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Vertical Angles

Two congruent, non-adjacent angles formed by intersecting lines.

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Vertical Angles

Vertical Angles

Vertical angles are two non-adjacent angles formed by intersecting lines. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 3\end{align*} are vertical angles and \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 4\end{align*} are vertical angles.

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

What if you were given two angles of unknown size and were told they are vertical angles? How would you determine their angle measures?

 

Examples

Example 1

Find the value of \begin{align*}x\end{align*} .

Vertical angles are congruent, so set the angles equal to each other and solve for \begin{align*}x\end{align*}.

\begin{align*}x+16&=4x-5\\3x&=21\\ x&=7^\circ\end{align*}

Example 2

Find the value of \begin{align*}y\end{align*} .

 

Vertical angles are congruent, so set the angles equal to each other and solve for \begin{align*}y\end{align*}.

\begin{align*}9y+7&=2y+98\\7y&=91\\y&=13^\circ\end{align*}

Example 3

Find \begin{align*}m\angle 1\end{align*}.

\begin{align*}\angle 1\end{align*} is vertical angles with \begin{align*}18^\circ\end{align*}, so \begin{align*}m\angle 1 = 18^\circ\end{align*}.

Example 4

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

Vertical angles are congruent, so set the angles equal to each other and solve for \begin{align*}x\end{align*}. Then go back to find the measure of each angle.

\begin{align*}4x+10&=5x+2\\ x&=8\end{align*}

So, \begin{align*}m\angle ABC = m\angle DEF=(4(8)+10)^\circ =42^\circ\end{align*}

Example 5

True or false: vertical angles are always less than \begin{align*}90^\circ\end{align*}.

This is false, you can have vertical angles that are more than \begin{align*}90^\circ\end{align*}. Vertical angles are less than \begin{align*}180^\circ\end{align*}.

Review

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

  1. Name one pair of vertical angles.
  1. If \begin{align*}m\angle INJ = 63^\circ\end{align*}, find \begin{align*}m\angle MNL\end{align*}.

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

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.10. 

Resources

 

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Vocabulary

Vertical Angles

Vertical angles are a pair of opposite angles created by intersecting lines.

Vertical Angles Theorem

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

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