 Hello and welcome to the session. I am Asha and I am going to help you with the following question which says in parallelogram ABCD two points P and Q are taken on diagonal BD such that DP is equal to BQ and we have to show these five parts. So let us begin by writing what is given to us. So we are given a parallelogram ABCD in which two points P and Q lie on diagonal BD BQ is equal to DP and firstly we have to prove the triangle A PD is component to triangle CQB. So let us begin with the proofs and in triangle A PD and triangle CQB PD is equal to BQ this is given and BC is equal to AD since in a parallelogram opposite sides are equal and also angle ADP is equal to angle C BQ is parallel to BC since ABCD is a parallelogram BD is a transversal therefore alternate interior angles are equal side angle and side condition triangle A PD component to triangle C Q P so this completes the first part. The second part we have to prove that AP is equal to CQ now in the first part we prove that triangle APD is component to triangle CQB this is my part first so this implies AP is equal to CQ and this is by CPCT that is corresponding parts of congruent triangles are equal hence we have AP is equal to CQ so this completes the second part and in the third part we have to prove that triangle A QB is component to triangle CPD now in triangle AQB and CPD we have BQ equal to PD this is given to us also AB is equal to CD since opposite sides also parallelogram are equal and now since AB is parallel to CD BD is a transversal this implies that ABQ is equal to angle CDP since if two parallel lines are intersected by a transversal then the alternate interior angles are equal angle ABD is equal to angle CDP now by side angle and side congruence condition triangle AQB component to triangle CPD so this completes the third part and now in the fourth part we have to prove that AQ is equal to CPD now triangle AQB is congruent to triangle CPD this is part three so this implies that AQ is equal to CPD by CPCT that is corresponding parts of congruent triangles are equal so this proves the fourth part and now proceeding on to the last part where we have to prove that APCQ is a parallelogram AQ is equal to CQ this is my part two and AQ is equal to CPD this is my part four in a correlator both the pairs of opposite sides are equal then it is a parallelogram so this implies that APCQ is a parallelogram and in this correlator both the pairs of opposite sides are equal so this is the last part we have to prove so this completes the session hope you enjoyed it take care and bye for now