September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a crucial topic for everyone who desires to learn arithmetic or any other subject that utilizes it. It's hard work, but we'll assure you get a good grasp of these theories so you can achieve the grade!

Don’t feel disheartened if you don’t remember or don’t understand these theories, as this blog will help you understand all the essentials. Furthermore, we will help you understand the secret to learning faster and increasing your grades in arithmetic and other popular subjects today.

The Theorem

The vertical angle theorem stipulates that whenever two straight lines bisect, they form opposite angles, called vertical angles.

These opposite angles share a vertex. Moreover, the most essential thing to keep in mind is that they also measure the same! This refers that irrespective of where these straight lines cross, the angles opposite each other will consistently 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 possible to discover the others using proportions.

Proving the Theorem

Proving this theorem is moderately easy. Primarily, let's pull a line and label it line l. After that, we will draw another line that intersects line l at some point. We will name 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 labeled pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:

We are aware that angles A and B are vertically opposite due to the fact they share the same vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A is identical angle B.

If you observe angles B and C, you will notice that they are not linked at their vertex but next to one another. They share a side and a vertex, meaning they are supplementary angles, so the sum of both angles will be 180 degrees. This situation 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 canceling out 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 studied about the theorem and how to prove it, let's discuss explicitly about vertically opposite angles.


As we said earlier, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.

Despite that, 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 total 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used in our proof of the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.

There are many characteristics of vertically opposite angles. Still, odds are that you will only require these two to secure your examination.

  1. Vertically opposite angles are always congruent. Therefore, 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 Use Opposite Angles in Real-Life Situations?

You may think where you can utilize these theorems in the real life, and you'd be amazed to observe that vertically opposite angles are very common! You can discover them in various daily objects and scenarios.

For instance, vertically opposite angles are formed when two straight lines cross. 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 alter the size of the angles. Road intersections are also a terrific example of vertically opposite angles.

Eventually, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.

Be sure to notice your surroundings, as you will discover an example next to you.

PuttingEverything Together

So, to summarize what we have considered so far, vertically opposite angles are created from two overlapping lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem explains that whenever two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and applying the concepts of congruent angles to finish measures.

Congruent angles means two angles that have identical measurements.

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 totals 90°. If the addition of both angles equals 180°, they are considered supplementary.

The sum of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will always equal 180°.

Vertically opposite angles are very common! You can find them in many everyday objects and scenarios, such as windows, doors, paintings, and trees.

Additional Study

Search for a vertically opposite angles practice questions on the internet for examples and problems to practice. Math is not a spectator sport; keep practicing until these theorems are ingrained in your mind.

Still, there is nothing humiliating if you require additional support. If you're struggling to understand vertical angles (or any other ideas of geometry), think about signing up for a tutoring session with Grade Potential. One of our professional teachers can guide you grasp the topic and ace your following examination.