How To Find if Triangles are Congruent

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   <i>(side, side, side)</i>

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

<i>(See <a href="http://www.mathsisfun.com/algebra/trig-solving-sss-triangles.html">Solving SSS Triangles</a> to find out more)</i>

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

2. SAS   <i>(side, angle, side)</i>

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

is congruent to:

<i>(See <a href="http://www.mathsisfun.com/algebra/trig-solving-sas-triangles.html">Solving SAS Triangles</a> to find out more)</i>

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

3. ASA   <i>(angle, side, angle)</i>

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to:

<i>(See <a href="http://www.mathsisfun.com/algebra/trig-solving-asa-triangles.html">Solving ASA Triangles</a> to find out more)</i>

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

4. AAS   <i>(angle, angle, side)</i>

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

<i>(See <a href="http://www.mathsisfun.com/algebra/trig-solving-aas-triangles.html">Solving AAS Triangles</a> to find out more)</i>

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

5. HL   <i>(hypotenuse, leg)</i>

This one applies only to <a href="http://www.mathsisfun.com/right_angle_triangle.html">right angled-triangles</a>!

or

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:

is congruent to:

<i>(See <a href="http://www.mathsisfun.com/pythagoras.html">Pythagoras' Theorem</a> to find out more)</i>

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution ! Don't Use "AAA" !

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.