September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is an essential topic for everyone who desires to master math or another subject that employs it. It's tough work, but we'll make sure you get a good grasp of these theories so you can attain the grade!

Don’t feel disheartened if you don’t remember or don’t understand these theories, as this blog will teach you all the fundamentals. Additionally, we will teach you the tricks to learning faster and increasing your scores in arithmetic and other common subjects today.

The Theorem

The vertical angle theorem states that whenever two straight lines bisect, they create 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 opposite 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 possible to work out 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 call this second line m.

After drawing these two lines, we will assume the angles formed by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:

We know that angles A and B are vertically opposite because 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 observe angles B and C, you will note that they are not linked at their vertex but adjacent to each other. They have in common a side and a vertex, meaning they are supplementary angles, so the sum 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 way:

∠B+∠C=180 and ∠A+∠C=180

Since both sums up to equal the same, we can sum up 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 identical measure.

Vertically Opposite Angles

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


As we said earlier, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite one another 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 never 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 whose sum will equal 180°, which we just used in our proof of the vertical angle theorem.

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

There are various characteristics of vertically opposite angles. Regardless, odds are that you will only require these two to nail 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 Locate Opposite Angles in Real-World Scenario?

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

For example, vertically opposite angles are made when two straight lines cross. Right in your room, the door installed to the door frame makes vertically opposite angles with the wall.

Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Road intersections are also a wonderful example of vertically opposite angles.

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

Be sure to observe your environment, as you will discover an example next to you.

Puttingit Together

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

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 depicting a straight line and another line overlapping it and implementing the concepts of congruent angles to complete measures.

Congruent angles refer to 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 sum of these angles equals 90°. If the sum of both angles totals 180°, they are assumed to be supplementary.

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

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

Further Study

Search for a vertically opposite angles questionnaire on the internet for examples and problems to practice. Mathematics is not a onlooker sport; keep applying until these concepts are well-established in your brain.

Still, there is nothing humiliating if you require further support. If you're having difficulty to grasp vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill teachers can assist you grasp the material and ace your next test.