Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a important topic for anyone who wants to study arithmetic or any related subject that employs it. It's hard work, but we'll assure you get a handle on these concepts so you can make the grade!
Don’t feel dispirited if you don’t remember or don’t understand these theories, as this blog will help you understand all the basics. Furthermore, we will teach you the secret to learning faster and increasing your scores in arithmetic and other prevailing subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines bisect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most essential thing to bear in mind is that they also measure the same! This applies that irrespective of where these straight lines cross, the angles converse each other will consistently share the exact value. These angles are referred 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 employing proportions.
Proving the Theorem
Proving this theorem is moderately easy. First, let's draw a line and name 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 assume the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically opposite because 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 observe angles B and C, you will notice that they are not linked at their vertex but next to one another. They share a side and a vertex, therefore they are supplementary angles, so the total 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 additions equal the same, we can sum up these operations as follows:
∠A+∠C=∠B+∠C
By canceling out 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 the same measurement.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk particularly regarding vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
Still, vertically opposite angles are never next to each other. 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 sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not satisfy 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 secure your examination.
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 at no time adjacent. They can share, at most, a vertex.
Where Can You Use Opposite Angles in Real-World Scenario?
You might wonder where you can utilize these concepts 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 situations.
For example, vertically opposite angles are formed when two straight lines cross. Right in your room, the door attached to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and modify the size of the angles. Road junctions are also a great example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered in nature. If you look at a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to observe your surroundings, as you will find an example next to you.
Puttingit All Together
So, to sum up what we have talked about, vertically opposite angles are formed from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem defines that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by depicting a straight line and another line intersecting it and applying the concepts of congruent angles to finish measures.
Congruent angles refer to two angles that have identical measurements.
When two angles share a side and a vertex, they cannot be vertically opposite. Nevertheless, they are complementary if the sum of these angles equals 90°. If the addition of both angles totals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can locate them in many daily objects and circumstances, such as doors, windows, paintings, and trees.
Further Study
Look for a vertically opposite angles practice questions on the internet for examples and problems to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are well-established in your head.
Despite that, there is no shame if you require additional help. If you're struggling to comprehend vertical angles (or any other ideas of geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional instructor can help you understand the material and nail your following examination.
