September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is an essential subject for anyone who wants to study mathematics or any related subject that uses it. It's hard work, but we'll ensure you get a handle on these theories so you can attain the grade!

Don’t feel discouraged if you don’t remember or don’t comprehend these theories, as this blog will help you study all the fundamentals. Furthermore, we will help you learn the secret to learning quicker and increasing your scores in mathematics and other prevailing subjects today.

The Theorem

The vertical angle theorem expresses that at any time two straight lines bisect, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Additionally, the most essential thing to remember is that they also measure the same! This applies that regardless of where these straight lines cross, the angles opposite each other will always share the same value. These angles are referred as congruent angles.

Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to discover the others using proportions.

Proving the Theorem

Proving this theorem is relatively easy. Primarily, let's pull a line and label it line l. After that, we will pull another line that goes through line l at some point. We will assume this second line m.

After drawing these two lines, we will label the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically opposite because they share the equivalent vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A equals angle B.

If we look at angles B and C, you will notice that they are not joined at their vertex but close to one another. They share a side and a vertex, therefore they are supplementary angles, so the sum of both angles will be 180 degrees. This instance repeats itself with angles A and C so that we can summarize this in the following way:

∠B+∠C=180 and ∠A+∠C=180

Since both additions equal the same, we can sum up these operations as follows:


By removing C on both sides of the equation, we will be left with:


So, we can conclude that vertically opposite angles are congruent, as they have the same measurement.

Vertically Opposite Angles

Now that we have learned about the theorem and how to prove it, let's talk particularly about vertically opposite angles.


As we mentioned, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.

However, vertically opposite angles are no way next to each other. Adjacent angles are two angles that have a common side and a common vertex. Vertically opposite angles at no time share a side. When angles share a side, these adjacent angles could be complementary or supplementary.

In case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized in our proof of the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not satisfy the properties of vertically opposite angles.

There are various characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to nail your test.

  1. Vertically opposite angles are at all time congruent. Hence, if angles A and B are vertically opposite, they will measure the same.

  2. Vertically opposite angles are never adjacent. They can share, at most, a vertex.

Where Can You Locate Opposite Angles in Real-World Circumstances?

You might wonder where you can use these theorems in the real life, and you'd be surprised to notice that vertically opposite angles are very common! You can locate them in many everyday things and situations.

For instance, vertically opposite angles are created when two straight lines overlap each other. Inside your room, the door installed to the door frame makes vertically opposite angles with the wall.

Open a pair of scissors to create two intersecting lines and adjust the size of the angles. Road intersections are also a wonderful example of vertically opposite angles.

In the end, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.

Be sure to watch your environment, as you will detect an example next to you.

Puttingit All Together

So, to summarize what we have talked about, vertically opposite angles are created from two intersecting lines. The two angles that are not next to each other have identical measurements.

The vertical angle theorem defines that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line overlapping it and applying the theorems of congruent angles to finish measures.

Congruent angles refer to two angles that measure the same.

When two angles share a side and a vertex, they can’t be vertically opposite. However, they are complementary if the sum of these angles equals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.

The total of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will at all time add up to 180°.

Vertically opposite angles are pretty common! You can locate them in several everyday objects and situations, such as doors, windows, paintings, and trees.

Further Study

Search for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a onlooker sport; keep practicing until these concepts are well-established in your head.

However, there is no problem if you require further support. If you're having a hard time to grasp vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you comprehend the topic and ace your following test.