Vertical Angles
When two lines intersect, they naturally form two pairs of vertical angles. Vertical angles share the same vertex or corner, and are opposite each other. These pair of angles are congruent which means they have the same angle measure.
Example Problems Involving Vertical Angles
Let’s get familiar with the characteristics of vertical angles by delving into a few examples.
Example 1: Name the angle vertical to [latex]\angle\textbf{5}[/latex].
Remember that vertical angles are angles that are across from each other. In this example, the angle opposite of [latex]\angle{5}[/latex] is [latex]\angle{3}[/latex].
Therefore, we can say that [latex]\angle\textbf{3}[/latex] is vertical to [latex]\angle\textbf{5}[/latex].
Example 2: Which pairs are vertical angles?
We see that four angles were formed by two intersecting lines. However, which pairs are vertical angles? Keep in mind that vertical angles are a pair of opposite angles that also share the same vertex.
So here, our two pairs of vertical angles are [latex]\angle \textbf{1}[/latex] and [latex]\angle\textbf{3}[/latex] as well as [latex]\angle\textbf{2}[/latex] and [latex]\angle\textbf{4}[/latex].
Example 3: Which angle has the same angle measure as [latex]\angle\textbf{4}[/latex]?
We know that vertical angles are always congruent. Since [latex]\angle{4}[/latex] and [latex]\angle{2}[/latex] are vertical angles, then both angles should have equal angle measures.
Hence, [latex]\angle\textbf{4}\cong\angle\textbf{2}[/latex].
Example 4: What is the measure of [latex]\angle\textbf{AOD}[/latex]?
In the diagram above, the measure for [latex]\angle COB[/latex] is given. This gives us a clue that the angle vertically opposite to [latex]\angle COB[/latex] will also have an angle measure of [latex]68^\circ [/latex].
Luckily, [latex]\angle COB[/latex] and [latex]\angle AOD[/latex] are vertical angles. So they should have the same angle measure.
Thus, the angle measure of [latex]\angle AOD[/latex] is [latex]\textbf{68}^\circ [/latex].
Example 5: Are [latex]\angle\textbf{SYR}[/latex] and [latex]\angle\textbf{ZYR}[/latex] vertical angles? Yes or No.
Looking at the diagram, we can easily tell that [latex]\angle{SYR}[/latex] is not vertically opposite to [latex]\angle{ZYR}[/latex]. In fact, the angle vertically opposite to [latex]\angle{SYR}[/latex] is actually [latex]\angle{ZYX}[/latex].
Therefore, the answer is No. [latex]\angle\textbf{SYR}[/latex] and [latex]\angle\textbf{ZYR}[/latex] are not vertical angles.
Example 6: What is the value of [latex]x[/latex]?
In this example, we are given the measures of two angles that are opposite each other. The first angle is [latex]112^\circ [/latex] while the other is equal to [latex]\left( {3x + 1} \right)^\circ[/latex]. Note that since these two angles are vertical angles, they are also congruent. In other words, since one of the angles is [latex]112^\circ [/latex] then the algebraic expression, [latex]3x + 1[/latex], should also equal to [latex]112[/latex].
Let’s proceed to set up our equation and solve for the variable [latex]x[/latex].
Now that we know the value of [latex]x[/latex], let’s verify if both sides of the equation will equal each other once we plug in this value.
And indeed, it does! So we can now safely say that the value of [latex]x[/latex] is [latex]\textbf{37}[/latex].
Example 7: What is the measure of [latex]\angle GOH[/latex]?
This next example again contains an algebraic expression. But this time, we are given not one but two angle measures that are expressed in algebraic expressions, given in degrees.
This may look challenging but it is actually not. As you can see, the angles are opposite each other. Therefore, early on, we can establish that they are vertical angles. Remember that vertical angles have the same angle measure on their mirrored side. So in this case, the measure of [latex]\angle GOH[/latex] should also be the measure of [latex]\angle EOF[/latex].
Since they are congruent, we’ll set both algebraic expressions equal to one another and solve for the unknown variable, [latex]x[/latex].
We now have the value of [latex]x[/latex] which is [latex]5[/latex]. Let’s plug it into either side of the equation to find the measure of angle GOH. Since both sides must have the same value, we can evaluate them simultaneously for completeness.
Perfect! Both sides equal each other which should be since they are vertical angles. More importantly, we now have the value of [latex]11x + 6[/latex] which is [latex]61[/latex] as well as [latex]5x + 36[/latex] which is also [latex]61[/latex]. Thus, [latex]\angle GOH \cong \angle EOF[/latex].
To answer our original question, [latex]\angle GOH[/latex] has an angle measure of [latex]\textbf{61}^\circ[/latex].
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