 Hello friends welcome again to another session on problem solving and proving theorems So in this theorem, it's given that there are three or more parallel lines and they are intersected by two Transversals we have to prove that intercepts made by them on the transversals are proportional Okay, so I have shown it in a diagram. So P and Q and R are parallel to each other So let's write given first So given what is given P is to P is parallel to Q is parallel to R three lines are parallel to each other Okay, now L and M L and M small m are transversals Transversals right now what is to be proven so to prove We need to prove that a b by b c is equal to d e by e f This is also called intercept theorem right so any I have shown only three lines But you can go for four lines five parallel lines and all that so the intercept So what is an intercept intercept is nothing but this is called an intercept So this is an intercept. This is also an intercept. So these are all intercepts. So these intercepts are proportional That's what we need to Prove okay, so let us do some construction. So again if there are ratios involved, you know the best tool in our quiver is the Basic proportionality theorem, so let's try and prove that using the basic proportionality theorem only okay, so how to Go about it. So, you know that in basic proportionality theorem, you need to have a triangle and There must be a line parallel to the base and then you can write the ratios equal, right? So here you can see ratios are there lines are parallel We have to achieve some ratios in equality. So hence we need to first find out a triangle Yeah appropriate triangle to establish or to apply dpd So what I'm going to do is I am going to do a construction where I am going to draw a line Like this which is parallel to M passing through Yeah, like that. So what is the construction? Construction is Let's call it LMN, right? So let's call this line as N. So N parallel to M drawn Right and it is intersecting the PQR at N points A and let's say G and H Okay, G and H now what can we what can we say? Clearly now you can see a triangle a C H appearing isn't it so in triangle in triangle a C H What can we say we can say? since BG is parallel to C H BG is parallel to C H therefore What can be very easily said a B upon BC is equal to a G upon G H and this is because of Dpd basic proportionality theorem Right now somehow if I prove that a G by G H is equal to D by EF Then my game is over Is it that's what we need to prove? So a G by G H somehow is equal to D by EF which is indeed. So why? because if you look carefully What's here if you see N is parallel to N is parallel to M as well as M P is parallel to Q therefore. What can we conclude we can conclude that? a D e g is a parallelogram Is a parallelogram? I Hope there's no doubt in this correct, so a b a D e g is a parallelogram therefore opposite sides of a parallelogram being equal I can write a G is equal to D E. I hope by now you would have guessed the rest of the proof Is it similarly write the word similarly? right P Again L is sorry N is parallel to M N is parallel to M and Q is parallel to R therefore again You can say G E F H Is a parallelogram? Again same logic opposite sides are equal So hence the quadrilateral becomes a parallelogram therefore You can conclude G H is equal to E F right now Look at this particular theorem or this part which you need to prove now if you see a G by G H Can be written as what is a G guys D e so write D e and And G H is equal to E F. So right E F So a G by G H is D e by E F. So let it be one and Let it be two So in one and two In one and two from one and two rather from one and two. What could you conclude? You can conclude that a b by bc is equal to a D e by E F That is what we intended to prove now you would say this is true only for two lines. No This is true for any number of parallel lines. You can see let's say there are lines like these one two three Four so you can you know Let's say these are the intercepts so you can very easily say this by this is equal to this by this Similarly This by this will be equal to this by that like that Right. So hence in that chain if you go you can prove that This by this by this will be equal to this by this By that right so one by one you have to prove so this by this is equal to this by this by the earlier proof proven theorem And again this by this will be equal to this by this So like that the chain will continue so you can see that All the intercepts will be proportional So it can be Validated or it can be established for more now more than three parallel lines as well Okay, so here was the proof of this particular theorem, which is also called intercept theorem