In these lessons, we will learn

### Congruent Triangles

### Side-Side-Side (SSS) Rule

### Side-Angle-Side (SAS) Rule

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

### Angle-Side-Angle (ASA) Rule

### Angle-Angle-Side (AAS) Rule

**Three ways to prove triangles congruent**

A lesson on SAS, ASA and SSS.

**1. SSS Postulate:** If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.

**2. SAS Postulate:** If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

**3. ASA Postulate:** If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

### Using Two Column Proofs to Prove Triangles Congruent

**Triangle Congruence by SSS **

How to Prove Triangles Congruent using the 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 using the 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.**Prove Triangle Congruence with ASA Postulate**

How to Prove Triangles Congruent using the 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.**
Prove Triangle Congruence by AAS Postulate**

How to Prove Triangles Congruent using the 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.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

- 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.**

For the two triangles below, if

Included Angle Non-included angle

*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*.

A lesson on SAS, ASA and SSS.

How to Prove Triangles Congruent using the Side Side Side Postulate?

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

How to Prove Triangles Congruent using the 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.

How to Prove Triangles Congruent using the 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.

How to Prove Triangles Congruent using the 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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.