Vertical Angles

Students will be able to learn and understand what are vertical angles and also how to calculate vertical angles with solved examples and fun facts and answers to the most frequently asked questions about vertical angles. Before we learn about vertical angles let us first understand a few basic concepts that are important to understand as well. 

What are Intersecting Lines? 

When two or more lines intersect each other on a plane, they are known as intersecting lines. All the intersecting lines will meet each other at one point as they are crossing each other, this common point on the intersecting lines is called an intersection point or point of intersection. 

Properties of Intersecting Lines

There will always be one common point even if there are two or more lines intersecting each other. These lines can cross each other at any angle but the angle is always more than 0° and less than 180°.  These two intersecting lines crossing each other on a plane form a pair of vertical angles and have a common vertex or a common meeting point. 

For example, scissors have two arms and these two arms form intersecting lines that have a common meeting point. Crossroads are also a great example of intersecting lines as they are two different roads meeting at a common point.

Fun Facts About Intersecting Lines

If there is a line that crosses more than one point or intersects more than one point are not straight lines but curved lines. 

When two lines intersect at an angle of 90 degree, which are drawn on the same plane, they are called perpendicular lines. Perpendicular lines are exceptional cases of intersecting lines.

What are Parallel Lines?

Parallel lines are opposite of intersecting lines. Where intersecting lines meet each other parallel lines do not meet each other on a plane and are at an equal distance from each other. Parallel lines could be horizontal, vertical or even diagonal lines. 

Properties of Parallel Lines

• Parallel lines have equal alternate interior and exterior. 

• The corresponding angles and the vertical angles of parallel lines are always equal. 

For example, the lines that we see in our notebooks are parallel lines. They do not meet each other and are at an equal distance from one another as well. Similarly, railway tracks and zebra crossing are also examples of parallel lines. 

Fun Facts about Parallel Lines

Every line can have multiple parallel lines. There are no set numbers to identify a parallel line. If the lines are not meeting each other and are at an equal distance they are parallel lines. 

Even if you keep extending a parallel line you can do that indefinitely and they would still not meet each other at any point or intersect with one another. 

What Are Complementary & Supplementary Angles?

Angles can be subcategorized into two major types based on rotation and based on their magnitude. Apart from these two major types of angles we also have another major category of angles called complementary and supplementary angles. 

Supplementary angles will always have a sum of 180° for two angles. Whereas, in complementary angles, the sum of two angles will be 90°.

What are Vertical Angles?

• In geometry, when two lines intersect they form four angles. Vertical angles are the angles that are opposite to each other. Any two intersecting lines form two pairs of vertical angles that are opposite to each other.

• The angles opposite to each other when two lines cross are known as vertical angles.

• The term “Vertical" in this case means they share the same Vertex or the same (corner point).

Example: Here a° and b° are vertical angles.

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The interesting thing is that vertical angles are equal:

 Angle a° = Angle b°

Facts About Vertical Angles-

The Theorem of Vertical Angles-

The Vertical Angles Theorem states that vertical angles, angles that 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 neighbouring 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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Applications of Vertical Angles

We see or experience many applications of vertical angles in our daily lives. 

For proper operation, the roller coasters are placed at a specific angle. These angles are so critical that if they shifted a degree to the left or right, an accident may occur. A roller coaster's maximum vertical angle (Mumbo Jumbo, Flamingo Land) is 112 degrees. 

We see two vapour trails crossing each other and making vertical angles at an airshow. 

For the safety of motorists, railroad crossing signs (X) are put on the roadways. 

A kite with two wooden sticks crossing each other and holding the kite in place. 

The bull's eye is a virtual vertex on the dartboard, which has ten pairs of vertical angles.

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.

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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°.

Fun Facts!

Conclusion 

A thorough understanding of the concepts of vertical angles in detail with  Solutions to questions as well is important for futuristic learning. Students can go through the concepts of intersecting lines, Complementary angles and much more. You should refer to the definitions and questions carefully and understand the concepts used to solve these questions. This will help students immensely in their examinations.