles are equal.
Always match the correct vertices so corresponding parts are compared
SSS Congruence
SSS stands for side side side.
If all three sides of one triangle are equal to all three sides of another triangle, then the triangles are congruent.
To use SSS, you must show:
• one pair of corresponding sides are equal
• a second pair of corresponding sides are equal
• a third pair of corresponding sides are equal
Once all three side equalities are established, you can conclude the triangles are congruent by SSS.
SAS Congruence
SAS stands for side angle side.
If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
To use SAS, you must show:
• one pair of corresponding sides are equal
• the angle between those sides is equal
• the second pair of corresponding sides are equal
The angle must be the angle between the two given sides.
The angle must be included, not a different angle
AAS Congruence
AAS stands for angle angle side.
If two angles and a corresponding side of one triangle are equal to two angles and the corresponding side of another triangle, then the triangles are congruent.
To use AAS, you must show:
• one pair of corresponding angles are equal
• a second pair of corresponding angles are equal
• one pair of corresponding sides are equal
The side does not need to be between the two angles.
Once two angles match, the third angle is fixed, so the triangle is determined fully when one side is also fixed.
RHS Congruence
RHS stands for right angle hypotenuse side.
This condition only applies to right angled triangles.
If two right angled triangles have:
• a right angle
• equal hypotenuse lengths
• and one other corresponding side equal
then the triangles are congruent.
To use RHS, you must show:
• each triangle has a right angle
• the hypotenuse in each triangle is equal
• one corresponding shorter side is equal
Always identify the hypotenuse as the side opposite the right angle
Using Formal Arguments
A formal argument should use clear logical statements.
For example, the reasoning style should look like this in words:
Side AB equals side DE.
Side AC equals side DF.
Angle A equals angle D.
Therefore triangle ABC is congruent to triangle DEF by SAS.
Then you may conclude further matching results, such as angle B equals angle E.
Common Errors to Avoid
Common mistakes include:
• using an angle that is not included for SAS
• mismatching corresponding sides and angles
• using A A A, which proves similarity but not congruence
• choosing RHS when the triangles are not right angled
Always check the condition fits the given information exactly.
Key Points to Remember
Congruent triangles are identical in shape and size.
SSS uses three matching sides.
SAS uses two matching sides and the included angle.
AAS uses two matching angles and a matching side.
RHS applies only to right angled triangles using hypotenuse and one other side.
Using recognised congruence conditions allows triangle proofs to be written clearly and ensures conclusions are logically justified.