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Facts About Vertical Angles-

Theorem of Vertical Angles-

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.

Are all Vertical Angles 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.

Are Vertical Angles Adjacent?

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.

Are all Vertical Angles Supplementary?

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°

Are Vertical Angles Complementary?

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. 

What are Vertical or Intersecting Lines?

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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Questions to be solved on Vertical Angles-

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

(image will be uploaded soon)

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°:

(image will be uploaded soon)

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°.

Fun Facts!