Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential topic for anyone who wishes to study math or any related subject that utilizes it. It's hard work, but we'll make sure you get a handle on these theories so you can attain the grade!
Don’t feel dispirited if you don’t recall or don’t have a good grasp on these concepts, as this blog will help you study all the essentials. Moreover, we will help you understand the secret to learning quicker and increasing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most important point to keep in mind is that they also measure the same! This refers that irrespective of where these straight lines cross, the angles opposite 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 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 intersects 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 labeled pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary reason being that they share the same vertex but don’t share a side. If you recall 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 share a side and a vertex, therefore 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 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 be left with:
∠A=∠B
So, we can say that vertically opposite angles are congruent, as they have identical measurement.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's discuss specifically regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Still, vertically opposite angles are at no time 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 add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized to prove the vertical angle theorem.
These theories are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, chances are that you will only require these two to ace your test.
Vertically opposite angles are always 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 Locate Opposite Angles in Real-World Scenario?
You may speculate where you can find these theorems in the real world, and you'd be surprised to notice that vertically opposite angles are very common! You can find them in several daily things and scenarios.
For example, vertically opposite angles are made when two straight lines cross. Inside your room, the door installed to the door frame makes 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 terrific 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 watch your environment, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have considered so far, vertically opposite angles are made from two intersecting lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem states that whenever two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting it and implementing the concepts of congruent angles to complete measures.
Congruent angles means two angles that have identical measurements.
When two angles share a side and a vertex, they can’t be vertically opposite. Nevertheless, they are complementary if the addition of these angles equals 90°. If the sum of both angles equals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are quite common! You can locate them in various daily objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles questionnaire on the internet for examples and sums to practice. Math is not a onlooker sport; keep practicing until these theorems are rooted in your head.
Despite that, there is no problem if you require further assistance. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), think about 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.
