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Vertical Angles

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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

CK-12 Vertical Angles

Watch the first part of this video.

James Sousa: Angle Relationships

Then watch the third part of this video.

James Sousa: Complementary Angles

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 .

CK-12 Linear Pairs

Guided Practice

Find the value of x or y .

1.

2.

3.

Answers:

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?

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