 Hello friends. So welcome again to this session on lines and angles and as we discussed in the last session We talked about corresponding angles action we talked about Transversal and two parallel lines and corresponding angles action now basis that action We are going to prove a few more theorems The first theorem is if a transversal intersects two parallel lines then each pair of alternate interior angles are equal That's what the theorem is and we are going to prove this. So first of all, what is a transversal? You know in this case line Mn is a transversal and intersects two parallel lines. So given is This is given given is a b is parallel to pq. Okay, and Mn Mn is a transversal is a Transversal, okay, what do you need to prove? to prove to prove what? angle 3 must be equal to angle 5 or Angle 4 must be equal to angle 6. This is what we have to prove Okay, so now proof. How do we do this proof? So, we know since a b is parallel to pq, right then by by corresponding corresponding Angles theorem or action. It's not a theorem. It's an action, right? corresponding angles action angle 1 is Equal to angle 5 Okay, let it be one and Angle Angle 1 is also equal to angle 3. Why? Because it is pair of vertically opposite vertically opposite Angles, okay, vertically opposite angles. So hence from 1 and 2 from 1 and 2 We can find out we can say what can we say we can say angle 5 is equal to angle 3 from from 1 and 2 from 1 and 2 we can say that angle 5 is and is equal to angle 3 Okay, so hence proved. So this is what we have to prove angle 5 is equal to angle 3. So hence proved hence proved hence proved Similarly, you can say similarly Similarly, what can we said what can we what can we say we can say that? angle 2 is equal to angle 6 and And and this is why chorus by corresponding angles equal by corresponding corresponding Angles axiom, isn't it and angle? 2 is equal to angle 4 again vertically opposite angles. So hence what do we learn? This implies Angle 6 is equal to angle 4. Okay Yeah, so hence proved again, so alternate angles interior angles are equal For a case where there are two parallel lines and there is a transversal cutting them Okay, so this is the proof of this theorem So let us see the converse of this theorem is that also true? So the converse of the given theorem is says that if a transversal intersects two lines in such a way That a pair of alternate interior angles are equal then the two lines are parallel in the previous case We saw that the lines are parallel and we proved that interior alternate angles are equal Now the converse of the theorem will be that Alternate interior angles are equal prove that the lines are parallel. Okay, so we will use the converse of Corresponding angles axiom, which we learned in the previous session. So let's now prove it. So given Let's say angle 1 Not angle 1 in that angle 3 is equal to let's say angle 5. This is given Okay, let's say this pair of alternate interior angles are equal to prove What do we what do we need to prove a b is parallel to pq? Let's prove it Very simple proof because we know that since angle 3 is equal to angle 5 So or rather angle since angle 3 is equal to angle 1, isn't it? Angle 3 is equal to angle 1. Why? vertically Opposite angles are equal Okay, so what can we say from let's say this is 1 and let us say this is 2 so from 1 and 2 we can say angle 1 is equal to angle 5 Okay, angle 1 is equal to 5 5 therefore by Converse Converse of corresponding Corresponding angles Axiom which says that if corresponding angles are equal then the lines are parallel, right? We can say a b is Parallel to pq Hence proved Okay, so if you see you just need to prove any one pair of alternate interior angles are equal and the lines will be parallel So this is the converse of the same theorem