September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a crucial subject for everyone who desires to learn arithmetic or any other subject that utilizes it. It's tough work, but we'll ensure you get a grip of these concepts so you can attain the grade!

Don’t feel discouraged if you don’t remember or don’t have a good grasp on these theories, as this blog will teach you all the basics. Moreover, we will teach you the tricks to learning quicker and improving your grades in mathematics and other common subjects today.

The Theorem

The vertical angle theorem states that when two straight lines intersect, they form opposite angles, known as vertical angles.

These opposite angles share a vertex. Moreover, the most crucial point to remember is that they are the same in measurement! This refers that irrespective of where these straight lines cross, the angles converse each other will constantly 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 easy. Primarily, let's draw a line and call it line l. Then, we will draw another line that intersects 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 prevent confusion, we labeled pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:

We understand that angles A and B are vertically contrary because 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 note that they are not joined at their vertex but next to each other. They share a side and a vertex, therefore they are supplementary angles, so the total of both angles will be 180 degrees. This scenario 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 add these operations as follows:


By canceling out C on both sides of the equation, we will end with:


So, we can conclude that vertically opposite angles are congruent, as they have the same measure.

Vertically Opposite Angles

Now that we have learned about the theorem and how to prove it, let's discuss particularly regarding vertically opposite angles.


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

Despite that, vertically opposite angles are never 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 the case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose addition 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 since supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.

There are various characteristics of vertically opposite angles. Still, chances are that you will only require these two to nail your exam.

  1. Vertically opposite angles are at all time congruent. Hence, 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 Find Opposite Angles in Real-World Situations?

You might wonder where you can find these concepts in the real world, and you'd be stunned to note that vertically opposite angles are fairly common! You can discover them in various daily objects and situations.

For instance, vertically opposite angles are formed when two straight lines cross. Inside 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 modify the size of the angles. Road junctions are also a great 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 environment, as you will find an example next to you.

Puttingit Together

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

The vertical angle theorem defines that whenever two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and using the concepts of congruent angles to finish measures.

Congruent angles refer to two angles that measure the same.

When two angles share a side and a vertex, they can’t be vertically opposite. Nevertheless, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are considered supplementary.

The total of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will always add up to 180°.

Vertically opposite angles are pretty common! You can locate them in various daily objects and circumstances, such as doors, windows, paintings, and trees.

Additional Study

Look for a vertically opposite angles practice questions on the internet for examples and problems to practice. Mathematics is not a spectator sport; keep practicing until these concepts are rooted in your brain.

However, there is no problem if you need additional help. If you're struggling to understand vertical angles (or any other ideas of geometry), contemplate signing up for a tutoring session with Grade Potential. One of our expert instructor can help you understand the material and ace your following test.