How do you calculate Vertical Angles?
There is no specific formula to calculate vertical angles but you can calculate vertical angles by applying supplementary and congruent angles theorems on different examples.
Let’s see the following examples to understand how vertical angles are calculated for different examples.
Vertical Angles Examples
Here are the examples of vertical angles to clearly understand their theorem with step by step solutions:
Example #1
For what value of y the corresponding angles are vertical angles?
Since the corresponding angles are congruent, so,
$$ 9y \;+\; 7 \;=\; 2y \;+\; 98 $$ $$ 9y \;-\; 2y \;=\; 98 \;-\;7 $$ $$ 7y \;=\; 91 $$ $$ y \;=\; \frac{91}{7} \;=\; 13 $$
Now using the value of y,
$$ 9(13) \;+\; 7 \;=\; 2(13) \;+\; 98 $$ $$ 117 \;+\; 7 \;=\; 26 \;+\; 98 $$ $$ 124 \;=\; 124 $$
Hence for y=13, the both angles are congruent angles and hence are vertical angles.
Example #2
Find the value of x and y if these are vertical angles.
Since the corresponding angles are congruent,
$$ 2x \;+\; 5 \;=\; 3x \;-\; 33 \; \; .......... \; (1) $$ $$ 3x \;-\; 2x \;=\; 33 \;+\; 5 $$ $$ x \;=\; 38 $$
Hence using the value of x in 1.
$$ 2(38) \;+\; 5 \;=\; 3(38) \;-\; 33 $$ $$ 81 \;=\; 81 $$
Now to find the value of y, we will use the linear pair of angles, so,
$$ 2x \;+\;5 \;+\; 4y \;-\;1 \;=\; 180 $$ $$ 2x \;+\; 4y \;=\; 176 \; \; .......... \; (2) $$
Using the value of x in equation 2.
$$ 2(38) \;+\; 4y \;=\; 176 $$ $$ 4y \;=\; 176 \;-\; 76 \;=\; 100 $$ $$ y \;=\; 1004 \;= \;25 $$
Now using the y
$$ 4(25) \;-\; 1 \;=\; 99 $$
Which is the unknown angle.
Example #3
Find the value of x and y and prove that the angles formed by the lines AB and CD are vertical angles.
Since the corresponding angles are congruent then,
$$ 4x \;+\; 10 \;=\; 2y \;-\; 30 $$ $$ 4x \;-\; 2y \;=\; -40 \; \; .......... \; (1) $$
Similarly,
$$ 2(y \;+\; 25) \;=\; 13x $$ $$ 2y \;+\; 50 \;=\; 13x $$ $$ 13x \;-\; 2y \;=\; 50 \; \; .......... \; (2) $$
Subtracting equations 1 and 2.
$$ 13x \;-\; 4x \;=\; 50 \;+\; 40 \;=\; 90 $$ $$ 9x \;=\; 90 $$ $$ x \;=\; 10 $$
Using the value x in equation 1.
$$ 4(10) \;-\; 2y \;=\; -40 $$ $$ 40 \;-\; 2y \;=\; -40 $$ $$ -2y \;=\; -80 $$ $$ y \;=\; 40 $$
Example #4
Find the value of x such that the angles form a linear pair.
By using the supplementary angles,
$$ 5x \;-\; 22 \;+\; 8x \;+\; 46 \;=\; 180 $$ $$ 13x \;+\; 24 \;=\; 180 $$ $$ 13x \;=\; 180 \;-\; 24 $$ $$ 13x \;=\; 156 $$ $$ x \;=\; 12 $$
Now by using the value of x,
$$ 5x-22=512-22=38 $$
And,
$$ 8x \;+\; 46 \;=\; 812 \;+\; 46 \;= \;142 $$
FAQ’s
What are 4 vertical angles?
When two lines cross each other, they meet at a single point called a vertex and form four angles. These are called vertical angles.
How do you find the missing angle?
You can find the missing angle by using the congruent angles theorems and linear pair of angles.
Do vertical angles add up to 180?
Yes, the vertical angles form a linear pair of supplementary angles and hence they add up to 180°.