September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial subject for everyone who wants to study arithmetic or any related subject that utilizes it. It's hard work, but we'll ensure you get a grip of these concepts so you can make the grade!

Don’t feel discouraged if you don’t remember or don’t have a good grasp on these concepts, as this blog will teach you all the fundamentals. Furthermore, we will teach you the secret to learning quicker and increasing your scores in math and other popular subjects today.

The Theorem

The vertical angle theorem stipulates that at any time two straight lines bisect, they make opposite angles, named vertical angles.

These opposite angles share a vertex. Moreover, the most important point to remember 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 exact 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 feasible to discover the others using proportions.

Proving the Theorem

Proving this theorem is somewhat easy. Primarily, let's draw a line and label 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 name the angles created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:

We are aware that angles A and B are vertically opposite 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 equals angle B.

If we look at angles B and C, you will notice that they are not joined at their vertex but adjacent to one another. They have in common a side and a vertex, meaning they are supplementary angles, so the total 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 eliminating 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 particularly regarding vertically opposite angles.


As we mentioned, 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 never share a side. When angles share a side, these adjacent angles could be complementary or supplementary.

In case of complementary angles, the sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.

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

There are various characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to ace your exam.

  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 Locate Opposite Angles in Real-World Circumstances?

You might speculate where you can find these concepts in the real life, and you'd be amazed to observe that vertically opposite angles are quite common! You can locate them in many daily things and situations.

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

Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Road intersections are also a wonderful example of vertically opposite angles.

In the end, 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 environment, as you will discover an example next to you.

Puttingit All Together

So, to sum up what we have discussed, vertically opposite angles are formed from two crossover lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem explains that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line intersecting it and using 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 cannot be vertically opposite. Despite that, they are complementary if the sum of these angles equals 90°. If the sum of both angles totals 180°, they are deemed supplementary.

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

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

Further Study

Search for a vertically opposite angles practice questions on the internet for examples and sums to practice. Math is not a spectator sport; keep applying until these concepts are rooted in your mind.

Still, there is no problem if you need additional help. If you're struggling to grasp vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our skill instructor can assist you grasp the material and nail your following test.