The Vertical Angles Theorem states that vertical angles, angles which are opposite to each other and are formed by two intersecting straight lines, are congruent. Vertical angles are always congruent.

According to the vertical angle theorem, no matter how we throw our pencils so that they cross, or how any two intersecting lines cross, vertical (opposite) angles will always be congruent, or in other words equal to each other. This is known as the Vertical Angles Theorem in Mathematics.

By definition, vertical angles cannot be adjacent (next to each other). Since vertical angles are opposite to each other they cannot be next to each other. Adjacent angles take one angle from one pair of vertical angles and another angle is taken from the other pair of vertical angles.

The angles that are opposite to each other when two lines intersect each other are known as vertical angles. The two pairs of vertical angles are equal to each other. The two pairs of neighboring angles are supplementary, meaning the angles add up to 180 degrees.

Let us consider the angles as a since vertical angles are always equal, both the angles can be written as 2a,

2a = 180° (Supplementary angles = 180°), a = 90°

Adjacent angles are always supplementary whereas vertical angles are not always supplementary. If we take any two adjacent angles from the four angles created by two intersecting lines then those two adjacent angles will always sum up to 180°.

Here’s a diagram for better understanding,

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In the diagram above, ∠x = 75° and ∠y = 105°.

From the diagram, we note that if two angles are vertical then they are equal (congruent). Notice that x and y are supplementary angles that sum up to 180 degrees.

∠x +∠y = 105° + 75° = 180°

Complementary Angles are the angles that sum up to 90 degrees. Let us consider the angles as a since vertical angles are always equal, both the angles can be written as 2a,

2a = 90° (Complementary angles =90°), a = 45°

Only when vertical angles measure 45°, they can be complementary.

When two lines meet at a point in a plane, they are known as intersecting lines or vertical lines. Parallel lines are the lines that do not meet at any point in a plane. The figure given below shows parallel lines and intersecting lines.

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Question 1) Find the value of x from the figure given below, when the value of y is 75.

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Solution) The value of angle y = 75° ,

Since x and y are supplementary angles (Sum of supplementary angles = 180°)

Therefore, the value of the angle x is equal to,

∠ x+∠y = 180⁰

∠ x = 180⁰- ∠y

∠ x = 180⁰- 75⁰

Therefore, the value of ∠ x = 105⁰

Question 2) Find the measure of the angles a°, b°, and c°:

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Since b° is vertically opposite to 40°, it must also be 40° (As vertical angles are congruent)

A full circle is equal to 360°,

So that means,

360° − 2×40° = 280°

Angle a° and angle c° are also vertical angles, so they are equal, which means the angles a and c are 140° each.

Therefore the angles measure, a = 140°, b = 40° and c = 140°.

FAQ (Frequently Asked Questions)

Question 1) What is a Vertical Angle Example?

Answer) The angles opposite each other when two lines cross are known as vertical angles. The angles are always equal. In the figure given example, a° and b° are vertical angles. The term “Vertical" generally refers to the vertex (where they cross) and they are also called vertically opposite angles. (image will be uploaded soon)

Question 2) Do all Vertical Angles Equal to 180?

Answer) The angles that are opposite to each other when two lines intersect each other are known as vertical angles. The two pairs of vertical angles are equal to each other. The two pairs of neighboring angles are supplementary, meaning the angles add up to 180 degrees.

Question 3) Do Vertical Angles Equal 360?

Answer) Vertical angles are always congruent or measure the same. Both pairs of vertical angles (all the four angles together) always sum to a full angle which is equal to 360°.

Question 4) Which Definition Best Describes Vertical Angles?

Answer) In geometry, if the angles are formed from two intersecting lines and the angles are not adjacent then the pair of angles is said to be vertical. The two angles share a common vertex. Such angles can be described as congruent as angles are equal in measure. Vertical refers to the vertex (where they cross or intersect).