How to Solve Congruent Triangles?

Geometrical figures can be either congruent or similar. If figures are congruent, then they are similar too but reverse is not always true. For 2 Triangles to be similar they must have their corresponding sides in proportion and all angles are equal. For triangles to be congruent, they must have same area and dimensions. Different properties we use to prove triangles congruent can be: SSS, SAS, ASA and HL.

Let us learn how to solve Congruent Triangles through an example:
 
 

In the given figure there is a triangle with its respective side lengths and AD as perpendicular bisector. Prove that area of ∆ABD is congruent to area of ∆ ACD.
Solution: As figure shows, triangle has two equal sides AB and AC. Thus it is an Isosceles Triangle which proves:
Angle ABD = Angle ACD,

In triangles ∆ ABD and ∆ ACD, we see that two sides AB and AC are congruent to each other. Next sides BD and CD are also congruent to each other. Angles ABD and ACD are equal in measure too. So, according to SAS property i.e. pair of two corresponding sides and angles inscribed in between two are congruent, we prove that triangles ∆ ABD and ∆ ACD are congruent to each other. Once triangles are congruent, they are similar too and so their areas are also equal i.e. area of ∆ABD is congruent to area of ∆ ACD.

To prove the areas equal and congruent mathematically we evaluate them as follows:
Area (∆ ABD) = 1 /2 * AD * BD and,
Area (∆ ACD) = 1 /2 * AD * CD,
As BD = CD, we have:
Area (∆ ACD) = 1 /2 * AD * BD,
Thus  Area (∆ ACD) ≠ Area (∆ ABD).