September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a crucial topic for anyone who desires to master mathematics or another subject that uses it. It's hard work, but we'll ensure you get a handle on these theories so you can attain the grade!

Don’t feel dispirited if you don’t recall or don’t understand these theories, as this blog will help you study all the basics. Additionally, we will help you learn the secret to learning faster and increasing your scores in mathematics and other common subjects today.

The Theorem

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

These opposite angles share a vertex. Additionally, the most important thing to keep in mind is that they are the same in measurement! This refers that regardless of where these straight lines cross, the angles converse each other will consistently share the same 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 find the others using proportions.

Proving the Theorem

Proving this theorem is relatively easy. First, let's pull a line and label 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 assume the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically contrary reason being that they share the equivalent vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A equals angle B.

If we look at angles B and C, you will notice that they are not joined 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 instance 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 sums up to equal the same, we can add these operations as follows:


By canceling out 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 measure.

Vertically Opposite Angles

Now that we have studied about the theorem and how to prove it, let's talk explicitly about 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.

Still, vertically opposite angles are at no time 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 the 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 utilized in our proof of the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the characteristics of vertically opposite angles.

There are several properties of vertically opposite angles. Regardless, chances are that you will only need these two to ace 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 Situations?

You may wonder where you can find these theorems in the real life, and you'd be surprised to observe that vertically opposite angles are fairly common! You can find them in many everyday things and scenarios.

For example, vertically opposite angles are created when two straight lines cross. Back of your room, the door attached to the door frame creates vertically opposite angles with the wall.

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

In the end, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.

Be sure to watch your surroundings, as you will detect an example next to you.

Puttingit Together

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

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 depicting a straight line and another line overlapping it and using the concepts 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. Despite that, they are complementary if the sum of these angles totals 90°. If the sum of both angles totals 180°, they are deemed supplementary.

The total of adjacent angles is consistently 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 discover them in many daily objects and situations, such as windows, doors, paintings, and trees.

Additional Study

Look for a vertically opposite angles practice questions on the internet for examples and problems to practice. Math is not a spectator sport; keep applying until these theorems are ingrained in your head.

Still, there is no problem if you need additional assistance. 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 teachers can assist you understand the topic and ace your following test.