September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial topic for anyone who wants to study arithmetic or any other subject that employs it. It's hard 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 comprehend these theories, as this blog will help you understand all the fundamentals. Furthermore, we will help you understand the secret to learning faster and improving your grades in mathematics and other common subjects today.

The Theorem

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

These opposite angles share a vertex. Furthermore, the most crucial thing to remember 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 equal 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 work out the others utilizing proportions.

Proving the Theorem

Proving this theorem is relatively straightforward. First, let's pull a line and call it line l. After that, we will pull another line that goes through line l at some point. We will assume this second line m.

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

We know 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 you observe angles B and C, you will note that they are not linked at their vertex but close to one another. They share a side and a vertex, therefore they are supplementary angles, so the total of both angles will be 180 degrees. This situation 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 end with:


So, we can conclude 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 specifically about vertically opposite angles.


As we mentioned, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.

Despite that, vertically opposite angles are at no time 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 total 90°. Supplementary angles are adjacent angles which will add up to 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 due to this reason supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.

There are many properties of vertically opposite angles. But, 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 Use Opposite Angles in Real-World Scenario?

You may speculate where you can find these concepts in the real world, and you'd be surprised to observe that vertically opposite angles are quite common! You can find them in various everyday things and circumstances.

For instance, vertically opposite angles are formed when two straight lines overlap each other. Right in 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 alter the size of the angles. Road crossings are also a terrific example of vertically opposite angles.

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

Be sure to notice your environment, as you will find an example next to you.

Puttingit All Together

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

The vertical angle theorem states that whenever 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 implementing the theorems of congruent angles to complete measures.

Congruent angles means 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 equals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.

The sum of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will at all time add up to 180°.

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

Further Study

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

Despite that, there is no shame if you require further support. If you're struggling to grasp vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our professional teachers can guide you comprehend the topic and nail your next examination.