Question

Prove that \[\vartriangle ABC \cong \vartriangle PQR\]using AAS congruence condition.

Answer 1

Hint:Here we take the help of diagrams of two triangles and show AAS(angle angle side ) convergence among the two triangles. We will try to make two angles and the non-included side of \[\vartriangle ABC\] equivalent to two angles and the non-included side of \[\vartriangle PQR\].

Complete step-by-step answer:
First we draw two triangles, \[\vartriangle ABC,\vartriangle PQR\]

We have to show \[\vartriangle ABC \cong \vartriangle PQR\] using the convergence rule of AAS.
We will take \[\angle A = \angle P,\angle B = \angle Q\]which will be the two angles of \[\vartriangle ABC\] equivalent to two angles of \[\vartriangle PQR\]
Also the sides\[AC,QR\] will be the set non-included sides. Where \[AC\] is the non-included side of \[\vartriangle ABC\] which is equivalent to the non-included side \[QR\] of \[\vartriangle PQR\].
So, we can write \[AC = QR\].
Now we have two angles of \[\vartriangle ABC\] equivalent to two angles of \[\vartriangle PQR\] and one side of \[\vartriangle ABC\] equivalent to one side of \[\vartriangle PQR\].
This fills our requirement of Angle Angle Side Convergence of two triangles.
Therefore the triangle \[\vartriangle ABC\] is said to be convergent to the triangle \[\vartriangle PQR\] by AAS rule of convergence.

Additional information:
There are many other ways to show two triangles congruent. Some other rules of convergence are
1) SSS – Side Side Side rule : If all the three sides of one triangle are equal to all the three sides of another triangle.
2) SAS - Side Angle Side rule: If two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle.
3) ASA – Angle Side Angle rule: If two angles and the included side of one triangle is equal to two angles and the included side of another triangle.

Note:Students are likely to make mistakes while making out which angle is equal to which angle so they should use the concept of writing down the equal angles from the name of the triangles which are given congruent. If \[\vartriangle ABC \cong \vartriangle PQR\] then we move from left to right and can name the angles in that sequence.