 Hello and welcome to another session on triangles and you know in the series of videos We are trying to prove various theorems related to triangles and yes other geometrical shapes like quadrilaterals trapezium as well So this is again related to a trapezium. So any line parallel to the parallel sides of a trapezium Divide the non-parallel sides proportionally Okay, so first of all, let's draw the figure. So let's Construct this diagram. So I've drawn a trapezium. Okay, any line now, let's first name the trapezium So a b c d is our trapezium a b c d is our trapezium. Okay, so a b c d is a trapezium and Any line parallel to the parallel side. So we have to draw a parallel line Along this. Okay, so here is a line parallel to the parallel sides of the Trapezium it divides the non-parallel side. So let's say this point is e and f So it divides the non-parallel sides proportionally. That means they're saying this parallel line a e by e d This parallel line is dividing the sides proportionality means a e by e d is Equal to b f by f c. This is what we have to achieve You have to prove so let's try and prove this So what is given? Let's first write that down given is a b is a trapezium a b c d is a trapezium, sorry a b c d is a Trapezium Fair enough Trapezium, you know opposite sides will be parallel then Which sides are parallel here a b is parallel to c d that's no right and Also given is e f is parallel to a b is parallel to c d Rdc is also given to prove What do you need to prove guys? You need to prove that a e upon e d That's how it is dividing it proportionally a e by e d is equal to b f by f c Okay, so again, it looks like that we have to prove some ratios to be equal Ratios that to you know, we can see there is a you know Quadrilateral four sides are there. So if I get ratios in a triangle Then I can use basic proportionality theorem or converse of it, isn't it? So let's do a construction. What do we do is we will construct or join this Okay, so here is the construction. What is the construction? We construction we do a construction and that is what is that a c joint a c Joint so we joined a c. So we get two triangles and we get parallel sides So now you can think of what we are going to do So guys, let's do the proof in triangle carefully see a DC, okay Now let's call this point g Okay, so clearly e g is a part of e f So hence e g is parallel to the base DC in that triangle without doubt. It should be clear So e g is parallel to DC, which will mean a e upon e d is equal to a g upon g c and why is this it is because of basic proportionality theorem a line Parallel to the base divides the other two sides proportionally. So this is true. Okay Similarly folks in triangle. So now I'm doing it here in triangle the others other triangle if you see C a b right look at like that such that a b is the base in triangle C a b What is given the code f g is parallel to a b by construction? Sorry, it's already given not by construction. So f g is parallel to a b Therefore by b pt again, you can say c g upon g a is equal to c f upon f b Is it it again by what by b pt? Now if that is so you can reciprocate it why Because you will see you need a g by gc, right? So here if you reciprocate this you will get g a by c g Is equal to f b by f c Right or c f whichever way so I've just rearranged the Point so instead you can write a g upon gc is equal to is equal to b f upon fc So let me call this one and let me call this two Right, so guys from one and two From One and Two what can you say you can say that a g by gc is equal to a G by gc Sorry, a g by gc. Anyways is a common equality term. So hence you can say a e by ed sorry a e by ed is Is equal to b f By fc because both of them are equal to a g by gc and this is exactly what we needed to establish That means the diagonals are You know or rather, you know the non-parallel sides are being divided proportionally by the parallel side of a or our line parallel to the Parallel sides of the Trapezium so once again, there is a trapezium a b and c d are parallel lines E f is another parallel line to the parallel lines of the trapezium therefore The ratio a e by ed will be equal to b f by fc or The parallel sides of a trapezium so a parallel line to the parallel sides of a trapezium divides the non-parallel sides Propulsionally, okay, that's the term