 Hello and welcome to the session. In this session we discuss the following question which says, in the given figure pqrs is a parallelogram and the line segments pt and rt dash by 60 angles pnr respectively show that pt is parallel to rt dash. Before moving on to the solution let's recall a fact which says a quadrilateral is a parallelogram one pair opposite sides this is the key idea to be used in this question. Let's proceed with the solution. Consider this figure here we have pqrs nnogram then is given that pt by 6 angle p rt dash by 6 angle r and we have to prove that pt is parallel to rt dash. Now in triangles st qt dash we have ps is equal to rq so these are the opposite sides of the parallelogram pqrs and we know that opposite sides of parallelogram equal. Also observe that in triangles pst and rqt dash we have angle s is equal to angle q since and the s and angle q are opposite angles of the parallelogram pqrs and we know that opposite angles of a parallelogram are equal. Now we have angle p is equal to angle r since they are the opposite angles of the parallelogram and we know that opposite angles of a parallelogram equal. So this would give us half of angle p is equal to half of angle r we know that pt by 6 angle p so half of angle p would be given as angle spt now this would be equal to half of angle r and we know that rt dash by 6 angle r so half of angle r would be given by angle qrt dash so now we get in triangles pst rqt dash we have ps is equal to rq angle s is equal to angle q and angle spt is equal to angle qrt dash so therefore we say that triangle pst is congruent to triangle rqt dash by congruent criteria now since these two triangles are congruent so this would give us st is equal to qt dash cpct that is they are the corresponding parts of congruent triangles then we have so this means that sides are equal that is pq is equal to rs that is now pq from the figure as you can see is equal to pt dash plus t dash q and this would be equal to rs which from the figure you can see is equal to rt plus ts now since we have got that st is equal to qt dash so just t dash q and st or ts cancels so we are left with pt dash is equal to rt now we also have pq is parallel to rs this would obviously mean that parallel to rt now in the quadrilateral rt we have pt dash is equal to rt to rt so from the key idea which says that a quadrilateral is the parallelogram if it has one pair of opposite sides equal and parallel we say that quadrilateral pt dash rt is a parallelogram one pair of opposite sides equal and parallel that is pt dash is equal to rt and pt dash is also parallel to rt now since pt dash rt is a parallelogram therefore this means that parallel to rt dash so health proved that pt is parallel to rt dash this completes this session hope you have understood the solution for this question