September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is an essential subject for everyone who wants to master mathematics or any related subject that utilizes it. It's hard work, but we'll ensure you get a good grasp of these concepts so you can attain the grade!

Don’t feel discouraged if you don’t recall or don’t have a good grasp on these concepts, as this blog will help you study all the essentials. Additionally, we will teach you the secret to learning quicker and improving your scores in mathematics and other common subjects today.

The Theorem

The vertical angle theorem stipulates that at any time two straight lines intersect, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Moreover, the most crucial point to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles opposite 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 find the others employing proportions.

Proving the Theorem

Proving this theorem is moderately easy. Primarily, let's pull a line and call it line l. After that, we will pull 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 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. Bear in mind 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 scenario 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 additions equal the same, we can add 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 measure.

Vertically Opposite Angles

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


As we stated, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite one another satisfy the vertical angle theorem.

Despite that, vertically opposite angles are never 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 sum will equal 180°, which we just utilized in our proof of 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 meet the properties of vertically opposite angles.

There are various properties of vertically opposite angles. But, chances are that you will only need 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 never adjacent. They can share, at most, a vertex.

Where Can You Locate Opposite Angles in Real-Life Circumstances?

You might think where you can use these concepts in the real world, and you'd be surprised to observe that vertically opposite angles are fairly common! You can locate them in many everyday things and scenarios.

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 produce two intersecting lines and modify the size of the angles. Track junctions are also a wonderful example of vertically opposite angles.

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

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

PuttingEverything Together

So, to sum up what we have discussed, vertically opposite angles are created from two crossover lines. The two angles that are not adjacent have identical measurements.

The vertical angle theorem explains that in the event of 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 implementing the theorems 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 cannot be vertically opposite. Despite that, they are complementary if the addition of these angles totals 90°. If the addition of both angles equals 180°, they are considered supplementary.

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

Vertically opposite angles are very common! You can find them in many everyday objects and situations, such as windows, doors, paintings, and trees.

Further Study

Look for a vertically opposite angles worksheet on the internet for examples and exercises to practice. Math is not a spectator sport; keep applying until these theorems are rooted in your brain.

Still, there is nothing humiliating if you require extra support. If you're having a hard time to understand vertical angles (or any other ideas of geometry), think about signing up for a tutoring session with Grade Potential. One of our skill tutors can assist you grasp the material and nail your next test.