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In these lessons, we will learn

- the SSS, SAS, ASA and AAS rules
- how to use two-column proofs to prove triangles congruent

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule. In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

*Side-Side-Side* is a rule used to prove whether a given set of triangles are congruent.

**The SSS rule states that **

*If three sides of one triangle are equal
to three sides of another triangle, then the triangles are congruent.
*

In the diagrams below, if *AB* = *RP*, *BC* = *PQ *and* CA = QR, *then triangle *ABC* is congruent to triangle *RPQ*.

*Side-Angle-Side* is a rule used to prove whether a given set of triangles are congruent.

**The SAS rule states that **

**If two sides and the included angle of one triangle are equal to two sides and included angle of
another triangle, then the triangles are congruent.**

**An included angle is an angle formed by two given sides.**

Included Angle Non-included angle

For the two triangles below, if *AC* = *PQ*, *BC* = *PR *and angle *C* = angle *P , *then using the SAS rule, triangle *ABC* is congruent to triangle *QRP
*

*Angle-side-angle *is a rule used to prove whether a given set of triangles are congruent.

**The ASA rule states that **

**If two angles and the included side of
one triangle are equal to two angles and
included side of another triangle, then the triangles are congruent.**

*Angle-angle-side *is a rule used to prove whether a given set of triangles are congruent.

**The AAS rule states that **

**If two angles and a non-included side of
one triangle are equal to two angles and
a non-included side of another triangle, then the triangles are
congruent.**

In the diagrams below, if *AC* = *QP*, angle *A =* angle *Q*, and angle *B =* angle *R, *then triangle *ABC* is congruent to triangle *QRP*.

The following video will explain three ways to prove triangles congruent - A lesson on SAS, ASA and SSS.

**Triangle Congruence by SSS **- How to Prove Triangles Congruent

Side Side Side Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

**Triangle Congruence by SAS** - How to Prove Triangles Congruent

SAS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Angle Side Angle Postulate

It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Triangle Congruence by AAS Postulate

Angle Angle Side Postulate

It two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.