September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a crucial subject for anyone who wants to learn math or any related subject that uses it. It's hard 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 remember or don’t have a good grasp on these concepts, as this blog will help you understand all the basics. 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 stipulates that at any time two straight lines bisect, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Moreover, the most crucial point to bear in mind is that they also measure the same! 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 work out the others employing proportions.

Proving the Theorem

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

After drawing these two lines, we will label 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 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 notice that they are not linked at their vertex but adjacent to one another. They have in common a side and a vertex, signifying they are supplementary angles, so the total of both angles will be 180 degrees. This case 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 add these operations as follows:


By eliminating 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 measurement.

Vertically Opposite Angles

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


As we stated, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.

Despite that, vertically opposite angles are at no time 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 add up to 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just used to prove the vertical angle theorem.

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

There are many characteristics of vertically opposite angles. Still, odds are that you will only need these two to ace your examination.

  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 Use Opposite Angles in Real-Life Circumstances?

You may think where you can find these theorems in the real life, and you'd be amazed to notice that vertically opposite angles are fairly common! You can locate them in several everyday objects and situations.

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

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

Eventually, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are created 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 made from two crossover lines. The two angles that are not next to each other have identical measurements.

The vertical angle theorem states that when 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 overlapping it and implementing the concepts of congruent angles to finish measures.

Congruent angles refer to 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 totals 90°. If the sum of both angles totals 180°, they are deemed supplementary.

The total of adjacent angles is always 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 find them in various daily objects and scenarios, such as doors, windows, paintings, and trees.

Further Study

Search for a vertically opposite angles questionnaire online for examples and exercises to practice. Math is not a onlooker sport; keep applying until these concepts are well-established in your brain.

However, there is nothing humiliating if you need extra assistance. If you're struggling to comprehend vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you grasp the topic and ace your next test.