Congruence and Triangles

What are congruent triangles?

If the corresponding sides and angles of two triangles are equal, then they are called as congruent triangles. In other words, triangles that have same size and shape are congruent triangles.

In the above figures, the corresponding sides are d and g, e and h, f and i. The corresponding angles are: x and s, y and r, z and u. We can conclude whether 2 triangles are congruent without testing all the sides and angles using the following rules:

Congruence of Triangles:

Two triangles are said to be congruent if and only if, they contain 6 pairs of congruent sides and congruent angles. But, most of the occasions there won't be luxury of having all the information. In such circumstances, the following postulates are used as shortcuts:

• SSS postulate (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are said to be congruent. Example:

  Let DEFG be a square and DF be one of its diagonals. Then, we can prove that triangles DEF and FGD are congruent as follows:

  

  In a square, all the four sides are equal/congruent. Therefore, sides DE and FG are congruent. Also, sides EF and GD are congruent.

  The two triangles also possess a common side DF. Triangles DEF has three sides congruent to the corresponding three sides in another triangle FGE.

  Therefore, according to the SSS postulate, the two triangles are congruent.

• SAS postulate (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are said to be congruent. An included angle formed by the two given sides.
• ASA postulate (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are said to be congruent.
• AAS Rule postulate If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are said to be congruent.