September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Learning vertical angles is a crucial topic for anyone who wants to master math or any related subject that employs it. It's hard work, but we'll ensure you get a good grasp of these concepts so you can attain the grade!

Don’t feel dispirited if you don’t remember or don’t understand these theories, as this blog will help you study all the fundamentals. Moreover, we will teach you the tricks to learning faster and enhancing your grades in mathematics and other common subjects today.

The Theorem

The vertical angle theorem expresses that whenever two straight lines intersect, they make opposite angles, known as vertical angles.

These opposite angles share a vertex. Moreover, the most important point to bear in mind is that they are the same in measurement! This means that regardless of where these straight lines cross, the angles opposite each other will constantly 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 work out the others employing proportions.

Proving the Theorem

Proving this theorem is somewhat easy. First, let's draw a line and name it line l. After that, we will draw another line that goes through line l at some point. We will assume this second line m.

After drawing these two lines, we will assume the angles created 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 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. Bear in mind 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 notice that they are not linked at their vertex but next 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 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 sum up these operations as follows:


By canceling out 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 know the theorem and how to prove it, let's discuss particularly 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 one another satisfy the vertical angle theorem.

However, vertically opposite angles are no way next to each other. 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 equal 90°. Supplementary angles are adjacent angles whose sum 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 meet the characteristics of vertically opposite angles.

There are many characteristics of vertically opposite angles. Still, chances are that you will only need these two to ace your exam.

  1. Vertically opposite angles are always 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 Use Opposite Angles in Real-Life Scenario?

You might think where you can use these theorems in the real world, and you'd be surprised to note that vertically opposite angles are fairly common! You can find them in many everyday objects and situations.

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

Open a pair of scissors to create two intersecting lines and modify the size of the angles. Road junctions are also a terrific example of vertically opposite angles.

Finally, 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 watch your environment, as you will discover an example next to you.

PuttingEverything Together

So, to sum up what we have considered so far, vertically opposite angles are created from two crossover lines. The two angles that are not adjacent 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 proven by depicting a straight line and another line intersecting it and implementing the theorems 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 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 deemed supplementary.

The sum of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.

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

Further Study

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

Still, there is no problem if you require further support. If you're having a hard time to understand vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our expert tutors can assist you understand the topic and nail your next test.