September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is an essential topic for anyone who wishes to master math or any related subject that employs it. It's tough work, but we'll make sure you get a handle on these theories so you can make the grade!

Don’t feel discouraged if you don’t recollect or don’t have a good grasp on these theories, as this blog will help you study all the essentials. Additionally, we will help you understand the secret to learning faster and increasing your grades in arithmetic and other prevailing subjects today.

The Theorem

The vertical angle theorem stipulates that whenever two straight lines intersect, they create 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 refers that irrespective of where these straight lines cross, the angles opposite each other will consistently 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 feasible to find the others employing proportions.

Proving the Theorem

Proving this theorem is relatively straightforward. Primarily, let's draw a line and name it line l. After that, we will draw 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 prevent confusion, we labeled pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:

We understand 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 is identical angle B.

If we look at angles B and C, you will notice that they are not joined at their vertex but close to each other. They share a side and a vertex, signifying 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 way:

∠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 conclude 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 stated, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.

However, vertically opposite angles are never next to each other. 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 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 used to prove the vertical angle theorem.

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

There are several characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to secure your examination.

  1. Vertically opposite angles are always congruent. Hence, 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 find these concepts in the real world, and you'd be surprised to observe that vertically opposite angles are quite common! You can locate them in various daily objects and circumstances.

For instance, vertically opposite angles are created when two straight lines overlap each other. Right in your room, the door installed to the door frame creates 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.

Finally, vertically opposite angles are also discovered 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 detect an example next to you.

Puttingit All Together

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

The vertical angle theorem states that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out 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. However, they are complementary if the addition of these angles equals 90°. If the sum of both angles equals 180°, they are deemed supplementary.

The total of adjacent angles is consistently 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 locate them in several daily objects and situations, 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 practicing until these theorems are rooted in your brain.

However, there is no problem if you need further support. If you're having a hard time to grasp vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our professional tutors can help you grasp the topic and ace your next test.