In this chapter of lines and angles, we will learn about pairs of angles which have been given specific names.
Lines and Angles
1) Adjacent Angles : Two angles in a plane are adjacent angles, if
i) They have a common vertex
ii) They have a common arm, and
iii) Their other arms lie on the opposite sides of the common arm.
Here, ∠ AOC and ∠BOC have the common vertex O. Also, they have a common arm OC and other arms OB and OA.
So, ∠AOC and ∠BOC are adjacent angles.
2) Linear Pair Angles : Two adjacent angles are said to form a linear pair of angles, if their non-common arms are two opposite rays.
Here, OA and OB are two opposite rays and ∠AOC and ∠BOC are the adjacent angles. Therefore, ∠AOC and ∠BOC form a linear pair.
The sum of the angles in a linear pair is 1800. ∠AOC + ∠BOC = 1800.
3) Vertically Opposite Angles : Two angles formed by two intersecting lines having no common arm are called vertically opposite angles.
Vertically opposite angles have the same measure. They are equal.
Here, AB and CD are intersecting lines and intersection point is O. So 4 angles are formed. Angles ∠1 and ∠3 form a pair of vertically opposite angles; while ∠2 and ∠4 form another pair of vertically opposite angles.
Proof of vertically opposite angles are equal.
∠1 and ∠2 form a linear pair.
∠1 + ∠2 = 1800
⇒ ∠1 = 180 - ∠2
Also ∠2 and ∠4 form a linear pair.
∠2 + ∠4 = 1800
⇒ ∠3 = 180 - ∠2
So from the above, its clear that
∠1 = ∠3
Similarly, ∠2 = ∠4
Angle at a point : Angles formed by a number of rays having a common initial point are called angles at a point.
In the given figure, O is the common point.
If you find the measures of these angles 1,2,3 and 4 then it will be