What are Vertical Angles?
Vertical angles are
Examples
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Vertical Angles are Congruent
When Will I Use Vertical Angles?
The Vertical Angles Theorem says that vertical angles are congruent. Congruent angles always have equal measurements, so the Vertical Angles Theorem also implies that vertical angles have equal measurements.
Vertical Angles Theorem
“Vertical angles are congruent”
“Vertical angles are equal in measurement”
If you don’t know what vertical angles are, I recommend reading this page first.
How to Use the Vertical Angles Theorem
- Make sure the angles are vertical angles.
- Are the angles formed by the intersection of two lines?
- Are the angles on opposite sides of the intersection?
- Use the Vertical Angles Theorem to write an equation.
- Identify what the problem is asking you to find and use the equation to find it.
Examples
Find the measurement of the missing angle.
Are the highlighted angles in the diagram vertical angles?
Yes, the angles are formed by the intersection of two lines and they are on opposite sides of the intersection.
How does the Vertical Angles Theorem apply to this problem?
The Vertical Angles Theorem says that the measurements of the two angles \(\green x^\circ\) and \(\green 30^\circ\) are equal to each other.
\[{\green x}={\green 30}\]
What is the problem asking for?
The problem asked me to find the measurement of the missing angle.
The equation I wrote in Step 2 tells me that \(\green x\) is equal to \(\green 30\).
\(\green x\) is also the measurement of the missing angle, so the missing angle is \(\green 30^\circ\).
The measurement of the missing angle is \(\green 30^\circ\).
Find the value of x.
Are the highlighted angles in the diagram vertical angles?
Yes, the angles are formed by the intersection of two lines and they are on opposite sides of the intersection.
How does the Vertical Angles Theorem apply to this problem?
The Vertical Angles Theorem says that the measurements of the two angles \(\purple (4x+3)^\circ\) and \(\purple 75^\circ\) are equal to each other.
\[{\purple 4x+3}={\purple 75}\]
What is the problem asking for?
The problem asked me to find the value of x.
I can find x by solving the equation I wrote in Step 2.
\[{\purple 4x+3}={\purple 75}\]
First, I will subtract 3 from both sides of the equation.
\[{\purple 4x}={\purple 72}\]
Then, I will divide both sides of the equation by 4.
\[{\purple x}={\purple 18}\]
The value of x is \(\purple 18\).
Find the measurements of the blue angles.
Are the highlighted angles in the diagram vertical angles?
Yes, the angles are formed by the intersection of two lines and they are on opposite sides of the intersection.
How does the Vertical Angles Theorem apply to this problem?
The Vertical Angles Theorem says that the measurements of the two angles \(\blue (8x-40)^\circ\) and \(\blue (3x+60)^\circ\) are equal to each other.
\[{\blue 8x-40}={\blue 3x+60}\]
What is the problem asking for?
The problem asked me to find the measurements of the blue angles.
Both of the angle measurements depend on x, so I have to know what x is before I can find the angle measurements.
I can use the equation I wrote in Step 2 to solve for x and then use that value to find the angle measurements.
Solving for x
\[{\blue 8x-40}={\blue 3x+60}\]
First, I will subtract 3x from both sides of the equation.
\[{\blue 5x-40}={\blue 60}\]
Next, I will add 40 to both sides of the equation.
\[{\blue 5x}={\blue 100}\]
Then I will divide both sides of the equation by 5.
\[{\blue x}={\blue 20}\]
Finding the Angle Measurements
The value of x is 20, but the question didn’t ask me to find x. It asked me to find the measurement of the blue angles.
I can find the measurement of the top angle by plugging \(\blue x=20\) into \(\blue 8x-40\).
\[\blue 8(20)-40\]
\[\blue 160-40\]
\[\blue 120\]
The measurement of the top angle is \(\blue 120^\circ\).
According to the Vertical Angles Theorem, the measurement of the bottom angle should also be \(\blue 120^\circ\) because the top and bottom angles are congruent.
I will still plug \(\blue x=20\) into \(\blue 3x+60\) just to double check my math.
\[\blue 3(20)+60\]
\[\blue 60+60\]
\[\blue 120\]
As expected, the measurement of the bottom angle is also \(\blue 120^\circ\).
The measurements of the blue angles are both \(\blue 120^\circ\).
Vertical Angles Theorem Proof
Theorem to Prove
Vertical angles are congruent.
Definitions
\(\green a\) = measurement of the green angle
\(\red b\) = measurement of the red angle
\(\yellow c\) = measurement of the yellow angle
\(\purple d\) = measurement of the purple angle
Axiom
The measurements of adjacent angles on straight lines always add up to \(180^\circ\).
In the diagram, there are four angles (green, yellow, red, and purple). The measurements of these angles were defined as \(\green a\), \(\red b\), \(\yellow c\), and \(\purple d\).
There are two pairs of vertical angles:
- green/yellow
- red/purple
To prove the Vertical Angles Theorem, I need to prove that both of these pairs of vertical angles are congruent.
Angles with the same measurement are congruent, so if I can prove that \({\green a}={\yellow c}\), that will prove that the green/yellow vertical angles are congruent. If I can prove that \({\red b}={\purple d}\) then that will prove that the red\purple vertical angles are congruent.
Proof Structure
- Use the axiom to write equations that show the relationships between \(\green a\), \(\red b\), \(\yellow c\), and \(\purple d\).
- Use the equations to prove that \({\green a}={\yellow c}\).
- Use the equations to prove that \({\red b}={\purple d}\).
The measurements of adjacent angles on a straight line always add up to \(180^\circ\).
So, that means I can write the following equations:
\[{\green a}+{\red b}=180^\circ\]
\[{\red b}+{\yellow c}=180^\circ\]
\[{\yellow c}+{\purple d}=180^\circ\]
\[{\purple d}+{\green a}=180^\circ\]
I will use the first two equations to prove that the green and yellow angles are congruent.
\[{\green a}+{\red b}=180^\circ\]
\[{\red b}+{\yellow c}=180^\circ\]
The right hand side of both equations is \(180^\circ\). If \({\green a}+{\red b}\) and \({\red b}+{\yellow c}\) both equal \(180^\circ\), then that means they also equal each other. So, I can write this equation…
\[{\green a}+{\red b}={\red b}+{\yellow c}\]
I can simplify the equation by subtracting \(\red b\) from both sides of the equation.
\[{\green a}={\yellow c}\]
The measurements of the green and yellow angles are equal, therefore the vertical angles are congruent.
I will use the middle two equations to prove that the red and purple angles are congruent.
\[{\red b}+{\yellow c}=180^\circ\]
\[{\yellow c}+{\purple d}=180^\circ\]
The right hand side of both equations is \(180^\circ\). If \({\red b}+{\yellow c}\) and \({\yellow c}+{\purple d}\) both equal \(180^\circ\), then that means they also equal each other. So, I can write this equation…
\[{\red b}+{\yellow c}={\yellow c}+{\purple d}\]
I can simplify the equation by subtracting \(\yellow c\) from both sides of the equation.
\[{\red b}={\purple d}\]
The measurements of the red and purple angles are equal, therefore the vertical angles are congruent.
Vertical angles are congruent because \({\green a}={\yellow c}\) and \({\red b}={\purple d}\) as proved above.
