 welcome folks so in the previous session we understood that if a ray stands on line then the sum of the adjacent angle so formed is 180 degrees so we learned in the previous session that if there is a line and a ray stands over it let's say this ray let's say the names are o and let's say this is a b and c right so we learned that angle a o c plus angle c o b c o b is equal to 180 degrees okay this is what we learned now we have to prove the converse of it converse is that if the sum of two adjacent angle is 180 degrees now that it is given let's say this is given then you have to prove that the non-common arms are two opposite rays okay so we will be using something called method of contradiction here okay the method is called method of contradiction contradiction what does this method mean let's say for anything to have only two possibilities let's say if if a statement is there let's say if a statement is there now the statement can either be true or it is false okay so somehow if you prove that it is not false okay let us or you start with let's say the statement is false and somehow you prove that logically that our assumption was wrong let's say if the statement cannot be false then automatically the statement becomes true okay this is what is the logic behind method of contradiction so here they are saying that if the adjacent sum of the adjacent angles is 180 then their non-common arms are two opposite rays so we'll start with the assertion that let the non-common arms non-common arms not be the opposite arms okay that means now they're not opposite so let us start with you know a diagram so let's say this was given okay so again we'll take O A B and C now this was given and it was given that angle A O C plus angle C O B is 180 degrees okay this is given so we had to prove that we had to prove that O A and O B are opposite rays opposite rays so we are saying we are we are saying that let they are they be not opposite that means if O A and O B are not opposite then then what happens if they are not opposite there must be some some ray which is opposite to it to to each one of them then let us say O D is opposite to let us say O D is opposite to O A okay and let me draw it now so let us say I'm saying this D right so don't go by the figure you know so that's what we are going to prove that you know it is appearing also not to be opposite but then just to prove for the argument sake we are saying let's say if you are saying OB is not opposite to O A then there must exist another ray O D which is opposite to O A right so OD is opposite to O A that means A O D A O and D all these three points lie lie on a straight line on a straight line okay this is what we can we can say why because now O A and O D are opposite now in the previous theorem we learned that if a ray stand on a straight line okay so now we are saying on a straight so that means A O D is a straight line is a straight line should be a straight line since we are claiming OD to be opposite to O A now O C stands on stands on A O D line A O D okay this implies therefore what is what is implied by that means angle A O C plus angle C O D must be equal to 180 degree by the previous theorem right so let me call this as 1 and this as 2 so hence I can say from 1 and 2 since both are equal to 180 degrees I can equate them so what can I say I can say angle A O C plus angle C O B is equal to angle A O C plus angle C O B okay both are equal to 180 degrees now clearly A O C and A O C gets cancelled this implies angle C O B is equal to angle C O D okay now two angles are equal two angles angles are equal right with common with C O as the common arm common arm and O B and O D on the same side of C O same side of C O if you see in this figure OD and OB OD and OB both are on the same side of O C and the way you are saying both the angles are equal how can that be possible this means it is possible it is possible only when D coincides with B right only when this D point D is on B only then the two angles will be equal that means that means this means that D and B are same point same point isn't it and you had you had said that OD was opposite and since OD was or is opposite to OA therefore OB is opposite to OA why because now D and B are the same point so understood this is what is the proof of the converse of this or this particular theorem which says that if the sum of two edges angle is 180 degrees then their non common arms are two opposite rays we just proved that and we used what method of contradiction so please keep this method in mind