September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a important topic for everyone who wants to learn math or any other subject that utilizes it. It's hard work, but we'll assure you get a handle on these concepts so you can attain the grade!

Don’t feel disheartened if you don’t recollect or don’t have a good grasp on these theories, as this blog will help you study all the basics. Moreover, we will help you learn the tricks to learning faster and increasing your scores in mathematics and other prevailing subjects today.

The Theorem

The vertical angle theorem stipulates that when two straight lines intersect, they form opposite angles, named vertical angles.

These opposite angles share a vertex. Furthermore, the most essential thing to remember is that they are the same in measurement! This means that irrespective of where these straight lines cross, the angles converse each other will consistently share the exact 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 discover the others utilizing proportions.

Proving the Theorem

Proving this theorem is somewhat easy. First, let's pull a line and name 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 label the angles formed by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:

We are aware that angles A and B are vertically contrary 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 see the angles B and C, you will notice that they are not connected at their vertex but adjacent to each other. 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 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 eliminating C on both sides of the equation, we will end with:


So, we can say 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 talk explicitly regarding vertically opposite angles.


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

Still, 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 case of complementary angles, the addition of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized to prove 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 many characteristics of vertically opposite angles. But, chances are that you will only require these two to nail your exam.

  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 at no time adjacent. They can share, at most, a vertex.

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

You may think where you can find these concepts in the real life, and you'd be stunned to notice that vertically opposite angles are very common! You can locate them in several daily objects and scenarios.

For example, vertically opposite angles are created when two straight lines overlap each other. Back of your room, the door installed 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 wonderful 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 observe your surroundings, as you will detect an example next to you.

PuttingEverything Together

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

The vertical angle theorem defines that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line overlapping it and applying 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 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 considered supplementary.

The total of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will always equal 180°.

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

Further Study

Search for a vertically opposite angles worksheet online for examples and sums to practice. Math is not a spectator sport; keep practicing until these concepts are ingrained in your brain.

However, there is nothing humiliating if you need extra assistance. If you're struggling to understand vertical angles (or any other concept in geometry), think about enrolling for a tutoring session with Grade Potential. One of our professional tutors can assist you comprehend the topic and ace your next examination.