September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a important topic for anyone who desires to master mathematics or another subject that uses it. It's hard work, but we'll make sure you get a good grasp of these theories so you can make the grade!

Don’t feel disheartened if you don’t recall or don’t comprehend these concepts, as this blog will help you study all the essentials. Moreover, we will help you understand the secret to learning faster and increasing your grades in arithmetic and other popular subjects today.

The Theorem

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

These opposite angles share a vertex. Moreover, the most essential thing to bear in mind is that they are the same in measurement! This applies that regardless of where these straight lines cross, the angles opposite each other will always share the exact value. These angles are known 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 using proportions.

Proving the Theorem

Proving this theorem is relatively simple. 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 name 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. Therefore, we label angle A, angle B, angle C, and angle D as follows:

We are aware that angles A and B are vertically opposite due to the fact they share the same vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A is the same as angle B.

If you see the angles B and C, you will note that they are not connected at their vertex but adjacent to each other. They have in common 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 manner:

∠B+∠C=180 and ∠A+∠C=180

Since both sums up to equal the same, we can add these operations as follows:


By removing 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 identical measure.

Vertically Opposite Angles

Now that we know the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.


As we mentioned, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.

However, 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 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 total 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized in our proof of the vertical angle theorem.

These theories are relevant 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 characteristics of vertically opposite angles. Regardless, odds are that you will only require these two to secure your exam.

  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 Use Opposite Angles in Real-World Scenario?

You might think where you can find these theorems in the real life, and you'd be stunned to notice that vertically opposite angles are very common! You can locate them in many everyday objects and situations.

For example, vertically opposite angles are created when two straight lines cross. Right in your room, the door installed 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. Road intersections are also a terrific example of vertically opposite angles.

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

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

Puttingit Together

So, to sum up what we have considered so far, vertically opposite angles are formed from two crossover lines. The two angles that are not adjacent have identical measurements.

The vertical angle theorem defines that whenever two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line overlapping it and implementing the theorems of congruent angles to complete measures.

Congruent angles refer to two angles that have identical measurements.

When two angles share a side and a vertex, they cannot be vertically opposite. Nevertheless, they are complementary if the sum of these angles equals 90°. If the sum of both angles equals 180°, they are deemed supplementary.

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

Vertically opposite angles are pretty common! You can find them in several everyday objects and scenarios, such as doors, windows, paintings, and trees.

Additional Study

Look for a vertically opposite angles questionnaire on the internet for examples and sums to practice. Math is not a spectator sport; keep applying until these concepts are rooted in your head.

Still, there is no shame if you need further support. If you're struggling to comprehend vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our expert teachers can guide you grasp the topic and ace your next examination.