September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a crucial topic for anyone who wants to learn math or another subject that utilizes it. It's tough work, but we'll assure you get a grip of these theories so you can attain the grade!

Don’t feel discouraged if you don’t remember or don’t understand these theories, as this blog will help you understand all the essentials. Furthermore, we will help you understand the secret to learning quicker and enhancing your scores in mathematics and other popular subjects today.

The Theorem

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

These opposite angles share a vertex. Furthermore, the most essential thing to remember is that they are the same in measurement! This means 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 employing proportions.

Proving the Theorem

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

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

We are aware that angles A and B are vertically contrary due to the fact 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 we look at angles B and C, you will notice that they are not joined at their vertex but close to one another. 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 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 sums up to 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 the same measure.

Vertically Opposite Angles

Now that we have studied about 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 intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.

Still, vertically opposite angles are no way adjacent. Adjacent angles are two angles that have 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 case of complementary angles, the addition of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose sum 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 since supplementary and complementary angles do not fulfill the properties of vertically opposite angles.

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

  1. Vertically opposite angles are at all time congruent. Consequently, if angles A and B are vertically opposite, they will measure the same.

  2. Vertically opposite angles are at no time adjacent. They can share, at most, a vertex.

Where Can You Use Opposite Angles in Real-Life Scenario?

You may speculate where you can use 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 daily objects and situations.

For example, 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 produce two intersecting lines and modify the size of the angles. Track intersections are also a terrific example of vertically opposite angles.

Finally, vertically opposite angles are also present 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 discover an example next to you.

Puttingit All Together

So, to sum up what we have talked about, vertically opposite angles are made from two crossover lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem explains that in the event of two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line intersecting it and using the concepts of congruent angles to complete measures.

Congruent angles means two angles that have identical measurements.

When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the addition of these angles totals 90°. If the addition of both angles totals 180°, they are considered supplementary.

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

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

Further Study

Search for a vertically opposite angles practice questions on the internet for examples and exercises to practice. Mathematics is not a onlooker sport; keep applying until these theorems are ingrained in your brain.

Despite that, there is no problem if you need additional support. If you're having a hard time to understand vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our professional instructor can assist you comprehend the material and ace your next test.