 Hello and welcome to the session. In this session we will discuss about lines and angles. Let's consider two different lines PQ and RS. The lines PQ and RS can be drawn in two different ways like here PQ and RS are two intersecting lines which are intersecting at a common point say O. Now in this figure PQ and RS are the parallel lines and the lengths of the common perpendicular at different points on these parallel lines is the same and this equal length is called the distance between two parallel lines. Now we discuss pairs of angles. Consider this line AB. Now we have a ray OC which stands on the line AB. Now as you can see two angles are formed at the point O. One is angle AOC and another is angle BOC. Let's find out the relation between the angles formed when a ray stands on a line. We state an axiom according to which we have if a ray stands on a line then the sum of two adjacent angles so formed is 180 degrees. So from this figure we can say that angle AOC plus angle BOC is equal to 180 degrees and we know that when the sum of two adjacent angles is 180 degrees then they are called a linear pair of angles. So these two angles are the linear pair of angles. Now we have another axiom according to which we have if the sum of two adjacent angles is 180 degrees then the non-common arms of the angles form a line. Now suppose an angle AOC plus angle BOC is 180 degrees and according to this axiom we have that the non-common arms of the angles form a line. So AO and BO form a line hence AB is the line. These two axioms together are called linear pair axiom. We have a very important theorem which states that if two lines intersect each other then the vertically opposite angles are equal. Now as you can see we have taken two lines AB and CD which are intersecting each other. Now angle AOD and angle BOC they form vertically opposite angles also angle AOC and angle BOD form vertically opposite angles. So according to this theorem we have that angle AOD is equal to angle BOC and angle AOC is equal to angle BOD. Let's consider this figure in which we have two lines AB and CD intersecting at a point O. It's given that angle AOC is equal to 50 degrees. Now as you can see angle AOC and angle BOC form a linear pair. So by linear pair axiom we have angle AOC plus angle BOC is equal to 180 degrees. So from here we get angle BOC is equal to 180 degrees minus 50 degrees since we have angle AOC is equal to 50 degrees. So this comes out to be equal to 130 degrees. That is angle BOC is 130 degrees. Now angle BOC and angle AOD they are vertically opposite angles. We know that vertically opposite angles are equal so angle AOD is also equal to 130 degrees. Now it's also evident that angle AOC and angle BOD are vertically opposite angles and angle BOD would be equal to angle AOC which is 50 degrees. So we have found out all the four angles in this figure. This completes the session. Hope you have understood the two axioms and the theorem which is related to the vertically opposite angles.