September 20, 2022

Vertical Angles: Theorem, Proof, Vertically Opposite Angles

Studying vertical angles is an essential topic for everyone who wants to master math or any related subject that employs it. It's tough work, but we'll assure you get a grip of these theories so you can attain the grade!

Don’t feel dispirited if you don’t remember or don’t comprehend these concepts, as this blog will help you study all the fundamentals. Additionally, we will teach you the tricks to learning faster and improving your scores in math and other common subjects today.

The Theorem

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

These opposite angles share a vertex. Furthermore, the most essential point to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles converse each other will consistently 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 possible to discover the others employing proportions.

Proving the Theorem

Proving this theorem is moderately straightforward. Primarily, let's draw a line and label it line l. After that, we will pull 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 prevent confusion, we labeled 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 opposite because they share the equivalent 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 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, signifying they are supplementary angles, so the sum of both angles will be 180 degrees. This situation 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:

∠A+∠C=∠B+∠C

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

∠A=∠B

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 discuss particularly regarding vertically opposite angles.

Definition

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

However, vertically opposite angles are never next to each other. 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 case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized in our proof of the vertical angle theorem.

These concepts are applicable 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. Regardless, odds are that you will only need these two to nail your exam.

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

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

You might wonder where you can utilize these theorems in the real life, and you'd be surprised to note that vertically opposite angles are very common! You can discover them in several daily things and scenarios.

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

Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Road crossings are also a great 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 made by the trunk and the branches.

Be sure to notice your environment, as you will find an example next to you.

Puttingit All Together

So, to sum up what we have considered so far, vertically opposite angles are formed from two intersecting lines. The two angles that are not adjacent have identical measurements.

The vertical angle theorem states that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line intersecting it and using 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 cannot be vertically opposite. Nevertheless, they are complementary if the sum of these angles totals 90°. If the sum of both angles equals 180°, they are assumed to be supplementary.

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

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

Further Study

Search for a vertically opposite angles questionnaire on the internet for examples and sums to practice. Math is not a onlooker sport; keep applying until these theorems are well-established in your mind.

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