Vertical Angles

Angles are formed by two lines or rays that meet at a common endpoint. When two angles are paired together, they are given unique names depending on how they are positioned together or how they are related to each other. One of which is the pair of vertically opposite angles.

Are you now getting excited to learn this type of angle? Get ready as we are going to dive deep into these remarkable pairs of angles!

Vertical Angles

Let’s start by defining vertical angles.

A pair of angles that are opposite or non-adjacent to each other are called vertical angles. These angles are formed by two distinct lines that intersect each other.

Now, take a look at the image below.

We can see that two distinct lines (line 1 and line 2) intersect at point A which forms four angles. Given the points B and E on line 2 and points C and D on line 1, we can name the angles formed as ∠BAC, ∠BAD,  ∠CAE, and ∠DAE. All these four angles have the same vertex at point A.

Looking at the figure, we can note two different colors around point A. These two colors, purple and green, denote that two pairs of vertical angles were created. Thus, we can conclude that ∠BAC is vertically opposite to ∠DAE, while ∠BAD is vertically opposite to ∠CAE.

We often define the word “vertical” as the opposite of horizontal – or something in an upright or standing position.

However, when we talk about vertical angles, we are more focused on the term “vertex” as each pair of vertical angles share a common vertex.

How to determine pairs of vertical angles?

There are certain things we need to consider to say that a pair of angles are vertically opposite. Let’s use the checklist below to determine pairs of vertically opposite angles.

• Are there two distinct straight lines?
• Do the lines intersect each other?
• Do the angles share the same vertex?
• Are the angles opposite to each other?

If you checked all the boxes on the list, then we can conclude that the angles are a pair of vertical angles. Now, let’s see if you can apply that to the image below.

Do you think ∠1 and ∠2 are a pair of vertical angles? Let’s use our checklist!

• Are there two distinct straight lines?
  – Yes, we can see two straight lines named as line m and line n.
• Do the lines intersect each other?
  – Yes, lines m and n intersects at point P.
• Do the angles share the same vertex?
  – Yes, angles 1 and 2 share the same vertex at point P.
• Are the angles opposite to each other?
  – Yes, angle 1 is positioned oppositely to angle 2.

Therefore, we can conclude that ∠1 and ∠2 are a pair of vertical angles.

Consider another example where line a and line b intersect each other.

Can you say that ∠3 and ∠4 are a pair of vertical angles? Let’s find out using our checklist!

• Are there two distinct straight lines?
  – Yes, lines a and b are two distinct straight lines.
• Do the lines intersect each other?
  – Yes, lines a and b intersect at point M.
• Do the angles share the same vertex?
  – Yes, the common vertex of line a and line b is at point M.
• Are the angles opposite to each other?
  – No, ∠3 is positioned next to ∠4.

Thus, ∠3 and ∠4 are not a pair of vertical angles because they are adjacent.

Naming vertical angles can be quite confusing for some, but once you practice and get the hang of it – it will be as easy as naming numbers and shapes.

Did you know that…

The vertical angles exist in the flag of Jamaica?

This is because of the yellow lines that form the letter X – thus creating two pairs of vertically opposite angles. Isn’t it amazing?

The Vertical Angle Theorem

The vertical angle theorem simply states that vertical angles are always equal.

But… where did this theorem come from? Let’s find out by proving.

The proof of the vertical angle theorem is very straightforward as it is based on linear pair. A linear pair is a pair of adjacent angles – and the linear pair postulate states that if two angles form a linear pair, then their sum is 180°.

Now, we are ready to prove the vertical angle theorem!

Suppose we have two lines – line m and line n that intersect at point P.

Then, we can say that ∠1 and ∠2 are linear pairs because they are adjacent angles. By linear pair postulate, it follows that the sum of ∠1 and ∠2 is 180°.

Hence, ∠1 + ∠2 = 180°.

Since ∠1 and ∠4 are two adjacent angles, then it also follows that
∠1 + ∠4 = 180°.

Using the transitive property of equality,
∠1 + ∠2 = ∠1 + ∠4
∠2 = ∠4

Similarly, if ∠1 and ∠2 and ∠2 and ∠3 are adjacent angles, we can already conclude that ∠1 = ∠3.

How to solve problems involving vertical angles?

Now that we’ve properly defined vertical angles, it’s time to solve problems about vertical angles.

Say, we have two pairs of vertical angles where “∠” a is vertically opposite to ∠c and ∠b is vertically opposite to ∠d. What will be the angle measure of ∠c if ∠a = 72?

How about the angle measure of ∠b and ∠d?

Since it is already stated that ∠a and ∠c are pairs of vertical angles, then by vertical angle theorem, the measure of their angles are equal. Thus, ∠c = 72.

Now, how are we going to find the angle measure of ∠b and ∠d?

∠b and ∠d are positioned adjacently to ∠a. Hence, we can say that ∠a and ∠b is linear pair. Using the linear pair postulate, ∠a + ∠b = 180°

By substitution,

∠72 + ∠b = 180°
∠b = 180°-72°
∠b = 108°

Since ∠b and ∠d are vertically opposite angles, we can conclude that ∠d is also equal to 108°.

Let’s try another example.

If ∠MOP and ∠NOQ are pairs of vertical angles. What is the value of x if ∠MOP measures 73° and ∠NOQ measures 4x + 13?

To answer this problem, we will apply the vertical angle theorem. Thus, ∠MOP = ∠NOQ. By substitution, we will have the equation
73° = 4x + 13

Now, let’s manipulate this equation to find the value of x.
73° – 13° = 4x
60° = 4x
$$\frac{60}4^\circ\;=\;\frac{4x}4$$
x=15°

Therefore, the value of x is 15°.

There is no step-by-step process to solve problems involving vertical angles. However, we must always determine if they are a pair of vertical angles before using the vertical angle theorem.

Vertical angles in real world

Vertical angles are not just a term that we use in mathematics. There are a lot of applications and objects that can be seen in the real world – and we’re sometimes just unaware of it.

Whenever we write the letter X or use scissors, we also create pairs of vertically opposite angles. More so, we can sometimes see a pair of vertical angles in our doors, windows, and tiles.

Another cool thing about vertical angles is that it is used to create a dartboard. In a dartboard, we can see 10 pairs of vertically opposite angles formed by the intersecting lines.

Can you name other objects that may have created vertical angles?

Take a quiz

Now that we’ve learned a lot of things about vertical angles. Are you ready to practice and apply your knowledge about vertical angles?