If two sides of a triangle are congruent then its two angles are 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.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

Side-Side-Side (SSS) Rule

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 (SAS) Rule

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 by 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 (AAS) Rule

Angle-side-angle 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.

Three Ways To Prove Triangles Congruent

A video 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.

• Show Video Lesson

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.

• Show Video Lesson

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.

• Show Video Lesson

Prove Triangle Congruence with ASA Postulate How to Prove Triangles Congruent using the Angle Side Angle Postulate?

If 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.

• Show Video Lesson

Prove Triangle Congruence by AAS Postulate How to Prove Triangles Congruent using the Angle Angle Side Postulate?

If 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.

• Show Video Lesson

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If two sides of a triangle are congruent , then the angles opposite to these sides are congruent.

∠ P ≅ ∠ Q

Proof:

Let S be the midpoint of P Q ¯ .

Join R and S .

Since S is the midpoint of  P Q ¯ , P S ¯ ≅ Q S ¯ .

By Reflexive Property ,

R S ¯ ≅ R S ¯

It is given that P R ¯ ≅ R Q ¯

Therefore, by SSS ,

Δ P R S ≅ Δ Q R S

Since corresponding parts of congruent triangles are congruent,

∠ P ≅ ∠ Q

The converse of the Isosceles Triangle Theorem is also true.

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If ∠ A ≅ ∠ B , then A C ¯ ≅ B C ¯ .

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

1. SSS   (side, side, side)

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

For example:

is congruent to:

(See Solving SSS Triangles to find out more)

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

2. SAS   (side, angle, side)

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:

(See Solving SAS Triangles to find out more)

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   (angle, side, angle)

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:

(See Solving ASA Triangles to find out more)

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   (angle, angle, side)

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:

(See Solving AAS Triangles to find out more)

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   (hypotenuse, leg)

This one applies only to right angled-triangles!

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:

(See Pythagoras' Theorem to find out more)

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.

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