September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is an essential topic for everyone who desires to master mathematics or any other subject that employs it. It's hard work, but we'll assure you get a grip of these concepts so you can achieve the grade!

Don’t feel dispirited if you don’t recall or don’t understand these theories, as this blog will teach you all the fundamentals. Furthermore, we will teach you the secret to learning faster and increasing your scores in arithmetic and other popular subjects today.

The Theorem

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

These opposite angles share a vertex. Moreover, the most essential thing to bear in mind is that they also measure the same! This applies that irrespective 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 work out the others employing proportions.

Proving the Theorem

Proving this theorem is moderately simple. First, let's draw a line and name it line l. After that, we will draw another line that intersects line l at some point. We will call this second line m.

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

We understand that angles A and B are vertically contrary due to the fact 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 is identical angle B.

If you observe angles B and C, you will note that they are not linked at their vertex but close to each other. They share a side and a vertex, signifying 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 way:

∠B+∠C=180 and ∠A+∠C=180

Since both sums up to equal the same, we can add these operations as follows:


By removing C on both sides of the equation, we will be left with:


So, we can say that vertically opposite angles are congruent, as they have identical measure.

Vertically Opposite Angles

Now that we have studied about the theorem and how to prove it, let's talk explicitly about vertically opposite angles.


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

Still, vertically opposite angles are never next to each other. Adjacent angles are two angles that share 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 addition will equal 180°, which we just utilized to prove the vertical angle theorem.

These concepts are applicable within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not fulfill the properties of vertically opposite angles.

There are many properties of vertically opposite angles. Regardless, odds are that you will only require these two to ace 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 at no time adjacent. They can share, at most, a vertex.

Where Can You Use Opposite Angles in Real-World Circumstances?

You may wonder where you can find these concepts in the real world, and you'd be amazed to observe that vertically opposite angles are very common! You can discover them in many daily objects and scenarios.

For example, vertically opposite angles are formed when two straight lines overlap each other. Back of your room, the door connected to the door frame produces vertically opposite angles with the wall.

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

In the end, vertically opposite angles are also present in nature. If you watch a tree, the vertically opposite angles are made 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 considered so far, vertically opposite angles are created from two intersecting lines. The two angles that are not adjacent have the same measure.

The vertical angle theorem defines that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line overlapping it and implementing 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. Nevertheless, they are complementary if the addition of these angles totals 90°. If the sum of both angles equals 180°, they are deemed supplementary.

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

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

Additional Study

Look for a vertically opposite angles practice questions online for examples and problems to practice. Math is not a onlooker sport; keep applying until these theorems are well-established in your brain.

Still, there is nothing humiliating if you need additional support. If you're having difficulty to understand vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our professional teachers can assist you comprehend the material and nail your next examination.