Conditions of Congruency of Triangles

# Conditions of Congruency of Triangles

## Congruence of Triangles

Geometrical figures which are of same shape and same size are called congruent figures. Regular polygons of same dimensions and circles of same radii are the examples of congruent figures.
Congruency is represented using symbol ‘’ and is read as ‘is congruent to’.

## Corresponding Sides and Corresponding Angles

Two triangles are said to be congruent if one triangle superimposes on the other or if both the triangles are identical. In congruent triangles, the sides which coincide with each other are called corresponding sides.

Sides PQ and AC, QR and BC, and PR and AB are the corresponding sides.
Similarly, the angles which coincide with each other are called corresponding angles in congruent triangles.
Angles P and A, Q and C, and R and B are corresponding angles.

## Conditions of Congruency of Triangles

### Side-Side-Side (SSS) Congruency Condition

Two triangles are congruent, if the three sides of one triangle are respectively equal to the three sides of the other triangle.
In the following figure, PQ = AB, PR = AC and RQ = CB.

Thus, by SSS congruency condition, Î”PQR  Î”ABC.

### Side-Angle-Side (SAS) Congruency Condition

Two triangles are congruent, if two sides and the included angle of one are respectively equal to the two sides and the included angle of the other.
In the following figure, PQ = AB, RPQ = CAB and PR = AC.

Thus, by SAS congruency condition, Î”PQR  Î”ABC.

### Angle-Side-Angle (ASA) Congruency Condition

Two triangles are congruent, if two angles and the included side of one are respectively equal to the two angles and the included side of the other.
In the following figure, PQ = AB, QPR = BAC and RQP = CBA.

Thus, by ASA congruency condition, Î”PQR  Î”ABC.

### Angle-Angle-Side (AAS) Congruency Condition

Two triangles are congruent, if two angles and a side opposite to one angle of one triangle are respectively equal to the two angles and a side opposite to one angle of other triangle.
In the following figure, PRQ = ACB, PQR = ABC and PQ = AB.

Thus, by AAS congruency condition, Î”PQR  Î”ABC.

### Right-angle-Hypotenuse-Side (RHS) Congruency Condition

Two right-angled triangles are congruent, if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other triangle.
In the following figure, AB = DE, ABC = DEF = 90° and BC = EF.

Thus, by RHS congruency condition, Î”ABC  Î”DEF.