Vertically Opposite Angles

When two lines intersect each other at a point, the angles opposite to each other are called vertically opposite angles. Moreover, vertically opposite angles are always equal to each other.

In the following figure, PQ and RS intersect each other. In this case, ∠ m and ∠ n are vertically opposite angles. Similarly, ∠ a and ∠ b are vertically opposite angles.

vertically opposite angle

Let us prove, how vertically opposite angles are equal to each other.

`m + b = 180°` (Linear pair of angles)
`b + n = 180°` (Linear pair of angles)

From above equations, it is clear that m = n
So, it is proved that vertically opposite angles are equal.


Example: Find the values of x and y in following figure.

vertically opposite angle

Solution: `x = 105°` (Vertically opposite angles)
Similarly, `y = 75°` (Vertically opposite angles)

Example:An angle is equal to its complementary angle. What is the value of this angle?

Solution: Let us assume, the angle = x

We have, `x + x = 90°`
Or, `2x = 90°`
Or, `x = 90° ÷ 2 = 45°`


Example: An angle is equal to its supplementary angle. What is the value of this angle?

Solution: Let us assume, the angle = x

We have, `x + x = 180°`
Or, `2x = 180°`
Or, `x = 180° ÷ 2 = 90°`

Example: An angle is double its supplementary angle. What is the value of this angle?

Solution: Let us assume that the smaller angle = x, then the larger angle = 2x

We have, `x + 2x = 180°`
Or, `3x = 180°`
Or, `x = 180° ÷ 3 = 60°`
Hence, `2x = 120°`

Example: An angle is double its complementary angle. What is the value of this angle?

Solution: Let us assume that the smaller angle = x, then the larger angle = 2x

We have, `x + 2x = 90°`
Or, `3x = 90°`
Or, `x = 90° ÷ 3 = 30°`
Hence, `2x = 60°`


Question: Find the value of x in each of the following figures.

vertically opposite angle

Solution: The angles in these figures are on the same side of a line, so their sum is equal to 180°.

Solution 1: `82° + 74° + x = 180°`
Or, `156° + x = 180°`
Or, `x = 180° - 156° = 24°`

Solution 2: `41° + 84° + x = 180°`
Or, `125° + x = 180°`
Or, `x = 180° + 125° = 55°`

Solution 3: `85° + 68° + 2x = 180°`
Or, `153° + 2x = 180°`
Or, `2x = 180° - 153° = 27°`
Or, `x = 13.5°`

Solution 4: `44° + 67° + x + 2x = 180°`
Or, `111° + 3x = 180°`
Or, `3x = 180° - 111° = 69°`
Or, `x = 23°`



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