 Hi friends, so I welcome you again to another session on lines and angles and we will continue with our concept lectures And in this lecture, we are going to understand how very important component of geometry vertically opposite angles Now if you remember when we were talking about straight lines so lines can have two straight lines have three options on a plane one is they can Be coincident on each other. So let's say this is one line and I have another line, which is let's say Coincident on this one. Okay, so another line on this one. So if you see Obviously, it will be very difficult to see two coincident lines But I have drawn through, you know one small arrow and if I change the color of this one so let me draw another one and Show you how? Yeah, so I'm just drawing right. So these are coincident lines two lines are coincident Other possibilities were two lines are parallel. So this is one line and another line is this one They are parallel and the third possibility is they intersect on the same plane, isn't it? So hence this is intersection To intersecting lines. So vertically opposite angles are generated when two lines intersect, okay So if you see here, let's say this is a and b and c and d which I have already shown here as well So there are two lines a b and c d. They are intersecting at point o Okay. Now angle a o c and angle b o d b o d form a pair of vertically opposite Angles Okay, similarly, so it's not that only one pair exists. There are one. There is one more pair. So every Whenever two lines intersect, there will be two pairs. One is I just show you another one if you see b o c and Angle a o d again, this is another pair, right? So there are two pairs of vertically opposite angles here So now you understood so whenever two lines intersect at point o or Whatever is the name of the point. So these are the pair of vertically opposite angle and the second pair is this one Okay, so please remember and please understand this concept because now what we are going to prove is That the vertically opposite angles are always equal. So what is our thing? Okay, so the theorem is vertically opposite angles are always equal. So we have to prove that so we have drawn here to intersecting Vertically opposite Sorry to intersecting lines. So this is a b and cd now. Let us say they are intersecting at point o We have to prove that let's say a o D this is x and let's say this is y So you have to we have to prove that x equals to y to prove to prove x equals to y and similarly if this is U and this is v So you is also equal to v So we will prove one And the other will automatically follow Okay, or you can repeat the same process to prove the other pair. So how to prove x plus y now Cd is a straight line Is a straight line Right and oa happens to be a ray on cd. So hence can I not say x plus u is 180 degrees Why again The reason is linear pair. We studied this in the previous sessions linear pair So a cd being the straight line oa being a ray over it So hence the they are adjacent Angles with opposite arms non common arms in the opposite direction. So hence it will be linear pair. So x plus u is 180 degrees similarly Similarly, if you see u plus y is also 180 degrees Okay, same reason what linear pair because ab is the straight line now. So from this one And from this two what can I say? From one and two I can equate both of them from one and two I can say since both are equal to 180 degrees. I can say x plus u Is equal to u plus y and now I can cancel this u both sides. So what do I get? I get x is equal to y. That's what we have to prove With the similar logic, you can prove that u is also equal to v Okay, so what is the conclusion conclusion is pair of vertically opposite angles are always equal Okay, so hence if you have like that two lines intersecting this line this angle will be equal to this angle And this angle will be equal to this angle Is that clear? So that's what Is the information about vertically opposite angles