 So in this theorem, it's been given that a transversal intersects to parallel lines Then we have to prove that each pair of conjugative interior angles are supplementary Just a quick recap. What are supplementary angles guys supplementary angles are angles Which sum up to 180 degrees, you know that right now what a transversal, you know the meaning of transversal Intersects to parallel lines. So in this case mn is the transversal Intersecting AB and PQ. So let's write given as AB parallel to PQ and Mn Mn is a transversal Mn is a transversal Okay to prove What do we need to prove that each pair each pair of conjugative interior Angles, okay, so each pair of conjugative interior angles, which all are conjugative interior angles So we have to prove that one angle three plus angle six is 180 degrees complementary and two Angle four plus angle five Is equal to 180 degrees Okay, so let's see how to prove So proof will be something like that Okay, what? So angle three plus angle six now we know see Angle three is equal to angle seven Okay, and you can write this is because they are corresponding angles corresponding angles Okay, and angle six plus angle seven is 180 degrees If you look closely these are nothing but linear pair linear pair isn't it? Six plus seven is 180 degrees. So you can say from one and two can say from One and Two what can I say? I can replace seven by six so hence I can say Seven by actually angle three. So this angle three can come here And you can say angle six plus angle three is equal to 180 degrees. So first one is proved This point is proved Very simple proof Now for the next pair again, what can you say you can say similarly? angle Similarly, you can write see me literally angle Four our angle one is equal to angle five. Why? Corresponding angles again chorus on ding Corresponding angles. This is number three. Let's say, okay and angle Angle Sorry What angle one is equal to angle five and we know that angle one plus angle four is equal to 180 degrees Yep, why again? This is four and the reason being a linear pair Linear pair, right? So now done from from three and four we can say angle Five plus angle four is 180 degrees and this is the second proof Hence proved Correct. So hence we saw we saw that the interior pair of in conjugative interior angles are supplementary right for a pair of conjugative interior angles are Supplementary so four plus five is 180 and three plus six is 180 Fair enough. Let's go to the converse of this Now the converse statement is something like that if a transversal inter intersects two lines in such a way that a pair of conjugative interior angles are Supplementary then the two lines are parallel. So hence you can start the proof by saying Given what's given guys? So given is chorus, you know, conjugative interior angles are supplementary This is given that means let's say angle three plus angle six is 180 degrees Angle three plus angle six is 180 degrees. Okay to prove You can also take four and five doesn't really matter the proof will be similar to prove AB is parallel to PQ in The previous theorem AB was given to be equal to PQ and we had to prove that angle three plus six is 180 180 degrees So in this theorem, it is the other way around angle three plus six is given as 180 degrees prove that AB is parallel to PQ Okay, let's see how to prove this so We'll again use the theorems that we have learned before so Angle three plus angle six is 180 degrees and you write this as one and it's given Okay, similarly angle two plus angle three is 180 degrees. Why clearly it is because of linear so hence if you Equate both of them one and two from one and two From one and two You can say angle three plus six Is angle two plus angle three So three three goes so hence angle two is equal to angle six Right, but then angle two is angle equal to six, but what are two and six guys if you see two and six are corresponding angles so hence by Converse of Converse of corresponding Converse of corresponding angles Right AB is parallel to PQ hence proved because converse of Axiom corresponding angles axiom said This is from axiom says that if corresponding angles are equal then the lines are parallel So hence proved so hence the converse of the given theorem is also proved