Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for everyone who wishes to study arithmetic or any related subject that employs it. It's tough work, but we'll assure you get a handle on these concepts so you can achieve the grade!
Don’t feel dispirited if you don’t recall or don’t understand these theories, as this blog will help you understand all the basics. Moreover, we will teach you the secret to learning quicker and increasing your scores in math and other popular subjects today.
The Theorem
The vertical angle theorem states that when two straight lines intersect, they form opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most important point 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 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 discover the others utilizing proportions.
Proving the Theorem
Proving this theorem is moderately easy. Primarily, let's draw a line and label 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 created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Thus, 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 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 note that they are not joined at their vertex but adjacent to each other. They share 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 additions equal the same, we can add these operations as follows:
∠A+∠C=∠B+∠C
By eliminating 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 measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's discuss explicitly regarding vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles made by the intersection of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Still, vertically opposite angles are no way 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 the case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are several characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to ace your exam.
Vertically opposite angles are at all time congruent. Therefore, if angles A and B are vertically opposite, they will measure the same.
Vertically opposite angles are never adjacent. They can share, at most, a vertex.
Where Can You Use Opposite Angles in Real-Life Scenario?
You might speculate where you can utilize these theorems in the real world, and you'd be stunned to observe that vertically opposite angles are very common! You can find them in many daily objects and scenarios.
For instance, vertically opposite angles are formed 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. Road crossings are also a terrific 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 formed by the trunk and the branches.
Be sure to observe your environment, as you will find an example next to you.
Puttingit Together
So, to sum up what we have talked about, vertically opposite angles are formed from two intersecting lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem explains that whenever 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 intersecting 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 can’t be vertically opposite. Nevertheless, they are complementary if the sum of these angles equals 90°. If the sum of both angles equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are very common! You can find them in various everyday objects and circumstances, such as paintings, doors, windows, and trees.
Further Study
Search for a vertically opposite angles practice questions on the internet for examples and sums to practice. Math is not a onlooker sport; keep applying until these concepts are rooted in your brain.
However, there is no problem if you require extra help. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional instructor can guide you grasp the topic and nail your following test.
