Proof of ASA , SAS , RHS , SSS congruency theorem I have tried searching in many places for some good proofs of these theorems but couldn't find them anywhere . Even my math teacher cannot explain it to me and says that these theorems just work.

I am a ninth grader so please try to explain in simple terms .
With "Angle-Side-Angle" we know the third angle and use the sine rule $\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}(=2R)$ to determine the unknown sides.

With "Side-Angle-Side" we use the cosine rule to determine the third side opposite the given angle $a^2=b^2+c^2-2bc \cos A$, and then the other angles from the same formula.

With "Side-Side-Side" we use the cosine rule to determine the angle opposite each side.

This always puzzled me until I realised that the sine and cosine rules effectively encoded the uniqueness of these constructions. No-one ever pointed out the connection, which is obvious when you see it.



In a right-angled triangle if we know the hypotenuse and one side, we find the other side using Pythagoras (a stripped down cosine rule), and then we have three sides and an angle and have reduced to one of the other cases.

In the general case SSA the cosine rule gives a quadratic for the third side, and there are two solutions. With a right-angled triangle the two solutions coincide.
With "Angle-Side-Angle" we know the third angle and use the sine rule $\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}(=2R)$ to determine the unknown sides.

With "Side-Angle-Side" we use the cosine rule to determine the third side opposite the given angle $a^2=b^2+c^2-2bc \cos A$, and then the other angles from the same formula.

With "Side-Side-Side" we use the cosine rule to determine the angle opposite each side.

This always puzzled me until I realised that the sine and cosine rules effectively encoded the uniqueness of these constructions. No-one ever pointed out the connection, which is obvious when you see it.



In a right-angled triangle if we know the hypotenuse and one side, we find the other side using Pythagoras (a stripped down cosine rule), and then we have three sides and an angle and have reduced to one of the other cases.

In the general case SSA the cosine rule gives a quadratic for the third side, and there are two solutions. With a right-angled triangle the two solutions coincide.