Vertical Line: (Coordinate Geometry) Definition, Properties

Vertical Angles: Vertical Line is a line parallel to the Y-axis in a coordinate plane. It’s a line that runs from top to bottom and from bottom to top. The x-coordinate for each location along this line will be the same. The points of vertical lines, for example, are (2,0), (3,0), (-4,0), and so on. A horizontal line, on the other hand, is a line that runs from left to right and is parallel to the x-axis.

There is no slope to the vertical lines. The slope is not specified since it runs parallel to the y-axis. As a result, the equation for a vertical line that crosses the x-axis at any point ‘a’ is x=a. where x is the point on the line where the lines cross the x-intercept, and an is the point where the lines cross the x-intercept. On this page, we will provide you with all the necessary information regarding Vertical Angles, examples, proof, theorem. Read on to find out more.

What Is a Vertical Angle?

Vertical Angle Definition: In Mathematics, two straight lines intersect a point (vertex) to form angles. The opposite pair of angles after the intersection of two straight lines are known as vertically opposite angles.  

The vertical angle and its adjacent angles make supplementary angles, which means they are always 180 degrees.

Vertical Angles Examples

Look at the revolving doors given below. If we consider the sides of the door as lines, two lines intersect at each other at the point and form angles. And this intersection of lines at one point is known as vertical angles.

Are Vertical Opposite Angles Are Equal?

Vertically Opposite angles are always equal. For example, look at the image given below. Two lines intersect at a point “O” and forms 4 angles and they are ∠AOB, ∠BOC, ∠COD, ∠DOA. ∠AOB & ∠COD make 1 set of vertically opposite angles and ∠BOC & ∠DOA make 1 set of vertically opposite angles.

Vertically Opposite Angles: Theorem & Proof

Look at the image given here. Two lines AC and DB intersect at a point “O”. There are 4 angles formed due to the intersection of 2 lines and they are:
∠AOB,
∠BOC,
∠COD,
∠DOA.

From the image, we can see that ∠AOB & ∠COD are vertically opposite angles. Similarly, ∠BOC & ∠DOA are vertically opposite angles.  So now we will have to prove ∠AOB and ∠COD are equal to know if vertically opposite angles are equal.

As we know, If we have a line and a ray from any point on the lines form two angles. The sum of two angles is always 180 degrees and this concept is known as linear pair. The total measurement of linear pair angles is 180 degrees.

In the image given above, we see two linear pair angles and they add up to 180 degrees. Keeping that in mind, we can consider ∠AOB + ∠BOC = 180 degrees and  ∠BOC + ∠COD = 180 degrees. Now we can write the equation as

•  ∠AOB + ∠BOC = 180 degrees —–> Equation 1 (based on the linear pair) 
• ∠BOC + ∠COD = 180 degrees ——> Equation 2 (based on the linear pair)

From equation 1 & 2, we can write that

∠AOB + ∠BOC = ∠BOC + ∠COD 
∠AOB + ∠BOC = ∠BOC + ∠COD (Cancelling common trems) 
∠AOB = ∠COD ——–> Equation 3

From equation 3, it is proved that vertically opposite angles are equal.

Vertically Opposite Angles Questions

Q1. Prove that vertically opposite angles ∠a and ∠b in the following image are equal.
Answer: To prove ∠a = ∠b,  
consider ∠a + ∠n = 180° —> Equation 1 (Linear Pair) 
Similarly, considering ∠n + ∠b = 180° —-> Equation 2 (Linear Pair) 
From equation 1 and 2, we can write  
∠a + ∠n = ∠b + ∠n 
∠a + ∠n = ∠n + ∠b (cancelling common terms) 
∠a = ∠b  
Thus, proved that vertically opposite angles ∠a is equal to ∠b. 

Q2. Find the values of x from the following figure.
Answer: According to the vertically opposite angles theorem, it is stated that vertically opposite angles are always equal. 
The vertically opposite angle of X is 50 degrees.
Thus, the value of x = 50 degrees. 

Q3. In the given figure, two lines intersect at a point. If a is 40 degrees, find the value of b, c, d. 
Answer: 
If a = 40°, then c = 40° (Based on the vertically opposite angles) 
Here, 2 linear pair angles exist which means a + b + c + d = 360°
40° + b + 40° + d = 360° 
Therefore 360° – 80° = b + d 
280° = b + d 
280° = b + b (b = d due vertically opposite angles)
280° = 2b 
280°/2 = b 
140° = b 
If b = 140°, then d = 140° (Based on the vertically opposite angles
Therefore, the value of c = 40°, b = 140°, d = 140° 

Q4. Identify 2 vertically opposite angles from the following figure.
Answer: The two pair of vertically opposite angles from the above figure are: 
1. ∠AOD & ∠COB 
2. ∠AOC & ∠DOB 

 What Did We Learn?

1. Two straight lines intersect at a point (vertex) to form angles. The opposite pair of angles after the intersection of two straight lines are known as vertically opposite angles.
2. Two angles are said to be linear if there are adjacent angles formed by the intersection of 2 lines. The measure of the linear angle is always 180 degrees.
3. Vertically opposite angles are always equal.

Vertically Opposite Angles – FAQs

The frequently asked questions on vertical angles are given below:

Now that you are provided with all the necessary information about Vertical Angles and we hope this detailed article is helpful to you. If you have any queries on Vertically Opposite angles, ping us through the comment section below and we will get back to you soon as possible.