### Theory:

Congruence of a triangle: Two triangles are congruence if their corresponding sides are equal in length and corresponding angles are equal in measure. That is if the two triangles are superimposed on each other, their sides and angles will coincide.

In the above figure, $$Δ ABC$$ and  $$Δ DEF$$ have the same size and shape. They are congruent. This can be expressed as  $$Δ ABC ≅ Δ DEF$$.

That is, if we superimpose $$A$$ on $$D$$, $$B$$ on $$E$$ and $$C$$ on $$F$$, they cover each other exactly.

Important!
Corresponding vertices:  $$A$$ and $$D$$, $$B$$ and $$E$$, and $$C$$ and $$F$$.

Corresponding sides: $$AB$$ and $$DE$$, $$BC$$ and $$EF$$, and $$CA$$ and $$FD$$.

Corresponding angles:  $$∠A$$ and $$∠D$$, $$∠B$$ and $$∠E$$, and $$∠C$$ and $$∠F$$.
Let us discuss the different types in the congruence of the triangle. They are:
• SSS congruence
• ASA congruence
• RHS congruence
SSS Congruence: The three sides of one triangle are equal to the three sides of another triangle.
Example:

In here the three sides of the first triangle are $$4$$ units, $$5$$ units and $$7$$ units which is congruent to the second triangle whose sides are the same $$4$$ units, $$5$$ units and $$7$$ units respectively.
Important!
SSS stands for "side, side, side" it means that we have two triangles with all three sides equal.
SAS congruence: Two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, and then the triangles are congruent.
Example:

In here the two sides of the first triangle are  $$3$$ units and  $$7$$ units, and the included angle is  $$40°$$  which is congruent to the second triangle whose sides are the same $$3$$ units and $$7$$ units with the included angle $$40°$$.
Important!
SAS stands for "side, angle, side" and means that we have two triangles with two sides and one angle are equal.
ASA congruence: Two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, and then the triangles are congruent.
Example:

In here the two angles of the first triangle are $$\angle P = 40°$$ and $$\angle R = 60°$$ units, and the included side is $$PR = 8$$ units which are congruent to the second triangle whose angles are the same $$\angle E = 40°$$ and $$\angle F = 60°$$ with the included side $$EF = 8$$ units.
Important!
ASA stands for "angle, side, angle" and means that we have two triangles with two angles and one side are equal.
Illustrating SSS and SAS congruence.
If $$∠E = ∠S$$ and $$G$$ is the midpoint of $$ES$$, prove that $$∆ GET ≡ ∆ GST$$.

 S. No Statement Reason $$1$$ $$∠E ≡ ∠S$$ given $$2$$ $$ET ≡ ST$$ if angle $$∠E ≡ ∠S$$, then by isosceles triangle condition $$ET≡ ST$$ for the triangle EST. $$3$$ $$G$$ is the midpoint of $$ES$$ given $$4$$ $$EG ≡ SG$$ Midpoint $$G$$ divides the line segment $$ES$$ into two parts such as $$EG$$ and $$GS$$. $$5$$ $$TG ≡ TG$$ By reflexive property, any shape is congruent to itself. $$6$$ $$∆ GET ≡ ∆ GST$$ by SSS (statements ($$2$$,$$4$$,$$5$$) & also by SAS (statements $$2$$,$$1$$,$$4$$) they are similar.