September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is an essential subject for anyone who wants to study arithmetic or any related subject that employs it. It's tough work, but we'll ensure you get a grip of these concepts so you can attain the grade!

Don’t feel disheartened if you don’t recollect or don’t comprehend 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 math and other common subjects today.

The Theorem

The vertical angle theorem states that whenever 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 are the same in measurement! This means that regardless of where these straight lines cross, the angles converse each other will consistently share the same value. These angles are called congruent angles.

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

Proving the Theorem

Proving this theorem is somewhat simple. First, let's pull a line and label it line l. After that, we will draw another line that goes through line l at some point. We will name this second line m.

After drawing these two lines, we will name the angles created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Accordingly, 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 same 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 observe angles B and C, you will notice that they are not linked at their vertex but adjacent 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 scenario 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 sums up to 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 identical measurement.

Vertically Opposite Angles

Now that we have studied 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 satisfy the vertical angle theorem.

Despite that, vertically opposite angles are never 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 equal 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized to prove the vertical angle theorem.

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

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

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

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

For example, vertically opposite angles are made when two straight lines cross. Inside your room, the door connected to the door frame makes vertically opposite angles with the wall.

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

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

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

PuttingEverything Together

So, to sum up what we have talked about, vertically opposite angles are made from two overlapping lines. The two angles that are not next to each other have identical measurements.

The vertical angle theorem defines that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line intersecting it and applying the theorems of congruent angles to complete measures.

Congruent angles means two angles that measure the same.

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

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

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

Additional Study

Look for a vertically opposite angles worksheet online for examples and sums to practice. Mathematics is not a onlooker sport; keep applying until these theorems are rooted in your brain.

However, there is nothing humiliating if you need additional help. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our expert instructor can help you comprehend the material and ace your next test.