Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for anyone who wants to learn math or any related subject that employs it. It's hard work, but we'll ensure you get a good grasp of these theories so you can attain the grade!
Don’t feel disheartened if you don’t recall or don’t understand these concepts, as this blog will help you study all the basics. Furthermore, we will teach you the tricks to learning quicker and increasing your scores in math and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they make opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most crucial 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 always 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 work out the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. Primarily, let's pull a line and call 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 assume the angles formed by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary because they share the equivalent vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If you see the angles B and C, you will note that they are not connected at their vertex but close to one another. 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 manner:
∠B+∠C=180 and ∠A+∠C=180
Since both sums up to equal the same, we can sum up these operations as follows:
∠A+∠C=∠B+∠C
By removing 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 identical measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's discuss specifically about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles created by the convergence 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 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 used in our proof of the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several properties of vertically opposite angles. Still, chances are that you will only require these two to nail your exam.
Vertically opposite angles are always congruent. Consequently, 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 Circumstances?
You might think where you can find these theorems in the real life, and you'd be surprised to notice that vertically opposite angles are fairly common! You can discover them in many everyday objects and situations.
For instance, vertically opposite angles are formed when two straight lines cross. Right in your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Track intersections 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 notice your environment, as you will discover an example next to you.
Puttingit All Together
So, to summarize what we have considered so far, vertically opposite angles are formed 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 tried out by depicting a straight line and another line intersecting it and using the concepts 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 totals 90°. If the addition of both angles equals 180°, they are considered 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 very common! You can locate them in several everyday objects and situations, such as doors, windows, paintings, and trees.
Additional Study
Search for a vertically opposite angles worksheet on the internet for examples and problems to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are ingrained in your mind.
Still, there is no problem if you need further assistance. If you're having a hard time to understand vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you comprehend the material and ace your next examination.
