 Hello and welcome to the session. In this session we discussed the following question which says prove that the sum of the angles of a triangle is 180 degrees. Before we move on to the solution, let's recall some facts. We know that the angles in a linear pair are supplementary and also we have if two parallel lines are intersected by a transversal then the angles of each pair of alternate interior angles are equal. This is the key idea for this question. Now we move on to the solution. Now since we need to prove that the sum of the angles of a triangle is 180 degrees. So first we consider a triangle say ABC. This is the triangle ABC. Now next through A we draw a line say PQ parallel to BC. So we have this PQ is parallel to BC. We have marked the angles as angle 1, angle 2, angle 3, angle 4 and angle 5. So now we need to prove that angle 1 plus angle 2 plus angle 3 is equal to 180 degrees. That is sum of the angles of triangle ABC is 180 degrees. Now next we have PQ is parallel to BC and AB is the transversal that cuts them. Therefore this would mean angle 1 is equal to angle 4 since they are the alternate angles and we know that if two parallel lines are intersected by a transversal then the angles of each pair of alternate interior angles are equal. So we take this as equation 1. Now next we have again PQ is parallel to BC and AC is the transversal. Therefore angle 2 is equal to angle 5 since they are the alternate interior angles or alternate angles so they are equal. Let this be equation 2 then adding equations 1 and 2 we have angle 1 plus angle 2 is equal to angle 4 plus angle 5. Let this be equation 3. Now again adding angle 3 to both sides of equation 3 we get angle 1 plus angle 2 plus angle 3 is equal to angle 4 plus angle 5 plus angle 3. Now from the figure you can see that angle 3, angle 4 and angle 5 they form a linear pair. So this means angle 3 plus angle 4 plus angle 5 is equal to 180 degrees. Since we know the angles in a linear pair are supplementary therefore some of the angles in a linear pair would be equal to 180 degrees. So this implies that angle 1 plus angle 2 plus angle 3 is equal to 180 degrees from this equation and we were supposed to prove this. So this means that the sum of the angles of a triangle is 180 degrees hence proved. So this completes the session hope you have understood the solution for this question.