Adjacent and Vertical Angle Formulas
This is how to identify adjacent and vertical angles that are formed when two straight lines intersect each other at one point.
Adjacent angles : ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠1.
Vertical angles: ∠1 and ∠3, ∠2 and ∠4
We will discuss these in detail in the next section.
How to use the Adjacent and Vertical Angle Formula?
Adjacent angle: Angles that have a common vertex and a common side are called adjacent angles.
In the given figure, rays OL and OM form the angle ∠LOM. ∠LOM is now cut by ray OX, resulting in the angle being split into two parts, which are referred to as adjacent angles. As a result, angle ∠LOX and angle ∠MOX are adjacent angles.
Vertical angles: Angles opposite to each other when two lines intersect.
In the above figure line AB and line CD intersects, so the angle opposite to each other, that is, ∠1 and ∠3, ∠2 and ∠4 are vertical angles. Vertical angles have the same measure because they are both supplementary to the same angle.
Solved Examples
Example 1: Name the adjacent and vertical angles in the given figure.
Solution:
Angles that have a common vertex and common side are called adjacent angles.
In the figure adjacent angles are:
- ∠AOE and ∠EOC
- ∠COB and ∠BOD
- ∠AOD and ∠DOB
- ∠DOA and ∠EOA
- ∠BOC and ∠COE
A pair of opposite angles formed by the two intersecting lines are called vertical angles.
In the figure vertical angles are:
- ∠AOC and ∠BOD
- ∠AOD and ∠BOC
Hence, we have found the vertical and adjacent angles that are given.
Example 2: Find the measure of missing angles.
Solution:
Angles that have a common vertex and common side are called adjacent angles. A pair of opposite angles formed by the two intersecting lines are called vertical angles.
∠AOB = 90° and
∠COD = 55°
∠AOE = ∠COD [As vertical angles are equal]
Therefore,
∠AOE = 55° [As, ∠COD = 55°]
∠BOD = ∠AOB = 90° [Linear pair of angles are supplementary]
∠BOD = ∠BOC + ∠COD
90° = ∠BOC + 55° [Substitute the values]
90° – 55° = ∠BOC + 55° – 55° [Subtract 55° from both sides]
90° – 55° = ∠BOC [Subtract]
∴ ∠BOC = 35°
∠DOE = ∠AOC [Vertical angles]
∠DOE = ∠AOB + ∠BOC
∠DOE = 90° + 35° [Add]
∴ ∠DOE = 125°
Hence the measures of angles are:
∠AOE = 55°,
∠BOC = 35°, and
∠DOE = 125°.
Example 3: Find the measure of angle x and y.
Solution:
Line l and line m intersect each other, so opposite angles formed by the two intersecting lines are called vertical angles.
∠x = 153° [As vertical angles are equal]
∠y = 27° [As vertical angles are equal]
Therefore measures of angle x and y are 153° and 27° respectively.
Example 4: Find if the angles are adjacent or not.
Solution:
We know that angles that have common vertex and common side are called adjacent angles.
In fig.(1):
∠55° and ∠45° are not adjacent angles, as they do not share a common side.
In fig.(2):
∠x and ∠y are not adjacent angles, as ∠y is a part of ∠x.
In fig.(3):
∠a and ∠b are adjacent angles, as they share common side and common vertex.
Therefore, angles in figure (1) and (2) are not adjacent angles and angles in figure (3) are adjacent angles.
Example 5: Look at the clock shown below. Name the angles made by the hour hand, minute hand, and second hand. Also, say if the angles are adjacent or not.
Solution:
Now, let the hour hand be OA, the minute hand be OB and the second hand be OC.
The angles formed are: ∠AOC, ∠BOC, and ∠AOB
As OA and OB share common side OC and a common vertex O.
Therefore, ∠AOC, and ∠BOC are adjacent angles.
Therefore, the angles made by the hour hand, minute hand, and second hand are adjacent angles.
Adjacent angles whose sum is always equal to 180° are called a linear pair of angles. They are also known as supplementary angles.
Two angles whose sum is 90° are called complementary angles.
Two angles whose sum is 180° are called supplementary angles.
Adjacent angles share a common side and a common vertex. However, vertical angles share a common vertex but not a common side, hence vertical angles are not adjacent.
Lines that meet each other are called intersecting lines. Alternately, lines that never meet each other are called non-intersecting or parallel lines. It should be noted that the distance between parallel lines always remains constant.
We have already seen that lines that never meet each other are called parallel lines.
In relation to parallel lines, a line that intersects the parallel lines at distinct points is called a transversal.
