Concept of Congruency of Triangles

Concept of Congruency
(From 9th Class Text)
(Used in Trigonometry Chapters of XI )

When two triangles have the same size, then they are said to be congruent triangles.

Two congruent triangles are equal in all respects and when one is placed on the other, both exactly coincide. This means each part of one triangle is equal to the corresponding part of the other.

If ABC and DEF are congruent triangles, when DEF is placed over ABC, both will coincide and this proves that they are congruent. This process of proof is known as proof by superposition.

But one need not check for all the six parameters (three sides and three angles) for proving congruency.

Conditions for Congruency

1. If two sides and the included angle of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

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

3. If the three sides of the first triangle are equal to the corresponding three sides of the second triangle, then the triangles are congruent.

4. In case of right angled triangles, if the hypotenuse and a side of a triangle are equal to the corresponding side and hypotenuse of another triangle, then the triangles are congruent.

More briefly the rules are
(i) Two sides and the included angle (S.A.S.)
(ii) Two angles and corresponding side (A.A.S.)
(iii) Three sides (S.S.S.)
(iv) Right angle, hypotenuse, and one side (R.H.S.)