September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is a crucial topic for everyone who wants to study arithmetic or any related subject that utilizes it. It's hard work, but we'll assure you get a good grasp of these concepts so you can make the grade!

Don’t feel dispirited if you don’t recall or don’t comprehend these concepts, as this blog will help you understand all the essentials. Moreover, we will help you learn the secret to learning faster and improving your scores in math and other prevailing subjects today.

The Theorem

The vertical angle theorem stipulates that when two straight lines meet, they create opposite angles, known as vertical angles.

These opposite angles share a vertex. Additionally, the most important thing to bear in mind is that they are the same in measurement! This refers that regardless of where these straight lines cross, the angles opposite each other will constantly share the same value. These angles are called congruent angles.

Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to work out the others utilizing 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 intersects line l at some point. We will assume this second line m.

After drawing these two lines, we will assume the angles formed by the intersecting lines l and m. To prevent confusion, we named 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 opposite reason being that 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 you see the angles B and C, you will note that they are not connected at their vertex but close to each other. They share a side and a vertex, meaning they are supplementary angles, so the sum 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 way:

∠B+∠C=180 and ∠A+∠C=180

Since both sums up to equal the same, we can add these operations as follows:


By canceling out 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 talk specifically regarding vertically opposite angles.


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

Despite that, vertically opposite angles are no way adjacent. Adjacent angles are two angles that have 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 sum 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 fulfill the characteristics of vertically opposite angles.

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

  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 never adjacent. They can share, at most, a vertex.

Where Can You Locate Opposite Angles in Real-Life Situations?

You may speculate where you can use these concepts in the real world, and you'd be surprised to note that vertically opposite angles are quite common! You can locate them in several everyday objects and scenarios.

For example, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door attached 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.

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

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

Puttingit Together

So, to summarize what we have considered so far, vertically opposite angles are formed from two overlapping lines. The two angles that are not next to each other have identical measurements.

The vertical angle theorem explains that when 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 overlapping it and applying the theorems of congruent angles to complete measures.

Congruent angles refer to two angles that have identical measurements.

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

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

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

Further Study

Look for a vertically opposite angles practice questions online for examples and sums to practice. Math is not a spectator sport; keep applying until these theorems are ingrained in your brain.

Despite that, there is no problem if you require extra assistance. If you're struggling to understand vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert teachers can help you understand the topic and ace your following test.