 Hello and welcome to the session. I am Asha and I am going to help you solve the following question which says A B C D is a trapezium in which A B is parallel to C D and A D is equal to B C. Show that first angle A is equal to angle B, second angle C is equal to angle D, third triangle A B C is congruent to triangle B A D, fourth diagonal A C is equal to diagonal B D. Also we are given a hint, extend A B and draw a line through C parallel to D A intersecting A B produced at E. So we are given a trapezium A B C D in which T is parallel to D C and right A D is equal to B C. Now let us write down the construction in which we will extend line A B draw a line through C such that C E is parallel to D A. So this is the required line which is parallel to D A and we have to prove firstly that angle A is equal to angle B. Let us now start with the proof B is parallel to C D this implies that A is parallel to C D and this is the construction of extended line A B and drawn a line through this point C such that it meets the extended line A B at a point E and A D is parallel to C E. So this implies that A E C D is a parallelogram. Let this be number one. Now C D is a trapezium so this implies angle this angle D is equal to 180 degree and angle A B C plus angle D C B is equal to 180 degree. Some of these two angles is 180 and A plus D is also 180 since two parallel lines by a transversal then side of a transversal are supplementary and let this be equation number two and this be equation number three. Also as we know in a parallelogram those angles are equal. So this implies angle D is equal to angle E and let this be equation number four and since A D is equal to B C this is given to us and A D is equal to E C this is since A E C D is a parallelogram and in a parallelogram opposite sides are equal so this implies B C is equal to E C or we can say that angle B E C is equal to angle E B C that is these two angles are equal since in a triangle sides opposite equal angles. So let this be equation number five. is parallel to C D B C is the transversal this implies angle B C D is equal to angle C B E since the two lines are intersected by a transversal then alternate interior angles are equal and now equal to this angle and angle C B E is equal to angle E C B so this implies angle B C E is equal to angle B C D so this implies angle D C B is equal to angle B E C so this is equation number six and and now we can further write it as angle D C B is equal to angle E now angle E is equal to angle D this is from equation number four so from these two equations it further implies that angle D C B is equal to angle D or we can say that in trapezium A B C D angle C is equal to angle D so let this be equation number seven and this is the second part we have to prove now further since angle A plus angle D is equal to 180 degree and angle B plus angle C is equal to 180 degree so from these two equations it implies that angle A plus angle D is equal to angle B plus angle C and since angle C is equal to angle D therefore this further implies that angle A is equal to angle B and this is since angle C is equal to angle D so let this be equation number eight and hence we have proved that angle A is equal to angle B which is the first part we have to prove and angle C is equal to angle D and now let us prove the third part where we have to show the triangle ABC is component to triangle BAD so let us draw the two diagonals and now in triangle BC and triangle BAD angle A is equal to angle B we have proved above in equation number eight also side AB is equal to BA this is common to both the triangles and side BC is equal to AD this is given to us so by SAS congruence it implies that triangle ABC is component to triangle BAD this completes the third part also and now proceeding on to the fourth part where we have to show that diagonal AC is equal to diagonal BD now triangle ABC is component to triangle BAD so this implies side AC is equal to BD since by CPCT which is corresponding parts of congruent triangles are equal so this completes the fourth part also so this completes the session take care and have a good day