Vertical Angles Theorem - Math Steps, Examples & Questions

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Vertical angles theorem

Here you will learn about the vertical angles theorem, including what it is and how to solve problems with it.

Students will first learn about the vertical angles theorem as part of geometry in $7$ th grade.

What is the vertical angles theorem?

The vertical angles theorem states that angles that are opposite one another at a specific vertex and created by two straight intersecting lines are called vertical angles.

For example,

Here the two angles labeled $a$ are congruent because they are vertical angles. This also applies to the angles labeled $b.$ You can try out the above rule by drawing two crossing lines and measuring the angles opposite to one another.

You will also notice that the adjacent angles $a$ and $b$ lie on a straight line. This means they are equal to $180^{\circ}$ when added together, so they are also supplementary angles.

What is the Vertical Angles Theorem?

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREE

Key words

Angle: defined as the amount of turn around a common vertex, measured most commonly in degrees or radians.

Vertex: the point created by two line segments meeting (plural is vertices).

How to label an angle

You can label angles in two main ways:

By giving the angle a ‘name’ which is normally a lower case letter such as $a, x$ or $y$ or the greek letter $\theta$ (theta).
By referring to the angle as the three letters that define the angle. The middle letter refers to the vertex at which the angle is.

Angles on a straight line equal 180°

Angles on one part of a straight line always add up to $180^{\circ}.$ See the diagram for an example where angles $a$ and $b$ are equal to $180^{\circ}.$

However, see the next diagram for an example of where $a$ and $b$ do not equal $180^{\circ}$ because they are not on one single part of a straight line, i.e. they do not share a vertex and are not adjacent to one another.

Note – you can try out the above rule by drawing out the above diagrams and measuring the angles using a protractor.

Angles around a point

Angles around a point will always equal $360^{\circ}.$ See the diagram for an example where angles $a,b$ and $c$ are sum to $360^{\circ}.$

Supplementary and complementary angles

Two angles are supplementary when they add up to $\bf{180^{\circ}},$ they do not have to be next to each other. Two angles are complementary when they add up to $\bf{90^{\circ}},$ they do not have to be next to each other.

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7 – Geometry (7.G.B.5)
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

How to use the vertical angles theorem to solve problems

In order to solve problems involving the vertical angles theorem:

Identify which angles are vertically opposite to one another.
Identify which of the unknown angles the question is asking you to find the value of.
Solve the problem and give reasons where applicable.
Clearly state the answer using angle terminology.

Vertical angles theorem examples

Example 1: two angles that are vertically opposite

Find the value of angle $x.$

Identify which angles are vertically opposite to one another.

The angle labeled $x$ and the angle with value $117^{\circ}$ are vertical angles, since their vertex is created by two straight lines intersecting.

2Identify which of the unknown angles the question is asking you to find the value of.

The angle labeled $x.$

3Solve the problem and give reasons where applicable.

$x=117$ because vertical angles are congruent.

4Clearly state the answer using angle terminology.

x=117^{\circ}

Example 2: two angles that are vertically opposite

Find the values of angles $x$ and $y.$

Identify which angles are vertically opposite to one another.

The angle labeled $x$ and the angle with value $93^{\circ}$ are vertical angles, since their vertex is created by two straight lines intersecting.

Identify which of the unknown angles the question is asking you to find the value of.

The angles labeled $x$ and $y.$

Solve the problem and give reasons where applicable.

$x=93^{\circ}$ because vertical angles are congruent.

$x$ and $y$ are not vertical angles, but they are supplementary angles because they are on a straight line at the same vertex.

\begin{aligned}& x+y=180 \\\\ & 93+y=180 \\\\ & y=87 \end{aligned}

Clearly state the answer using angle terminology.

x=93^{\circ}, y=87^{\circ}

Example 3: finding all the angles around a point with vertically opposite angles

Find the values of the angles labeled $a,b$ and $c$ .

Identify which angles are vertically opposite to one another.

The angle labeled $a$ and the angle with value $22^{\circ}$ are vertical angles.

The angle labeled $b$ and the angle labeled $c$ are vertical angles.

Identify which of the unknown angles the question is asking you to find the value of.

The angles labeled $a,b$ and $c.$

Solve the problem and give reasons where applicable.

$a=22^{\circ}$ because vertical angles are congruent.

$a$ and $b$ are not vertical angles, but they are supplementary angles, because they are on a straight line at the same vertex.

\begin{aligned}& a+b=180 \\\\ & 22+b=180 \\\\ & b=158 \end{aligned}

$c=158^{\circ}$ because vertical angles are congruent.

Clearly state the answer using angle terminology.

a=22^{\circ}, b=158^{\circ}, c=158^{\circ}

Example 4: vertically opposite angles with algebra

Use the vertical angles theorem to find the value of $x.$

Identify which angles are vertically opposite to one another.

The angle labeled $‘x+10’$ and the angle on the other side of the vertex are vertical angles.

The angle labeled $‘x+120’$ and the angle on the other side of the vertex are vertical angles.

Identify which of the unknown angles the question is asking you to find the value of.

You are not being asked to find an angle, you are being asked to find the value of $x.$

Solve the problem and give reasons where applicable.

The four angles total $360^{\circ}$ because they are all around a vertex.

Therefore:

\begin{aligned}& (x+10)+(x+120)+(x+10)+(x+120)=360 \\\\ & (x+x+x+x)+(10+120+10+120)=360 \\\\ & 4 x+260=360 \\\\ & 4 x=100 \\\\ & x=25 \end{aligned}

Clearly state the answer using angle terminology.

x=25^{\circ}

Example 5: vertical angles with more than two lines

What is the value of $b?$

Identify which angles are vertically opposite to one another.

The two lines that form the $81^{\circ}$ angle create vertical angles. So, the $81^{\circ}$ angle is congruent to angle $b$ and the $44^{\circ}$ angle.

Identify which of the unknown angles the question is asking you to find the value of.

The angle labeled $b.$

Solve the problem and give reasons where applicable.

Set the expressions equal to each other and solve for $x.$

\begin{aligned}& 81=b+44 \\\\ & 37=b \end{aligned}

Clearly state the answer using angle terminology.

x=37^{\circ}

Example 6: word problem

Two angles with values of $x+30$ and $4x -30$ are vertically opposite one another. Prove the two angles are both $50^{\circ}.$

Identify which angles are vertically opposite to one another.

The angles labeled $x+30$ and $4x-30$ vertically opposite to one another meaning they are equal to one another.

Clearly identify which of the unknown angles the question is asking you to find the value of.

You are being asked to prove the size of each angle is $50^{\circ}.$ This means show all your working to reach this conclusion.

Solve the problem and give reasons where applicable.

Set the expressions equal to each other and solve for $x.$

\begin{aligned}& x+30=4 x-30 \\\\ & x+60=4 x \\\\ & 60=3 x \\\\ & 20=x \end{aligned}

Therefore, the size of the two angles can be found by substituting $x=20$ into each angle.

Angle $1$

\begin{aligned}x+30&=20+30x \\\\ &=50^{\circ} \end{aligned}

Angle $2$

\begin{aligned}4x-30&=4(20)+30 \\\\ x&=50^{\circ} \end{aligned}

Clearly state the answer using angle terminology.

Therefore both angles are $50^{\circ}.$

Teaching tips for vertical angles theorem

Start with lesson plans that have pairs of vertical angles that do not involve variables and whose measure of angles is whole numbers. Also give students informal graphing opportunities to test vertical angle examples, before proving it with linear pairs.

Before sending students off to do a worksheet on their own, be sure that they have a few solved examples they can refer back to if they get stuck.

Easy mistakes to make

Assuming angles are vertical without proof
Angles that are formed by two straight lines crossing are congruent angles. However, you cannot assume congruence without this proof. Sometimes a set of angles may appear to be vertical angles, but this must be given information or proven.

For example,
If you are given just this image and no text or values, you should not assume they are vertical angles. The angles below are close to being vertical angles, but they are not.

Finding the incorrect angle due to misunderstanding the terminology
It’s easy to confuse the terms vertical angle, supplementary angle, complementary angle, straight angle, etc. Since these terms are often introduced and used together, it is important to always read problems carefully and consider the definition of each.

Practice questions on angle rules

x=113

x=23

x=67

x=22

The angle labeled $x$ and the angle with value $67^{\circ}$ are vertical angles since their vertex is created by two straight lines intersecting. This means the angles are equivalent.

x=146

x=56

x=34

x=214

The angle labeled $x$ and the angle with value $146^{\circ}$ are vertical angles since their vertex is created by two straight lines intersecting. This means the angles are equivalent.

x=72

x=86

x=82

x=98

The angle labeled $x$ and the angle with value $98^{\circ}$ are vertical angles since their vertex is created by two straight lines intersecting. This means the angles are equivalent.

x=68, y=68

x=112, y=112

x=112, y=68

x=68, y=112

Angle $x$ is vertically opposite the given angle $112^{\circ},$ so it is the same.

x=112

Angle $x$ and angle $y$ lie on a straight line, so they must add up to $180^{\circ}.$

\begin{aligned}& 112+y=180 \\\\ & y=68 \end{aligned}

x=50

x=25

x=100

x=75

Angle $2x$ is vertically opposite the given angle of $50^{\circ}$ so it is equivalent.

\begin{aligned}& 2 x=50 \\\\ & x=25 \end{aligned}

x=20

x=40

x=70

x=10

Angle $6x+10$ is vertically opposite angle $10x-70,$ so they are equivalent. Solving the equation, $6x+10=10x-70,$ leads to the correct value for $x.$

\begin{aligned}& 6 x+10=10 x-70 \\\\ & 10=4 x-70 \\\\ & 80=4 x \\\\ & 20=x \end{aligned}

Vertical angles theorem FAQs

What are opposite angles?

A pair of opposite angles is a pair of non-adjacent angles that have an equal measure because they are formed by two lines crossing. This term is used interchangeably with vertical angles.

Are perpendicular lines considered vertical angles?

Yes, because they are formed by two lines crossing, making their vertical angles equivalent.

What are exterior angles?

Angles on the outside of a figure formed by the side of a figure and an extended line; the opposite of interior angles.

What is a transversal?

A line that intersects two other lines at a distinct place.

What is a bisector?

A line that splits something geometrical in half.

What is a quadrilateral?

A polygon with four sides.

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