September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is an essential subject for anyone who desires to study mathematics or any related subject that employs it. It's tough work, but we'll assure you get a good grasp of these concepts so you can achieve the grade!

Don’t feel discouraged if you don’t recollect or don’t have a good grasp on these concepts, as this blog will help you understand all the basics. Moreover, we will teach you the secret to learning quicker and increasing your scores in arithmetic and other common subjects today.

The Theorem

The vertical angle theorem states that when two straight lines intersect, they make opposite angles, known as vertical angles.

These opposite angles share a vertex. Furthermore, the most crucial point to keep in mind is that they also measure the same! This applies that regardless of where these straight lines cross, the angles opposite each other will constantly 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 possible to find the others using proportions.

Proving the Theorem

Proving this theorem is relatively simple. Primarily, let's pull a line and call 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 formed 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 know that angles A and B are vertically contrary reason being that they share the equivalent vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A is the same as angle B.

If we look at angles B and C, you will note that they are not joined at their vertex but close to each other. They have in common a side and a vertex, therefore 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 manner:

∠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 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 talk particularly regarding vertically opposite angles.


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

Despite that, vertically opposite angles are at no time 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 the case of complementary angles, the sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose addition 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 due to this reason supplementary and complementary angles do not fulfill the properties of vertically opposite angles.

There are several properties of vertically opposite angles. But, odds are that you will only need these two to nail your exam.

  1. Vertically opposite angles are always 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 Use Opposite Angles in Real-Life Scenario?

You might wonder where you can find these theorems in the real life, and you'd be surprised to observe that vertically opposite angles are quite common! You can find them in several daily objects and circumstances.

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

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

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

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

PuttingEverything Together

So, to summarize what we have discussed, vertically opposite angles are formed from two intersecting lines. The two angles that are not adjacent have identical measurements.

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

Congruent angles means two angles that measure the same.

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

The sum of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will at all time add up to 180°.

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

Additional Study

Look for a vertically opposite angles questionnaire online for examples and problems to practice. Math is not a spectator sport; keep applying until these concepts are ingrained in your brain.

Despite that, there is nothing humiliating if you require additional support. If you're having a hard time to understand vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our expert instructor can guide you understand the topic and nail your following test.