 Hello and welcome to the session. In this session, we discussed the following question that says proof that the bisector of the vertical angle of an isosceles triangle bisects the base at right angles. Before we move on to the solution, let's recall the SAS congruence condition. According to this we have that two triangles are congruent if the two sides and the included angle of one are respectively equal to the two sides and the included angle of the other. This is the key idea to be used for this question. Now we move on to the solution. Consider this triangle PQR that is triangle PQR is an isosceles triangle. So we have in triangle PQR, side PQ is equal to the side PR and PL is the bisector of angle P and we need to prove that the bisector of the vertical angle of an isosceles triangle bisects the base at right angles that is we need to prove angle PLQ is equal to angle PLR is equal to 90 degrees and QL is equal to LR. We consider triangles PLQ and PLR in this we have PQ is equal to PR it's given to us then PL is equal to PL that is the common side and angle QPL is equal to triangle RPL since PL is the bisector of angle P. So therefore we get triangle PLQ is congruent to the triangle PLR by the SAS congruence condition. Now since both these triangles are congruent therefore QL is equal to LR CPCT that is the corresponding parts of congruent triangles are equal and also angle PLQ is equal to angle PLR again CPCT as they are the corresponding parts of congruent triangles so they are equal. Now angle PLQ plus angle PLR is equal to 180 degrees since they form a linear pair and both these angles are equal therefore 2 times angle PLQ is equal to 180 degree or angle PLQ is equal to 180 degrees by 2 equal to 90 degrees thus we get angle PLQ is equal to angle PLR is equal to 90 degrees thus we now get that angle PLQ is equal to angle PLR is equal to 90 degrees and QL is equal to LR. Hence we say that the bisector of the vertical angle of an isosceles triangle bisects the base at right angles. So this completes the session hope you have understood the solution for this question.