 Hi and welcome to the session. Let us discuss the following question. Question says, in figure 6.21, A, B and C are points on O, B, O, Q and O, R respectively. Such that A, B is parallel to P, Q and A, C is parallel to P, R. Show that B, C is parallel to Q, R. This is the given figure 6.21. First of all, let us recall basic proportionality theorem. It states that, if a line is drawn parallel to one side of a triangle, intersecting other two sides at two distinct points, then the other two sides are divided in the same ratio by this line. If we are given DA is parallel to BC in this triangle ABC, then AD upon DB is equal to AE upon BC. DE is parallel to BC and it is intersecting AB and AC at two distinct points, D and E. So, by basic proportionality theorem, AD upon DB is equal to AE upon EC. Now, let us understand converse of basic proportionality theorem. Converse of basic proportionality theorem states that, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. That is, if we are given in triangle PQR, DE is dividing PQ and PR in the same ratio. That is, PD upon DQ is equal to PE upon ER. This is given to us. Then, DE is parallel to QR. This is converse of basic proportionality theorem. This is the key idea to solve the given question. Now, let us start the solution. First of all, let us write what is given in the question. In figure 6.21, AB is parallel to PQ and AC is parallel to PR. Now, we have to prove that BC is parallel to QR. Now, let us start the proof. First of all, let us consider triangle OPQ. In triangle OPQ, AB is parallel to PQ. This is given in the question. So, by basic proportionality theorem, we get OA upon AP is equal to OB upon BQ. Let us name this expression as 1. Now, we will consider triangle ORP. In triangle ORP, AC is parallel to PR. So, by basic proportionality theorem, we can write OC upon CR is equal to OA upon AP in triangle OPR. AC is parallel to PR. This is given in the question. So, by basic proportionality theorem, we get OA upon AP is equal to OC upon CR. Let us name this expression as 2. Now, in expression 1, OA upon AP is equal to OB upon BQ. And in expression 2, OA upon AP is equal to OC upon CR. So, from expression 1 and 2, we get OB upon BQ is equal to OC upon CR. Since both the ratios are equal to OA upon AP. Now, let us consider triangle OQR. In triangle OQR, we are having OB upon BQ is equal to OC upon CR. We have proved it just now. So, by converse of basic proportionality theorem, we get BC is parallel to QR. So, we can write in triangle OQR, OB upon BQ is equal to OC upon CR. This we have proved above. Now, by converse of basic proportionality theorem, we get BC is parallel to QR. We know converse of basic proportionality theorem states that if a line divides any two sides of a triangle in the same ratio, then line is parallel to the third side. Here BC is dividing OQ and OR in the same ratio. So, BC is parallel to QR. This is our required answer. Hence, proved. This completes the session. Hope you understood the session. Take care and have a nice day.