 Hello friends, so let's continue with another theorem in this session and You know in this The theorem says if a transversal intersects two parallel lines, right? So there are there are two parallel lines and let us start by writing the given fact So given what is given a b is parallel to cd Two lines which are parallel to each other pq is the transversal pq is The transversal so you know transversal is nothing but a line which cuts two lines on a plane pq is a trans is a transversal and mr bisects mr bisects angle a rs and So this is one fact another fact is sn bisects sn bisects angle rs d r s D right so that means I have denoted here the angles by you know if you can see There's this is small x here. So this is small x this one and This one so these are the same right see mr bisects A rs so x is equal to x and here y is equal to y you can check that out. Okay now So if that is so What is to be proven so we have to prove We have to prove that RM is Parallel to sn Okay, this is what you have to prove Okay, so let's begin the proof so Angle a rs is equal to angle D s r and why is that because you can write a b is parallel to cd and Chorus alternate alternate Interior Alternate interior angles Okay, so alternate interior angles are equal. We know so this is true So you can write a R s by 2 is equal to angle B s r by 2 Correct. So now we know that RM is the bisector. So you can say angle m r s is Equal to angle N s r right because MRS is ARS by 2 and NSR is DSR by 2 Okay, now if you see look closely, so this is nothing but x is equal to y. Isn't it? This is x equals to y so that means what? if Pq is Transversal to MR and SN then Interior Angles That is alternate interior angles right alternate Interior angles angle MRS and Angle N SR are equal Are equal therefore By the converse of this Theorem which theorem that if alternate interior angles are equal then the lines are parallel we can say MR is parallel to SN And screwed Right, what is the underlying philosophy underlying thing is that if Two lines are given and transversal is there intersecting the two lines then if the alternate interior angles are equal Then the lines are parallel if you have difficulty in understanding this See this line is can we extend it like that? Isn't it? so this is nothing but these two are the Parallel lines and Pq happens to be the transversal here and This angle is equal to this angle alternate interior angles and hence The two lines are parallel. I hope you understood the theorem