Congruence and Triangles

  Introduction

 What is congruence?

Two line segments are said to be congruent if and only if, they have the same measurement/length. Similarly, two angles are said to be congruent if and only if, they have the same degree measurement. The congruence of angles and segments are reflexive, transitive and symmetric.

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): This means, if there are 2 triangles and the three pairs of corresponding sides are congruent can be shown, then, they are said to be congruent. This is only a postulate, that is, it cannot be proved, but appears to be true and everybody accepts it.
• SAS postulate (Side-Angle-Side): This means, if there are two triangles and their pair of congruent corresponding sides along with a pair of congruent corresponding angles are found, then they are said to be congruent. This also is only a postulate.
• ASA postulate (Angle-Side-Angle): This means that if there are two triangles and their two pairs of congruent corresponding angles and a pair of congruent corresponding sides are found, then they are said to be congruent. Once again this is a postulate only.

Unfortunately, in acute triangles, the SSA postulate does not always work

Triangle Non-Congruences:

We may have to draw various equilateral triangles to convince, since there is no AAA (Angle-Angle-Angle) triangle congruence theorem. However, there exists AAA (Angle-Angle-Angle) similarity theorem. In reality, as you know two angles, the third angle is fixed as 180 degrees, it is referred to as AAA similarity theorem.