Defining Vertical Angles
Let us consider the intersection of two lines. In the image there are two lines AB and CD intersecting at the point O.
Also in the image, ∠AOC and ∠BOD, and, ∠COB and ∠AOD are vertical angles and are also known as vertically opposite angles. So, whenever two lines intersect there are two pairs of vertical angles that are formed at the point of intersection of the two lines.
Explaining the Vertical Angles Theorem
The vertical angle theorem states that the vertical angles that are formed when two lines intersect are congruent.
From the image we can understand that:
- ∠AOC = ∠BOD = ∠x
- ∠COB = ∠AOD = ∠y
These two equations are proved by the vertical angle theorem.
Proving the Vertical Angle Theorem
In order to prove the vertical angle theorem let us consider the example mentioned earlier in this article.
In the image lines AB and CD intersect at point O.
Angles formed:
- ∠AOC = ∠1
- ∠AOD = ∠2
- ∠DOB = ∠3
Statement 1:
∠1 + ∠2 = 180° – Linear pair of angles are supplementary
Statement 2:
∠2 + ∠3 = 180° – Linear pair of angles are supplementary
Statement 3:
∠1 + ∠2 = ∠2 + ∠3 = 180° – As mentioned in Statements 1 and 2
Statement 4:
∠1 = ∠3 – Subtracting ∠2 from both sides of the equation
Hence, the opposite pairs of vertical angles are equal:
∠AOC = ∠DOB.
Rapid Recall
Solved Examples
[Note:The images are representative and the actual angles may be different.]
Example 1:
Find the value of ‘x’ and the value of the vertical angles in the following images:
1. If ∠KNH is 2x + 20 and ∠KNI is 3x – 15
2. If ∠QFS is 2x + 22 and ∠QFR is 2x + 14
Solution
- If ∠KNH is 2x + 20 and ∠KNI is 3x – 15
Solving to find x:
∠KNH + ∠KNI = 180° [Pair of linear angles]
(2x + 20) + (3x – 15) = 180° [Substituting the values]
2x + 3x + 20 – 15 = 180° [Grouping like terms]
5x + 5 = 180° [Simplify]
5x = 175 [Subtracting 5 from both sides]
x = 35 [Simplify]
So, 2x + 20 = 90° = ∠KNH
3x – 15 = 90° = ∠KNI
Hence all four vertical angles are 90° in measure.
2. If ∠QFS is 2x +22 and ∠QFR is 2x + 14
Solving to find x:
∠QFS + ∠QFR = 180° [Pair of linear angles]
(2x + 22)+ (2x + 14) = 180° [Substituting the values]
2x + 2x + 22 + 14 = 180° [Grouping like terms]
4x + 36 = 180° [Simplify]
4x = 144 [Subtracting 36 from both sides]
x = 36 [Simplify]
So, 2x + 22 = 94° = ∠QFS
2x + 14 = 86°= ∠QFR
Hence:
∠QFS = ∠RFP = 94°
∠QFR = ∠PFS = 86°
Example 2:
Find the value of ‘a’ and ‘b’ from from the following images:
A.
B.
Solution
A.
36 + b = 180° [Pair of linear angles]
b = 144° [Simplify]
So, a = 144° [Pair of vertical angles]
Hence:
∠a = ∠b = 144°
B.
∠a = 112° [Pair of vertical angles]
112 + b = 180° [Pair of linear angles]
b = 68° [Simplify]
Hence:
∠a = 112°
∠b = 68°
When two rays or lines meet at a point called the vertex, an angle is formed. Angles are measured in either degrees or radians.
When the sum of the measure of two angles is 90°, the two angles are called complementary angles. On the other hand, when the sum of the measures of two angles is 180°, the two angles are called supplementary angles.
Two lines intersecting at a point results in the formation of vertically opposite or just vertical angles. Such angles are equal and this is proved by the ‘Vertical Angle Theorem’.