September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial subject for anyone who wants to master mathematics or another subject that employs it. It's tough work, but we'll ensure you get a handle on these concepts so you can make the grade!

Don’t feel disheartened if you don’t recollect or don’t comprehend these theories, as this blog will help you study all the fundamentals. Additionally, we will teach you the secret to learning faster and enhancing your scores in arithmetic and other popular subjects today.

The Theorem

The vertical angle theorem states that when two straight lines meet, they make opposite angles, called vertical angles.

These opposite angles share a vertex. Moreover, 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 converse each other will always share the equal 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 possible to work out the others using proportions.

Proving the Theorem

Proving this theorem is moderately easy. First, let's pull a line and call it line l. Then, we will pull another line that intersects line l at some point. We will name this second line m.

After drawing these two lines, we will name the angles created by the intersecting lines l and m. To avoid confusion, we labeled 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 contrary due to the fact they share the equivalent vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A is the same as angle B.

If we look at angles B and C, you will notice that they are not joined at their vertex but adjacent to each other. They have in common a side and a vertex, signifying they are supplementary angles, so the sum of both angles will be 180 degrees. This case 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 discuss explicitly regarding vertically opposite angles.


As we mentioned, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite one another fulfill the vertical angle theorem.

However, vertically opposite angles are never 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 case of complementary angles, the addition of two adjacent angles will total 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized in our proof of the vertical angle theorem.

These concepts are appropriate 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 many properties of vertically opposite angles. Still, chances are that you will only need these two to ace 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 at no time adjacent. They can share, at most, a vertex.

Where Can You Find Opposite Angles in Real-Life Circumstances?

You might wonder where you can utilize these concepts in the real life, and you'd be stunned to note that vertically opposite angles are very common! You can discover them in several everyday objects and situations.

For example, vertically opposite angles are formed 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 make two intersecting lines and alter the size of the angles. Road junctions are also a terrific example of vertically opposite angles.

Eventually, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are created by the trunk and the branches.

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

PuttingEverything Together

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

The vertical angle theorem defines that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting it and applying the concepts of congruent angles to finish measures.

Congruent angles refer to two angles that have identical measurements.

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 totals 90°. If the sum of both angles totals 180°, they are deemed supplementary.

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

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

Further Study

Look for a vertically opposite angles worksheet on the internet for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these theorems are ingrained in your mind.

Still, there is no shame if you need further help. If you're having difficulty to grasp vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our skill instructor can assist you understand the material and nail your following test.