September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is an essential topic for everyone who desires to study math or another subject that utilizes it. It's tough work, but we'll make sure you get a good grasp of these concepts so you can achieve 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 help you study all the fundamentals. Moreover, we will help you understand the tricks to learning faster and increasing your grades in math and other prevailing subjects today.

The Theorem

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

These opposite angles share a vertex. Additionally, the most crucial thing to bear in mind is that they are the same in measurement! This means that irrespective of where these straight lines cross, the angles converse each other will always share the exact 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 find the others utilizing proportions.

Proving the Theorem

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

After drawing these two lines, we will assume the angles created 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 are aware that angles A and B are vertically contrary due to the fact they share the same 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 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 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 additions equal the same, we can add these operations as follows:


By removing 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 measurement.

Vertically Opposite Angles

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


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

Still, vertically opposite angles are at no time next to each other. 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 addition will equal 180°, which we just utilized to prove 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 satisfy the characteristics of vertically opposite angles.

There are various characteristics of vertically opposite angles. But, chances are that you will only require these two to secure your exam.

  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 may speculate where you can use these concepts in the real world, and you'd be surprised to observe that vertically opposite angles are fairly common! You can find them in various everyday objects and circumstances.

For instance, vertically opposite angles are formed when two straight lines cross. 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. Track intersections are also a wonderful example of vertically opposite angles.

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

Be sure to watch your environment, as you will detect an example next to you.

PuttingEverything Together

So, to sum up what we have discussed, vertically opposite angles are made from two overlapping lines. The two angles that are not next to each other have the same measure.

The vertical angle theorem explains that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting it and implementing the theorems 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. Nevertheless, they are complementary if the sum of these angles totals 90°. If the addition of both angles totals 180°, they are considered 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 quite common! You can locate them in many everyday objects and situations, such as paintings, doors, windows, and trees.

Additional Study

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

Still, there is no shame if you require further support. If you're struggling to grasp vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our professional tutors can assist you comprehend the material and nail your following examination.