September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a important topic for everyone who wants to learn arithmetic or another subject that uses it. It's tough work, but we'll ensure you get a handle on these concepts so you can attain the grade!

Don’t feel disheartened if you don’t recall or don’t have a good grasp on these theories, as this blog will teach you all the essentials. Additionally, we will teach you the secret to learning faster and enhancing your grades in math and other popular subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines intersect, they form opposite angles, named vertical angles.

These opposite angles share a vertex. Additionally, the most crucial thing to keep in mind is that they also measure the same! This refers that regardless of where these straight lines cross, the angles opposite each other will consistently share the equal 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 discover the others using proportions.

Proving the Theorem

Proving this theorem is somewhat straightforward. 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 label the angles formed by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically contrary because they share the same vertex but don’t share a side. Bear in mind 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 close to each other. They have in common a side and a vertex, signifying 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 additions equal the same, we can sum up these operations as follows:


By removing 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 identical measurement.

Vertically Opposite Angles

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


As we said earlier, vertically opposite angles are two angles formed 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 adjacent. 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 addition of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.

These concepts are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.

There are many properties of vertically opposite angles. Still, chances are that you will only require these two to ace your examination.

  1. Vertically opposite angles are at all time congruent. Therefore, 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-World Situations?

You might wonder where you can find these concepts in the real life, and you'd be amazed to note that vertically opposite angles are very common! You can discover them in many daily things and scenarios.

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 adjust the size of the angles. Track junctions are also a wonderful example of vertically opposite angles.

In the end, vertically opposite angles are also discovered 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.

Puttingit All Together

So, to summarize what we have discussed, vertically opposite angles are made from two crossover lines. The two angles that are not adjacent have the same measure.

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

Congruent angles means two angles that measure the same.

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

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

Vertically opposite angles are quite common! You can discover them in several everyday objects and circumstances, such as paintings, doors, windows, and trees.

Further Study

Search for a vertically opposite angles practice questions online for examples and problems to practice. Math is not a onlooker sport; keep applying until these theorems are well-established in your brain.

Despite that, there is no problem if you need additional help. If you're struggling to comprehend 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 understand the material and nail your next examination.