September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Understanding vertical angles is a crucial subject for anyone who desires to learn math or any other subject that uses it. It's hard work, but we'll make sure you get a handle on these theories so you can make the grade!

Don’t feel discouraged if you don’t recollect or don’t have a good grasp on these theories, as this blog will help you understand all the essentials. Furthermore, we will help you learn the secret to learning quicker and improving your grades in math and other common subjects today.

The Theorem

The vertical angle theorem stipulates that whenever two straight lines intersect, they create opposite angles, called vertical angles.

These opposite angles share a vertex. Furthermore, the most essential point to keep in mind is that they are the same in measurement! This means that regardless of where these straight lines cross, the angles converse each other will consistently 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 feasible to find the others utilizing proportions.

Proving the Theorem

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

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

We understand that angles A and B are vertically opposite 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 notice that they are not joined at their vertex but close to one another. They share a side and a vertex, therefore 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 manner:

∠B+∠C=180 and ∠A+∠C=180

Since both additions equal the same, we can add these operations as follows:

∠A+∠C=∠B+∠C

By canceling out C on both sides of the equation, we will end with:

∠A=∠B

So, we can say 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 talk explicitly regarding vertically opposite angles.

Definition

As we said earlier, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite one another 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 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 which will add up to equal 180°, which we just used to prove the vertical angle theorem.

These theories are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the properties 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 test.

  1. Vertically opposite angles are always congruent. Therefore, 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 Use Opposite Angles in Real-World Scenario?

You might speculate where you can utilize these concepts in the real world, and you'd be amazed to notice that vertically opposite angles are very common! You can find them in many daily objects and circumstances.

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

Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Track crossings are also a terrific example of vertically opposite angles.

Eventually, vertically opposite angles are also present 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 detect an example next to you.

Puttingit Together

So, to summarize what we have talked about, vertically opposite angles are made from two intersecting lines. The two angles that are not adjacent have identical measurements.

The vertical angle theorem defines that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line overlapping 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. Nevertheless, they are complementary if the addition of these angles equals 90°. If the addition of both angles equals 180°, they are assumed to be supplementary.

The total 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 discover them in several daily objects and circumstances, such as doors, windows, paintings, and trees.

Additional Study

Look for a vertically opposite angles questionnaire on the internet for examples and exercises to practice. Math is not a spectator sport; keep practicing until these theorems are well-established in your brain.

However, there is no shame if you need extra assistance. If you're struggling to understand vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our professional teachers can help you understand the material and ace your next examination.