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.

two lines intersect and form angles 1, 2, 3, and 4. the pairs of vertical angles are angles 1 and 3, and angles 2 and 4.

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 \angle\textbf{5}.

two intersecting lines forming four angles, namely, angles 3, 4, 5, and 6 with angle 5 highlighted.

Remember that vertical angles are angles that are across from each other. In this example, the angle opposite of \angle{5} is \angle{3}.

angle 3 and angle 5 are vertical angles.

Therefore, we can say that \angle\textbf{3} is vertical to \angle\textbf{5}.


Example 2: Which pairs are vertical angles?

angles 1, 2, 3, and 4 are formed by two lines crossing each other.

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.

angle 1 and angle 3 are vertically opposite and so are angle 2 and angle 4.

So here, our two pairs of vertical angles are \angle \textbf{1} and \angle\textbf{3} as well as \angle\textbf{2} and \angle\textbf{4}.


Example 3: Which angle has the same angle measure as \angle\textbf{4}?

angle 4 highlighted among the four angles formed by the two lines intersecting each other.

We know that vertical angles are always congruent. Since \angle{4} and \angle{2} are vertical angles, then both angles should have equal angle measures.

Hence, \angle\textbf{4}\cong\angle\textbf{2}.


Example 4: What is the measure of \angle\textbf{AOD}?

diagram showing angle AOD, angle COD, angle BOC, and angle BOA with angle BOC having an angle measure of 68 degrees.

In the diagram above, the measure for \angle COB is given. This gives us a clue that the angle vertically opposite to \angle COB will also have an angle measure of 68^\circ .

Luckily, \angle COB and \angle AOD are vertical angles. So they should have the same angle measure.

since angle BOC and angle AOD are vertical angles, they both have an angle measure of 68 degrees.

Thus, the angle measure of \angle AOD is \textbf{68}^\circ .


Example 5: Are \angle\textbf{SYR} and \angle\textbf{ZYR} vertical angles? Yes or No.

diagram showing four angles, namely, angles SYR, ZYR, SYX and ZYX.

Looking at the diagram, we can easily tell that \angle{SYR} is not vertically opposite to \angle{ZYR}. In fact, the angle vertically opposite to \angle{SYR} is actually \angle{ZYX}.

angle SYR and angle ZYX are vertical angles.

Therefore, the answer is No. \angle\textbf{SYR} and \angle\textbf{ZYR} are not vertical angles.


Example 6: What is the value of x?

a pair of vertical angles with angle measure of 112 degrees and 3x plus 1

In this example, we are given the measures of two angles that are opposite each other. The first angle is 112^\circ while the other is equal to \left( {3x + 1} \right)^\circ. Note that since these two angles are vertical angles, they are also congruent. In other words, since one of the angles is 112^\circ then the algebraic expression, 3x + 1, should also equal to 112.

Let’s proceed to set up our equation and solve for the variable x.

3x plus 1 is equal to 112 and x is equal to 37.

Now that we know the value of x, let’s verify if both sides of the equation will equal each other once we plug in this value.

3 times 37 plus 1 is equal to 112.

And indeed, it does! So we can now safely say that the value of x is \textbf{37}.


Example 7: What is the measure of \angle GOH?

Line EH intersects line GF at O

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 \angle GOH should also be the measure of \angle EOF.

Since they are congruent, we’ll set both algebraic expressions equal to one another and solve for the unknown variable, x.

11x plus 6 equal to 5x plus 36; x is equal to 5

We now have the value of x which is 5. 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.

11 times 5 plus 6 is equal to 61 and so is 5 times 6 plus 36 is equal to 61.

Perfect! Both sides equal each other which should be since they are vertical angles. More importantly, we now have the value of 11x + 6 which is 61 as well as 5x + 36 which is also 61. Thus, \angle GOH \cong \angle EOF.

To answer our original question, \angle GOH has an angle measure of \textbf{61}^\circ.