September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a crucial subject for anyone who wants to learn arithmetic or any other subject that utilizes it. It's hard work, but we'll assure you get a good grasp of these theories so you can achieve 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 help you understand all the basics. Additionally, we will help you learn the secret to learning faster and increasing your grades in mathematics and other popular subjects today.

The Theorem

The vertical angle theorem expresses that at any time two straight lines intersect, they form opposite angles, known as vertical angles.

These opposite angles share a vertex. Furthermore, the most crucial thing to remember is that they also measure the same! This applies 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 relatively simple. Primarily, let's draw a line and call it line l. Then, we will draw another line that goes through line l at some point. We will assume this second line m.

After drawing these two lines, we will name the angles created by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically opposite due to the fact 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 is the same as angle B.

If you see the angles B and C, you will notice that they are not connected at their vertex but next to each other. They share a side and a vertex, signifying they are supplementary angles, so the sum 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 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 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 know the theorem and how to prove it, let's talk particularly regarding vertically opposite angles.


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

However, vertically opposite angles are no way 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 satisfy 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 secure your examination.

  1. Vertically opposite angles are always congruent. Hence, 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 may speculate where you can find these theorems in the real world, and you'd be stunned to observe that vertically opposite angles are fairly common! You can find them in many everyday objects and situations.

For instance, vertically opposite angles are created when two straight lines overlap each other. Right in 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 crossings are also a wonderful example of vertically opposite angles.

In the end, 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 observe your environment, as you will detect an example next to you.

PuttingEverything Together

So, to summarize what we have considered so far, 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 states that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and using 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. However, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are assumed to be supplementary.

The total of adjacent angles is consistently 180°. Thus, 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 several daily objects and scenarios, such as windows, doors, paintings, and trees.

Further Study

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

Despite that, there is no shame if you require additional help. If you're having a hard time to grasp vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our expert teachers can help you grasp the material and ace your next test.