Congruent Triangles

Overview

Two triangles are congruent if all corresponding sides and angles are equal. Congruence ensures that the shape and size of the two triangles are identical, even if their orientation is different.

To prove that triangles are congruent, one of the following rules must be satisfied:

Congruence Rules

SSS (Side-Side-Side):

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

SAS (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, the triangles are congruent.

ASA (Angle-Side-Angle):

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

AAS (Angle-Angle-Side):

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

RHS (Right-Angle-Hypotenuse-Side):

If the hypotenuse and one other side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, the triangles are congruent.

Congruence Rules Diagram

Worked Examples

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Example 1: Proving Congruence Using SAS

Problem:

Given triangles $\triangle ABC$ and $\triangle DEF$ , where $AB = DE$ , $AC = DF$ , and $\angle BAC = \angle EDF$

Prove that the triangles are congruent.

Solution:

Step 1: Identify the given values:

$AB = DE$ (corresponding sides).
$AC = DF$ (corresponding sides).
$\angle BAC = \angle EDF$ (corresponding included angles).

Step 2: Apply the SAS rule:

Since the two sides and the included angle are equal

\triangle ABC \cong \triangle DEF

Answer: The triangles are congruent by SAS.

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Example 2: Proving Congruence Using RHS

Problem:

Two right-angled triangles $\triangle PQR$ and $\triangle XYZ$ have $PQ = XY$ and $PR = XZ$ .

Prove that they are congruent.

Solution:

Step 1: Identify the given values:

$\triangle PQR$ and $\triangle XYZ$ are right-angled.
$PR = XZ$ (hypotenuse).
$PQ = XY$ (one side).

Step 2: Apply the RHS rule:

Since the hypotenuse and one other side are equal, .

\triangle PQR \cong \triangle XYZ

Answer: The triangles are congruent by RHS.

Summary

Congruence Rules: SSS, SAS, ASA, AAS, and RHS.
Congruence guarantees that corresponding sides and angles are equal.
Congruence is useful in geometric proofs, constructions, and problem-solving.

Congruent Triangles Simplified Revision Notes for Leaving Cert Mathematics