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# Vertical Angles

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# Vertical Angles

What if you were given two angles of unknown size and were told they are vertical angles? How would you determine their angle measures? After completing this Concept, you'll be able to use the definition of vertical angles to solve problems like this one.

### Watch This

Watch the first part of this video.

Then watch the third part of this video.

### Guidance

Vertical angles are two non-adjacent angles formed by intersecting lines. $\angle 1$ and $\angle 3$ are vertical angles and $\angle 2$ and $\angle 4$ are vertical angles.

The Vertical Angles Theorem states that if two angles are vertical angles, then they are congruent.

#### Example A

Find $m\angle 1$ .

$\angle 1$ is vertical angles with $18^\circ$ , so $m\angle 1 = 18^\circ$ .

#### Example B

If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(4x+10)^\circ$ and $m \angle DEF=(5x+2)^\circ$ , what is the measure of each angle?

Vertical angles are congruent, so set the angles equal to each other and solve for $x$ . Then go back to find the measure of each angle.

$4x+10&=5x+2\\ x&=8$

So, $m\angle ABC = m\angle DEF=(4(8)+10)^\circ =42^\circ$

#### Example C

True or false: vertical angles are always less than $90^\circ$ .

This is false, you can have vertical angles that are more than $90^\circ$ . Vertical angles are less than $180^\circ$ .

### Guided Practice

Find the value of $x$ or $y$ .

1.

2.

3.

1. Vertical angles are congruent, so set the angles equal to each other and solve for $x$ .

$x+16&=4x-5\\3x&=21\\ x&=7^\circ$

2. Vertical angles are congruent, so set the angles equal to each other and solve for $y$ .

$9y+7&=2y+98\\7y&=91\\y&=13^\circ$

3. Vertical angles are congruent, so set the angles equal to each other and solve for $y$ .

$11y-36&=63\\11y&=99\\y&=9^\circ$

### Practice

Use the diagram below for exercises 1-2. Note that $\overline{NK} \perp \overleftrightarrow{IL}$ .

1. Name one pair of vertical angles.
1. If $m\angle INJ = 63^\circ$ , find $m\angle MNL$ .

For exercise 3, determine if the statement is true or false.

1. Vertical angles have the same vertex.
1. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(9x+1)^\circ$ and $m \angle DEF=(5x+29)^\circ$ , what is the measure of each angle?
2. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(8x+2)^\circ$ and $m \angle DEF=(2x+32)^\circ$ , what is the measure of each angle?
3. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(x+22)^\circ$ and $m \angle DEF=(5x+2)^\circ$ , what is the measure of each angle?
4. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(3x+12)^\circ$ and $m \angle DEF=(7x)^\circ$ , what is the measure of each angle?
5. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(5x+2)^\circ$ and $m \angle DEF=(x+26)^\circ$ , what is the measure of each angle?
6. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(3x+1)^\circ$ and $m \angle DEF=(2x+2)^\circ$ , what is the measure of each angle?
7. If $\angle ABC$ and $\angle DEF$ are vertical angles and $m\angle ABC =(6x-3)^\circ$ and $m \angle DEF=(5x+1)^\circ$ , what is the measure of each angle?