Congruence of Triangles

Two figures, which have the same shape and same size are called congruent figures and this property is called congruence. Congruent figures, when placed one over another, exactly coincide with one another or cover each other. In other words, two figures will be congruent, if parts of one figure are equal to the corresponding parts of the other.

Two line segments are congruent when they are of equal length. Two squares are congruent if their sides are equal. Two circles are congruent, if their radii are equal, implying their circumferences are also equal.

Congruence of Triangles

Two triangles are congruent, if all the sides and all the angles of one are equal to the corresponding sides and angles of other.

Δ PQR is congruent to Δ XYZ and written as Δ PQR ≅ Δ XYZ. P corresponds to X, Q corresponds to Y and R corresponds to Z.

In order to prove, whether two triangles are congruent or not, you need to know that all the six parts of one triangle are equal to the corresponding parts of the other triangle. However, it is possible to prove the congruence of two triangles, even if you are able to know the equality of three of their corresponding parts.

Criterion 1: SAS

If any two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle, the two triangles are congruent. This criterion is referred to as SAS (Side Angle Side).

Criterion 2: ASA or AAS

If any two angles and one side of a triangle are equal to corresponding angles and the side of another triangle, then the two triangles are congruent. This criterion is referred to as ASA or AAS (Angle Side Angle or Angle Angle Side).

Criterion 3: SSS

If the three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are congruent. This is referred to as SSS (Side, Side, Side) criterion.

Criterion 4: RHS

If the hypotenuse and a side of one right triangle are respectively equal to the hypotenuse and a side of another right triangle, then the two triangles are congruent. This criterion is referred to as RHS (Right Angle Hypotenuse Side). It is applicable for two right triangles only.