Congruent Triangles

Congruent triangles are those two triangles which are said to be congruent if and only if one of them can be made to superpose on the other so as to cover it exactly.

$Congruent Triangles$

Let ∆ABC and ∆DEF be two congruent triangles, then we can superpose ∆ABC on ∆DEF so as to cover it exactly. The vertices of ∆ABC fall on the vertices of ∆DEF in the following order A ↔ D, B ↔ E, C ↔ F.

Thus, the order in which vertices match automatically determines the correspondence between the sides and angles of the triangle. Corresponding parts are also called matching parts of triangles.

So, we have six equalities

Corresponding sides are congruent: AB = DE BC = EF CA = FD

Corresponding angles are congruent: ∠A = ∠D ∠B = ∠E ∠C = ∠F

In the congruent triangles we will observe six correspondences between their verities. The symbol used to denote correspondence is ‘↔’

A ↔ D B ↔ E C ↔ F written as ABC ↔ DEF

A ↔ E B ↔ F C ↔ D written as ABC ↔ EFD

A ↔ F B ↔ D C ↔ E written as ABC ↔ FDE

A ↔ D B ↔ F C ↔ E written as ABC ↔ DFE

A ↔ E B ↔ D C ↔ F written as ABC ↔ EDF

A ↔ F B ↔ E C ↔ D written as ABC ↔ FED

Therefore, ∆ABC ≅ ∆DEF

Note:

The order of letters in the name of two triangles will indicated the correspondence between the vertices of two triangles. Thus, two triangles are congruent only if there exist a correspondence between their vertices such that the correspondence sides and correspondence angles of two triangles are equal.

Congruent Shapes

Congruent Line-segments

Congruent Angles

Congruent Triangles

Conditions for the Congruence of Triangles

Side Side Side Congruence

Side Angle Side Congruence

Angle Side Angle Congruence

Angle Angle Side Congruence

Right Angle Hypotenuse Side congruence

Pythagorean Theorem

Proof of Pythagorean Theorem