Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for anyone who desires to study math or another subject that employs it. It's tough work, but we'll make sure you get a good grasp of these concepts so you can achieve the grade!
Don’t feel dispirited if you don’t remember or don’t understand these concepts, as this blog will help you study all the essentials. Moreover, we will teach you the secret to learning quicker and improving your scores in arithmetic and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines bisect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most crucial point to bear in mind is that they are the same in measurement! This refers that irrespective of where these straight lines cross, the angles converse each other will always 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 feasible to find the others employing proportions.
Proving the Theorem
Proving this theorem is moderately straightforward. Primarily, 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 assume 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. Accordingly, 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 equivalent vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A is the same as angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but next to each other. They have in common a side and a vertex, meaning they are supplementary angles, so the total 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 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 be left with:
∠A=∠B
So, we can say that vertically opposite angles are congruent, as they have identical measure.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's discuss particularly about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles created 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 case of complementary angles, the addition of two adjacent angles will equal 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used 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 satisfy the properties of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, odds are that you will only need these two to secure your examination.
Vertically opposite angles are at all time congruent. Hence, 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 world, and you'd be surprised to observe that vertically opposite angles are fairly common! You can locate them in several daily objects and scenarios.
For example, vertically opposite angles are created when two straight lines overlap each other. Back of your room, the door installed to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and alter the size of the angles. Road crossings are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also found 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 find an example next to you.
Puttingit Together
So, to sum up what we have talked about, vertically opposite angles are created from two intersecting lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem defines that whenever two intersecting straight lines, the angles formed 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 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 addition of these angles totals 90°. If the addition 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 at all time add up to 180°.
Vertically opposite angles are quite common! You can discover them in many daily objects and situations, such as doors, windows, paintings, and trees.
Additional Study
Look for a vertically opposite angles worksheet online for examples and problems to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are rooted in your mind.
However, there is no shame if you need extra help. If you're having a hard time to understand vertical angles (or any other ideas of geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional tutors can help you grasp the material and nail your next examination.
