September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a important subject for anyone who wishes to learn arithmetic or any other subject that uses it. It's tough work, but we'll ensure you get a grip of these theories so you can attain the grade!

Don’t feel disheartened if you don’t remember or don’t understand these concepts, as this blog will teach you all the fundamentals. Moreover, we will help you learn the tricks to learning faster and improving your grades in mathematics and other prevailing subjects today.

The Theorem

The vertical angle theorem stipulates that when two straight lines meet, they make opposite angles, known as vertical angles.

These opposite angles share a vertex. Furthermore, the most important point to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles converse each other will constantly share the equal 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 feasible to discover the others utilizing proportions.

Proving the Theorem

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

After drawing these two lines, we will name the angles formed 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 same vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A equals angle B.

If you see the angles B and C, you will note that they are not connected at their vertex but close to each other. They share a side and a vertex, therefore 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 sum up 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 identical measurement.

Vertically Opposite Angles

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


As we said earlier, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.

However, vertically opposite angles are never adjacent. 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 the case of complementary angles, the sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.

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

There are various characteristics of vertically opposite angles. Still, odds are that you will only need these two to nail your exam.

  1. Vertically opposite angles are always congruent. Consequently, 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 Scenario?

You may speculate where you can find these theorems in the real world, and you'd be stunned to notice that vertically opposite angles are very common! You can locate them in various daily things and situations.

For instance, vertically opposite angles are made when two straight lines cross. Back of your room, the door connected to the door frame makes vertically opposite angles with the wall.

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

Finally, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are formed 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 sum up what we have discussed, vertically opposite angles are created from two intersecting lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem explains that when 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 overlapping it and applying 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. However, they are complementary if the sum of these angles totals 90°. If the sum of both angles totals 180°, they are considered 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 locate them in many daily objects and situations, such as doors, windows, 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 concepts are well-established in your mind.

Still, there is nothing humiliating if you require extra help. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our expert instructor can guide you understand the material and nail your next test.