The vertical angle theorem tells us that the pairs of vertical angles formed by the intersection of two lines have the same size. In this article, we will look at a summary of the vertical angles theorem.
Then, we will apply this theorem by solving various practice problems.
Summary of the vertical angles theorem
Vertical angles are the angles formed by the intersection of two lines. For example, in the diagram below, we have two pairs of vertical angles.
Angles a and b and angles c and d are pairs of vertical angles.
The vertical angles theorem tells us that pairs of vertical angles have the same size. Therefore, in the diagram shown above, we know that angles a and b and angles c and d are equal.
You can look at the proof of this theorem in this article.
Vertical angles theorem – Examples with answers
The following examples are solved using the vertical angles theorem. Try to solve the exercises yourself before looking at the solution.
EXAMPLE 1
Determine the size of each of the angles in the diagram below.
Solution
Angles ∠62° and ∠b are a pair of vertical angles. Thus, we have:
∠b = 62°
The value of ∠a can be found considering that it is a supplementary angle to the angle of 62°. That is, these angles add up to 180° and we have:
62° + ∠a = 180°
∠a = 118°
Finally, we can see that angles ∠a and ∠c are another pair of vertical angles, so we have:
∠c = 118°
EXAMPLE 2
What is the value of Y in the diagram below?
Solution
Angles ∠100° and ∠X are part of a straight line, so they are supplementary. Therefore, we have:
100° + ∠X = 180°
∠X = 80°
Since the angles (Y + 30)° and X are vertical angles, we can form the following equation and solve for Y:
Y + 30 = 80
Y = 50°
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EXAMPLE 3
What is the value of Z in the diagram below?
Solution
This problem is similar to the previous one. Therefore, we know that angles ∠X and ∠110° form a straight line, so they are supplementary. Then, we have:
110° + ∠X = 180°
∠X = 70°
We also know that angles (Z + 10)° and X are opposite verticals, so we have:
Z + 10 = 70
Z = 60°
EXAMPLE 4
We have the angles (5x-11)° and (3x+23)°, which are opposite vertical angles. Determine the value of x and the size of the given angles.
Solution
Opposite vertical angles are equal, so we have:
5x-11 = 3x + 23
5x-3x = 23 + 11
2x = 34
x = 17
The value of the given angles is:
3(17)+23 = 74°
Vertical angles theorem – Practice problems
Solve the following problems using the vertical angles theorem. If you need help with this, look at the solved examples above.
See also
Interested in learning more about vertical angles and other types of angles? Take a look at these pages:
