September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is an essential topic for everyone who desires to study mathematics or another subject that utilizes it. It's tough work, but we'll assure you get a good grasp of these theories so you can achieve the grade!

Don’t feel dispirited if you don’t recollect or don’t have a good grasp on these concepts, as this blog will help you understand all the essentials. Furthermore, we will help you learn the secret to learning quicker and increasing your grades in math and other popular subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines bisect, they make 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 irrespective of where these straight lines cross, the angles opposite each other will consistently share the exact value. These angles are known as congruent angles.

Vertically opposite angles are congruent, so if you have a value for one angle, then it is possible to discover the others using proportions.

Proving the Theorem

Proving this theorem is somewhat simple. First, let's pull a line and name it line l. Then, 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 name the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Therefore, we label 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 is the same as angle B.

If we look at angles B and C, you will note that they are not joined at their vertex but next to one another. They share a side and a vertex, signifying 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 way:

∠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 identical measure.

Vertically Opposite Angles

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


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

Despite that, vertically opposite angles are never adjacent. 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 total 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized in our proof of the vertical angle theorem.

These concepts are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.

There are several properties of vertically opposite angles. But, odds are that you will only require these two to ace your exam.

  1. Vertically opposite angles are at all time congruent. Consequently, 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 Find Opposite Angles in Real-World Circumstances?

You may speculate where you can find these theorems in the real world, and you'd be surprised to note that vertically opposite angles are quite common! You can find them in various everyday objects and situations.

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

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

Eventually, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created 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 considered so far, vertically opposite angles are created from two crossover lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem defines that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line intersecting it and applying the theorems of congruent angles to complete measures.

Congruent angles refer to two angles that have identical measurements.

When two angles share a side and a vertex, they can’t be vertically opposite. However, they are complementary if the sum of these angles totals 90°. If the addition of both angles equals 180°, they are considered supplementary.

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

Vertically opposite angles are quite common! You can discover them in various everyday objects and scenarios, such as doors, windows, paintings, and trees.

Additional Study

Look for a vertically opposite angles practice questions on the internet for examples and exercises to practice. Mathematics is not a spectator sport; keep applying until these theorems are well-established in your brain.

However, there is no shame if you require further help. If you're struggling to grasp vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill teachers can guide you grasp the topic and ace your following test.