 Hello and welcome to the session. In this session we discuss the following question which says if two straight lines intersect each other then prove that the ray opposite to the bisector of one of the angles so formed bisects the vertically opposite angle. We know that that if lines intersect then the vertically opposite angles are equal. This is the key idea to be used in this question. Now we move on to the solution. Consider this figure where we have lines A B and C D intersect at point O E that is the ray O E bisects angle A O C that is we have angle 1 is equal to angle 2. We need to prove that the ray opposite to the bisector of angle A O C that is the opposite to ray O E bisects the vertically opposite angle that is we need to prove that angle 3 is equal to angle 4 that is ray O F bisects the vertically opposite angle angle B O D. Now we have ray O E bisects angle A O C ray O F is opposite to then we say that this E O F a straight line is a straight line and we have that E F intersects C D this would mean that angle 1 is equal to angle 4 since they are the vertically opposite angles and we know that when two lines are intersecting the vertically opposite angles are equal. Also E F and A B intersects 2 is equal to angle 3 as you can see there also the vertically opposite angles and so they are equal. But we already have angle 1 is equal to angle 2 this would mean angle 3 is equal to angle 4. So we were supposed to prove that angle 3 is equal to angle 4 so hence proved this completes the session and we have understood the solution for this question.