September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial topic for anyone who wants to study math or another subject that utilizes it. It's hard work, but we'll make sure you get a handle on these concepts so you can attain the grade!

Don’t feel disheartened if you don’t remember or don’t understand these theories, as this blog will help you study all the fundamentals. Moreover, we will help you learn the secret to learning faster and enhancing your scores in arithmetic and other common subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines intersect, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Moreover, the most important point to keep in mind 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 possible to find the others employing proportions.

Proving the Theorem

Proving this theorem is relatively simple. First, let's draw a line and name it line l. Then, we will pull another line that intersects 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 avoid confusion, we named pairs of vertically opposite angles. Accordingly, 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 same vertex but don’t share a side. Remember 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 notice that they are not joined at their vertex but next to each other. They share a side and a vertex, meaning they are supplementary angles, so the sum of both angles will be 180 degrees. This instance 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 be left with:


So, we can say that vertically opposite angles are congruent, as they have the same measurement.

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 formed by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.

However, vertically opposite angles are never adjacent. 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 case of complementary angles, the sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just used in our proof of the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the properties of vertically opposite angles.

There are many properties of vertically opposite angles. Still, chances are that you will only need these two to nail 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-Life Situations?

You may think where you can use these concepts in the real life, and you'd be stunned to observe that vertically opposite angles are fairly common! You can locate them in several everyday things and situations.

For example, vertically opposite angles are created when two straight lines overlap each other. Inside your room, the door connected to the door frame makes vertically opposite angles with the wall.

Open a pair of scissors to make two intersecting lines and modify the size of the angles. Road junctions are also a wonderful example of vertically opposite angles.

In the end, vertically opposite angles are also discovered 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 find an example next to you.

Puttingit All Together

So, to summarize what we have considered so far, vertically opposite angles are created from two intersecting lines. The two angles that are not adjacent have the same measure.

The vertical angle theorem explains that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting 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 can’t be vertically opposite. Nevertheless, they are complementary if the addition of these angles totals 90°. If the sum of both angles equals 180°, they are assumed to be 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 pretty common! You can locate them in various everyday objects and scenarios, such as doors, windows, paintings, and trees.

Further Study

Look for a vertically opposite angles practice questions online for examples and sums to practice. Mathematics is not a spectator sport; keep practicing until these concepts are well-established in your head.

However, there is no shame if you require further support. If you're having a hard time to understand vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our professional instructor can assist you comprehend the topic and ace your following test.