 Hello friends in the previous session we had seen the validation of internal angle bisector theorem and then we also did the proof of the theorem. In this case we are going to prove the converse of that theorem right and what does the converse says it says if a line through one vertex of a triangle divides the opposite side in the ratio of the other two sides then the line bisects the angle at the vertex. So let's understand the theorem first okay so what it is saying is abc is a triangle so it's shown here abc is a triangle and ad is such that it is bisecting so let's understand the theorem first the theorem says that if a line through one vertex of a triangle so let that vertex be a and let that line be ad now ad divides the opposite side in this case the opposite side is bc correct so if ad divides bc in the ratio of other two sides that is abn ac so what is given that bd upon dc this is what the line is doing the line ad is doing this bd upon dc is equal to ab upon ac this is given okay so you have to prove that angle b ad that is this is x and if this is y then you have to prove to prove x is equal to y or ad is angle bisector ad is angle bisector of angle a now there is a construction here again you can see there is ratios involved in this question isn't it bd by dc ab by ac so hence somehow if we try to use therese theorem what is also called as basic proportionality theorem then there is a scope so we see we sense a scope over there so I have done some construction the construction is I am drawing a e a e is equal to ac right so I have produced b a so e is nothing but e lies on b a produced e lies on b a produced enough got it so a e this side a e is equal to ac the moment I say that what happens the opposite angle becomes equal is it so let's say this angle is z and this angle is z okay now how to go about it so we know that from the given condition bd by dc is equal to now we are going to further proof so I am saying bd upon dc is equal to ab upon ac this is given and since since angle ac e is equal to angle a e c by construction we just did that construction hence all right why is this construction because a e is equal to so you can write a e is equal to ac so hence can I not write that angle um yeah so I can say and it's not angle so I can very well say that instead of ac I can write a e so bd upon dc is equal to b a upon ac or a e because ac was equal to a is equal to ac correct therefore actually there was no need of equating this angle but later on right now it's not new later on it is going to be you know needed in this theorem so this is okay bd by dc so this side this side by that side is equal to this side by that side right so bd by dc is equal to b a by a e so this means by converse of so you can write by converse of bpt so if the ratios are same then what can we say we can say that ad is parallel to c e right by converse of bpt so if you remember in bpt there was a triangle and a line was um cutting the two sides then a let's say this b this c this is d and this is e so in that case these two are parallel right so ad by db was equal to a e by ac and converse is also true that means if this is true then the line de is parallel to bc we have seen that in the previous session so de is parallel to bc in this case so equivalently here ad will be parallel to c because the ratios were same so the moment i say that what do i get i will get y is equal to z right and why is that alternate interior angles and also x is equal to z y corresponding angles corresponding angles right convincing enough so that means this implies if y is z and x is z then clearly y is equal to x y is equal to x therefore ad bisects angle b ac right proved so hence if the if a line from vertex divides the opposite side in the ratio of the adjacent sides right so bd bd upon dc is equal to ba upon ac then ad bisects the angle a that's what the converse of this theorem says