September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial topic for everyone who wishes to master mathematics or any other subject that uses it. It's hard work, but we'll assure you get a handle on these concepts so you can achieve the grade!

Don’t feel dispirited if you don’t recollect or don’t understand these theories, as this blog will help you understand all the basics. Additionally, we will teach you the secret to learning quicker and enhancing your scores in mathematics and other common subjects today.

The Theorem

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

These opposite angles share a vertex. Furthermore, the most crucial thing to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles opposite each other will consistently share the same 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 possible to work out 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 pull 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 named pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:

We are aware that angles A and B are vertically opposite because 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 equals angle B.

If you observe angles B and C, you will notice that they are not linked at their vertex but next to one another. They share a side and a vertex, meaning 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 way:

∠B+∠C=180 and ∠A+∠C=180

Since both additions equal the same, we can sum up 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 know the theorem and how to prove it, let's talk explicitly about vertically opposite angles.


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

Despite that, vertically opposite angles are at no time next to each other. 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 sum will equal 180°, which we just used to prove the vertical angle theorem.

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

There are many characteristics of vertically opposite angles. But, odds are that you will only need these two to nail 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 Find 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 note that vertically opposite angles are very common! You can locate them in many everyday objects and scenarios.

For instance, vertically opposite angles are created when two straight lines cross. Back of your room, the door connected to the door frame produces vertically opposite angles with the wall.

Open a pair of scissors to produce two intersecting lines and adjust the size of the angles. Road crossings are also a terrific example of vertically opposite angles.

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

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

Puttingit 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 next to each other have identical measurements.

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

Congruent angles means two angles that measure the same.

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

The total of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will at all time equal 180°.

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

Further Study

Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Math is not a onlooker sport; keep applying until these theorems are rooted in your head.

Still, there is nothing humiliating if you need additional support. If you're struggling to grasp 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 topic and ace your following examination.