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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. In the picture below, \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.

Notice that these angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.

Investigation: Vertical Angle Relationships

  1. Draw two intersecting lines on your paper. Label the four angles created \begin{align*}\angle 1, \ \angle 2, \ \angle 3,\end{align*} and \begin{align*}\angle 4\end{align*}. See the picture above.
  2. Take your protractor and find \begin{align*}m \angle 1\end{align*}.
  3. What is the angle relationship between \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*}? Find \begin{align*}m \angle 2\end{align*}.
  4. What is the angle relationship between \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*}? Find \begin{align*}m \angle 4\end{align*}.
  5. What is the angle relationship between \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*}? Find \begin{align*}m \angle 3\end{align*}.
  6. Are any angles congruent? If so, write down the congruence statement.

From this investigation, hopefully you found out that \begin{align*}\angle 1 \cong \angle 3\end{align*} and \begin{align*}\angle 2 \cong \angle 4\end{align*}. This is our first theorem. That means it must be proven true in order to use it.

Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.

We can prove the Vertical Angles Theorem using the same process we used above. However, let’s not use any specific values for the angles.

\begin{align*}&\text{From the picture above:}\\ &\angle 1 \ \text{and} \ \angle 2 \ \text{are a linear pair} && m \angle 1 + m \angle 2 = 180^\circ\\ &\angle 2 \ \text{and} \ \angle 3 \ \text{are a linear pair} && m \angle 2 + m \angle 3 = 180^\circ\\ &\angle 3 \ \text{and} \ \angle 4 \ \text{are a linear pair} && m \angle 3 + m \angle 4 = 180^\circ\\ \\ &\text{All of the equations} = 180^\circ, \ \text{so set the} \ && m \angle 1 + m \angle 2 = m \angle 2 + m \angle 3\\ &\text{first and second equation equal to} && AND\\ &\text{each other and the second and third.} && m \angle 2 + m \angle 3 = m \angle 3 + m \angle 4\\ \\ &\text{Cancel out the like terms} && m \angle 1 = m \angle 3, \ m \angle 2 = m \angle 4\end{align*}

Recall that anytime the measures of two angles are equal, the angles are also congruent.

 

 

Measuring Vertical Angles 

Find \begin{align*}m \angle 1\end{align*} and \begin{align*}m \angle 2\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*}. \begin{align*}\angle 2\end{align*} is a linear pair with \begin{align*}\angle 1\end{align*} or \begin{align*}18^\circ\end{align*}, so \begin{align*}18^\circ + m \angle 2 = 180^\circ. \ m \angle 2 = 180^\circ - 18^\circ = 162^\circ\end{align*}.

Identifying Vertical Angles 

Name one pair of vertical angles in the diagram below.

One example is \begin{align*} \angle INJ\end{align*} and \begin{align*} \angle MNL\end{align*}.

Example C

If \begin{align*}\angle ABC\end{align*} and \begin{align*} \angle DBF\end{align*} are vertical angles and \begin{align*}m\angle ABC =(4x+10)^\circ\end{align*} and \begin{align*}m \angle DBF=(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 DBF=(4(8)+10)^\circ =42^\circ\end{align*}

 

 

Examples

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

Example 1

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

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

 

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

\begin{align*}11y-36&=63\\11y&=99\\y&=9^\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 = 53^\circ\end{align*}, find \begin{align*}m\angle MNL\end{align*}.

For exercises 3-5, determine if the statement is true or false.

  1. Vertical angles have the same vertex.
  2. Vertical angles are supplementary.
  3. Intersecting lines form two pairs of vertical angles.

Solve for the variables \begin{align*}x\end{align*} and \begin{align*}y\end{align*} for exercises 6-10.

  1. If \begin{align*}\angle ABC\end{align*} and \begin{align*} \angle DBF\end{align*} are vertical angles and \begin{align*}m\angle ABC =(4x+1)^\circ\end{align*} and \begin{align*}m \angle DBF=(3x+29)^\circ\end{align*}, what is the measure of each angle?
  2. If \begin{align*}\angle ABC\end{align*} and \begin{align*} \angle DBF\end{align*} are vertical angles and \begin{align*}m\angle ABC =(5x+2)^\circ\end{align*} and \begin{align*}m \angle DBF=(6x-32)^\circ\end{align*}, what is the measure of each angle?
  3. If \begin{align*}\angle ABC\end{align*} and \begin{align*} \angle DBF\end{align*} are vertical angles and \begin{align*}m\angle ABC =(x+20)^\circ\end{align*} and \begin{align*}m \angle DBF=(4x+2)^\circ\end{align*}, what is the measure of each angle?
  4. If \begin{align*}\angle ABC\end{align*} and \begin{align*} \angle DBF\end{align*} are vertical angles and \begin{align*}m\angle ABC =(2x+10)^\circ\end{align*} and \begin{align*}m \angle DBF=(4x)^\circ\end{align*}, what is the measure of each angle?

Review (Answers)

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

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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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