 Hello friends We are going to prove another theorem in this session and the theorem says that if the diagonals of a quadrilateral Divide each other proportionately or proportionally Then it is a trapezium Okay, so the dinals of a quadrilateral. So let us draw a quadrilateral for the simplicity sake I will draw, you know Trapezium only because the end result is going to be trapezium though We will be starting with as your machine assumption that it's not a trapezium just for the ease of understanding. Let's draw a Trapezium only So this is a trapezium So but we are not going to assume that it is a trapezium. Let's say this is a b c d these are The four vertices of this quadrilateral so a b c d is a quadrilateral. I'm saying it's not a trapezium Okay, now let's draw the diagonals. So this is the first diagonal and The other one is Okay, let us say they are intersecting at O Okay, now Let's prove So what is given given is a b c d is a quadrilateral a b c d again a b c d is a quadrilateral So I'm not assuming it to be a trapezium though. It looks like Okay, a b c d is a quadrilateral and What is given it's given that a o by o c is equal to o b by O d Okay, so this by this is equal to this by this All right Here we have to prove that to prove to prove That a b c d is a trapezium a b c d is a trapezium. So how do I prove that? How do we prove that a b c d is a trapezium? So, you know that if two sides are proven to be parallel in any given Equilateral any given quadrilateral Then it has to be a trapezium, right? So we have to somehow prove that the two sides opposite sides of A quadrilateral are So let's try to prove that now we know But before that, yes, we have to do some construction So what is the logic of the construction construction is we know that we can prove something to be parallel if we apply Converse of basic proportionality theorem, isn't it? So somehow if we get a triangle where The base is parallel to another side another line cutting the other two sides Then we know that that line and the base are parallel. That's converse of Bpt or basic proportionality theorem. So let's first draw a line like this Okay, so let's say this line is or this point is e. So we have done a construction where we are saying a e parallel to a b drawn or dc drawn. So let's say a b parallel to dc drawn Dc drawn. Okay, so this is the construction Okay, so if that construction is done now, let's try and prove this so proof should not be the difficult one now So a a e oh, I'm sorry. This is not a e and this is o e o e o e parallel to dc drawn o e parallel to dc drawn now so in triangle in triangle a dc a dc right We have what e o is parallel to dc is it this this particular line is parallel to dc by construction So let's write by construction By construction therefore applying Bpt. So by Bpt basic proportionality theorem I'm writing in short form Bpt. You can say what a e by e d is equal to a o by o c A o by o c Correct, which in turn is equal to o b by o d given see o b by o d was equal to a o by o c You know, so the moment this part is by Bpt, you know by Bpt and This part is given this part is given. So what do we conclude we conclude that a e by e d is equal to o b by o d Correct, look carefully a e by e d. So this part, so let me use this color e this part a e by e d is equal to o b by O d Correct. So if you see the moment this happens We know by converse of Bpt. So by Converse of Bpt right converse of Bpt says a Line parallel to base cuts the other two sides in equal proportion then the line is parallel to the base So a b happens to be the base here check. This is the base and this line happens to be Cutting in same ratio, right? No. So hence what can I say? I can say a e I can say e o. Sorry e o is parallel to a b Right, I can say e o is parallel to a b, but E o was parallel to Cd by construction Right. No, we had constructed this Therefore, what can I say I can say? a b itself is parallel to CD two lines parallel to one given line Are parallel to each other. So a b is parallel to CD the moment I say that opposite sides of a quadrilateral being parallel We can declare a b cd is a is a trapezium trapezium Is it so it was very very simple. We had to prove that a b cd is a trapezium So we had to prove that the opposite sides are parallel and one by one We proved each one of these lines to be parallel to a third side and hence they are parallel to each other while doing so We used basic proportionality theorem as well as converse of basic proportionality theorem