September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a important topic for everyone who wishes to learn arithmetic or another subject that utilizes it. It's hard work, but we'll ensure you get a handle on these concepts so you can make the grade!

Don’t feel disheartened if you don’t recall or don’t comprehend these theories, as this blog will help you study all the basics. Additionally, we will teach you the tricks to learning faster and increasing your scores in arithmetic and other popular subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines intersect, they make opposite angles, named vertical angles.

These opposite angles share a vertex. Furthermore, the most important thing to bear in mind is that they also measure the same! This refers that regardless of where these straight lines cross, the angles converse each other will always share the same value. These angles are known as congruent angles.

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

Proving the Theorem

Proving this theorem is moderately straightforward. Primarily, let's pull a line and label it line l. Then, we will pull another line that intersects line l at some point. We will call this second line m.

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

We understand that angles A and B are vertically opposite reason being that 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 you see the angles B and C, you will note that they are not connected at their vertex but close to each other. They share a side and a vertex, meaning they are supplementary angles, so the total 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 manner:

∠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 end with:


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

Vertically Opposite Angles

Now that we know the theorem and how to prove it, let's discuss explicitly regarding vertically opposite angles.


As we stated, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite one another satisfy the vertical angle theorem.

Still, vertically opposite angles are never adjacent. Adjacent angles are two angles that have a common side and a common vertex. Vertically opposite angles never 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 total 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.

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

There are many properties of vertically opposite angles. Still, chances are that you will only require these two to ace your examination.

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

  2. Vertically opposite angles are at no time adjacent. They can share, at most, a vertex.

Where Can You Find Opposite Angles in Real-World Situations?

You may speculate where you can utilize these concepts in the real world, and you'd be surprised to observe that vertically opposite angles are quite common! You can locate them in several daily things and situations.

For instance, vertically opposite angles are formed when two straight lines overlap each other. Right in your room, the door attached to the door frame makes vertically opposite angles with the wall.

Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Track crossings are also a great example of vertically opposite angles.

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

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

PuttingEverything Together

So, to sum up what we have talked about, vertically opposite angles are created from two intersecting lines. The two angles that are not adjacent have identical measurements.

The vertical angle theorem states that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line intersecting it and implementing the theorems of congruent angles to finish measures.

Congruent angles refer to two angles that have identical measurements.

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

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

Vertically opposite angles are quite common! You can find them in several daily objects and scenarios, such as doors, windows, paintings, and trees.

Additional Study

Search for a vertically opposite angles questionnaire online for examples and exercises to practice. Mathematics is not a spectator sport; keep practicing until these concepts are ingrained in your mind.

However, there is no problem if you need extra support. If you're having a hard time to comprehend vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our professional instructor can guide you understand the topic and nail your next test.