Congruence of Triangles

Congruence of Triangles

If two geometrical figures are identical in shape and size then they are said to be congruent.

The Method of Superposition.

Step 1 : Take a trace copy of the Fig. 1. We can use Carbon sheet.

Step 2 : Place the trace copy on Fig. 2 without bending, twisting and stretching.

Step 3 : Clearly the figure covers each other completely.

Therefore the two figures are congruent.

Congruence of Triangles

Two triangles are said to be congruent, if the three sides and the three angles of one triangle are respectively equal to the three sides and three angles of the other.

Conditions for Triangles to be Congruent

We know that, if two triangles are congruent, then six pairs of their corresponding parts (Three pairs of sides, three pairs of angles) are equal.

But to ensure that two triangles are congruent in some cases, it is sufficient to

verify that only three pairs of their corresponding parts are equal, which are given as axioms.

There are four such basic axioms with different combinations of the three pairs of corresponding parts. These axioms help us to identify the congruent triangles.

If ‘S’ denotes the sides, ‘A’ denotes the angles, ‘R’ denotes the right angle and ‘H’ denotes the hypotenuse of a triangle then the axioms are as follows:

(i) SSS axiom          (ii) SAS axiom          (iii) ASA axiom        (iv) RHS axiom

(i) SSS Axiom (Side-Side-Side axiom)

If three sides of a triangle are respectively equal to the three sides of another triangle then the two triangles are congruent.

(ii) SAS Axiom  (Side-Angle-Side Axiom)

If any two sides and the included angle of a triangle are respectively equal to any two sides and the included angle of another triangle then the two triangles are congruent.

 (iii) ASA Axiom (Angle-Side-Angle Axiom)

If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another triangle then the two triangles are congruent.

(iv) RHS Axiom (Right angle - Hypotenuse - Side)

If the hypotenuse and one side of the right angled triangle are respectively equal to the hypotenuse and a side of another right angled triangle, then the two triangles are congruent.