Vertical Angles – Visual Fractions

Vertical angles are simply vertically opposite angles formed when two lines intersect and meet at a common point, which we call the vertex. These are one of the first types of angles that you’ll learn in geometry and you’ll end up working with more advanced topics. By understanding the definition and the properties exhibited by vertical angles, you’ll have access to more properties, applications, and tools.

In this article, we’ll cover the fundamentals needed to identify vertical angles from a given diagram. We’ll also show you why vertical angles are equal and demonstrate the techniques needed to find unknown measures of vertical angles.

What Are Vertical Angles?

Vertical angles are two pairs of opposite angles formed when two lines intersect each other and meet at a vertex. Each pair of vertical angles will always be equal to each other. Here’s an example of two pairs of vertical angles formed by the intersecting lines, $m$ and $n$ .

These two lines form an “X” shape and the pairs of angles facing opposite each other are what we call the vertical angles. This means that each pair of angles will have equal measures:

\begin{aligned}\angle 1 &= \angle 2\\\angle 3 &= \angle 4 \end{aligned}

Some examples of real-world objects that form vertical angles are 1) crossroad signs like the one shown below and 2) the vertical angles formed by a pair of open scissors. At the intersecting or pivot point, the intersecting lines will form vertical angles where each pair will always be equal.

We’ve shown you the basic definition and concept you need to know about vertical angles. Before we dive into more complex examples and problems, let’s try identifying vertical angles given different examples.

Problem 1

Write down all pairs of vertical angles you can find in the following diagrams.

Recall that vertical angles are opposite angles meeting at the intersection of two intersecting lines.

Let’s focus on the upper half of our diagram first: we can see two intersecting lines forming four angles. Identify the pairs of vertical angles facing opposite each other, so we have the following vertical angles:

\begin{aligned}\angle a &= \angle d\\\angle b &= \angle c \end{aligned}

Now, apply a similar process to find the vertical angles found in the lower half of the diagram. Hence, we have the following vertical angles:

\begin{aligned}\angle e &= \angle h\\\angle f &= \angle g \end{aligned}

This means that from the diagram, we can name four pairs of vertical angles:

$\angle a$ and $\angle d$
$\angle b$ and $\angle c$
$\angle e$ and $\angle h$
$\angle f$ and $\angle g$

Understanding the Vertical Angle Theorem and Its Proof

At this point, you already know that when angles are vertical, their measures are equal. This statement is in fact the vertical angle theorem. According to the theorem, when two opposite angles share a common vertex formed by two intersecting lines, their angle measures will always be equal.

“Vertical angles are always equal.”

Proving this theorem is straightforward – we’ll just have to apply the property of linear angles and the transitive property in math. Let’s begin with the fact that since these are lines, the sum of the angles forming a line will always be equal to 180^o.

This means that we have the following equations:

\angle 1 + \angle 4 = 180\degree

\angle 1 + \angle 3 = 180\degree

Now, by transitive property ( $a =b$ and $b =c$ then $1 =c$ ), we can equate the right-hand side of each of the equations to each other.

\begin{aligned}\angle 1 + \angle 4 &= 180\degree\\\angle 1 + \angle 3 &= 180\degree\\\angle 1 + \angle 4 &= \angle 1 + \angle 3\\\angle 4 &= \angle 3 \end{aligned}

Hence, we’ve shown that the pair of vertical angles, $\angle 3$ and $\angle 4$ , are equal – confirming half of the vertical angle theorem. We simply apply the same process to confirm that $\angle 1 = \angle 2$ . This establishes the vertical angle theorem and confirms that each pair of vertical angles has equal angle measures.

How To Use Vertical Angles?

We can apply the definition and properties of vertical angles to solve word problems and find unknown angles in a given diagram. When using vertical angles, keep the following pointers in mind:

Confirm first if the two angles we’re working on are in fact vertical angles.
Vertical angles will always be equal.

When solving problems involving vertical angles, identify the measures of the vertical angles then equate them to each other to solve for any unknowns. This section will focus on more examples of problems that involve vertical angles.

Problem 2

Determine the measures of the three angles: $\{\angle 1, \angle 2, \angle 3\}$ .

First, let’s find the measure of $\angle 1$ using the fact that it forms a line with $\angle 4 = 80\degree$ . This means that they share a sum of $180\degree$ .

\begin{aligned} \angle 1 + \angle 4 &= 180\degree \,\,{\color{Orchid} \text{Linear Angles}}\\\angle 1 + 80\degree &= 180\degree\\\angle &- 180\degree – 80\degree\\&=100\degree \end{aligned}

Since lines $a$ and $b$ are intersecting lines, they form two pairs of vertical angles: $\angle 1$ and $\angle 2$ as well as $\angle 3$ and $\angle 4$ . This means that each pair of vertical angles will have equal measures.

\begin{aligned} \angle 1 &= \angle 2\\&= 100\degree\\\angle 3 &= \angle 4\\&= 80\degree \end{aligned}

We will actually be using a similar thought process when solving a more difficult problem involving vertical angles.

Problem 3

Suppose that lines $\overline{AB}$ and $\overline{CD}$ intersect each other at point, $O$ . If $\angle BOC = 140\degree$ , what is the angle measure of $\angle AOC$ ?

The first step when solving problems like this is to sketch a diagram that shows our angles and given like the one shown below.

We know that $\angle BOD$ and $\angle BOC$ are linear angles and form a line, so $\angle BOD + \angle BOC = 180\degree$ . Since $\angle BOC = 140\degree$ , we have:

\begin{aligned} \angle BOD + \angle BOC &= 180\degree\\\angle BOD + 140\degree&= 180\degree\\\angle BOD&= 180-140\degree\\&=40\degree \end{aligned}

Now, we know that $\angle BOD$ and $\angle AOC$ are vertical angles, so their angle measures must be equal. This means that $\angle AOC$ must be equal to $40\degree$ as well.

There are instances when we’re solving an algebra problem that involves angles, particularly, vertical angles. Use the same method when identifying vertical angles and equate the appropriate vertical angles to solve for any unknown values. Here are some sample problems you can work on to test your understanding.

Problem 4

Determine the value of the unknowns for each of the diagrams shown below.

Since the two lines intersect each other to form two pairs of vertically opposite angles. This means that each pair of vertical angles will share the same measure.

We only need to focus on the pair of vertical angles that contain our unknown value: the measures of $4x$ and $6x -12$ will be equal. Equate each algebraic expression then solve for $x$ .

\begin{aligned} 6x - 12 &= 4x\\ 6x - 4x &= 12\\2x &= 12\\\dfrac{2x}{2} &= \dfrac{12}{2}\\x&= 6\end{aligned}

This means that $x$ is equal to 6.

This time, we’re given two unknowns. Don’t worry, we’ll apply a similar approach to find the values of $x$ and $y$ . First, let’s identify the pairs of vertical angles from the diagram:

$\angle DOB = \angle AOC$
$\angle AOD = \angle BOC$

Now, equate the algebraic expressions corresponding to each of the angles then solve for the two resulting equations. The table below summarizes the steps to find $x$ and $y$ .

$\begin{aligned}\boldsymbol{x}\end{aligned}$	$\begin{aligned}\boldsymbol{y}\end{aligned}$
$\begin{aligned} \angle DOB &= \angle AOC\\12x + 5 &= 44 - x\\12x + x &= 44 - 5\\13x &= 39\\\dfrac{13x}{3} &= \dfrac{39}{3}\\x&= 3 \end{aligned}$	$\begin{aligned} \angle AOD &= \angle BOC\\6y +24 &= 4y +56\\6y - 4y&= 56 - 24\\2y &= 12\\\dfrac{2y}{2} &= \dfrac{12}{2}\\y &= 6 \end{aligned}$

Hence, have $x = 3$ and $y = 6$ .

At this point, we know that we have the following pairs of vertical angles for the diagram shown.

$\angle DOB = \angle AOC$
$\angle AOD = \angle BOC$

In addition, the linear angles, $\angle DOB$ and $\angle BOC$ will have a sum of $180\degree$ .

\begin{aligned} \angle DOB + \angle BOC &= 180\degree\\\angle DOB + 36\degree&= 180\degree\\\angle DOB&= 180-36\degree\\&=144\degree \end{aligned}

Now, let’s equate the expressions of each pair of vertical angles to solve for $x$ and $y$ . This means that we have the two equations: $x - 6 = 36$ and $3y + 6 = 144$ . Solve the equations to find the values of $x$ and $y$ .

$\begin{aligned}\boldsymbol{x}\end{aligned}$	$\begin{aligned}\boldsymbol{y}\end{aligned}$
$\begin{aligned} \angle AOD &= \angle BOC\\x - 6 &= 36\\x&= 36 + 6\\x&= 42 \end{aligned}$	$\begin{aligned}\angle AOD &= \angle DOB\\3y +6 &= 144\\3y&= 144 -6\\3y &= 138\\\dfrac{3y}{3} &= \dfrac{138}{3}\\y &= 46 \end{aligned}$

This shows that $x = 42$ and $y= 46$ .

Now, how do you check if the values are correct? Just substitute the values of $x$ and $y$ into the algebraic expressions then see if the angles satisfy the definitions of vertical and linear angles. Apply a similar process when solving for unknowns involving vertical angles in the future. For now, review your notes and try out the problems you’ve missed the first time around!