Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial subject for everyone who wishes to study mathematics or another subject that uses it. It's tough work, but we'll ensure you get a grip of these theories so you can achieve the grade!
Don’t feel discouraged if you don’t recollect or don’t comprehend these theories, as this blog will help you study all the fundamentals. Additionally, we will help you understand the secret to learning quicker and enhancing your scores in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial thing to keep in mind is that they are the same in measurement! This applies that irrespective of where these straight lines cross, the angles opposite each other will constantly share the exact value. These angles are called congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is possible to work out the others using proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. Primarily, let's pull a line and name it line l. Then, 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 prevent confusion, we labeled pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically opposite due to the fact 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 identical angle B.
If you observe angles B and C, you will notice that they are not linked at their vertex but adjacent 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 case 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 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 conclude that vertically opposite angles are congruent, as they have identical 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 formed by the intersection of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Despite that, 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 sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose addition 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 characteristics of vertically opposite angles.
There are several characteristics of vertically opposite angles. Regardless, chances are that you will only require these two to nail your test.
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 Locate Opposite Angles in Real-Life Situations?
You might wonder where you can use these concepts in the real world, and you'd be amazed to note that vertically opposite angles are fairly common! You can discover them in various everyday things and situations.
For instance, vertically opposite angles are formed when two straight lines overlap each other. Right in your room, the door installed to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Track intersections are also a great example of vertically opposite angles.
In the end, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to watch your environment, as you will detect 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 adjacent have identical measurements.
The vertical angle theorem states that when 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 applying 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. Despite that, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are assumed to be supplementary.
The sum 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 pretty common! You can find them in many daily objects and circumstances, such as windows, doors, paintings, and trees.
Additional Study
Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Mathematics is not a spectator sport; keep applying until these concepts are rooted in your brain.
Despite that, there is no shame if you require extra support. If you're struggling to understand vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional teachers can guide you grasp the topic and nail your following examination.
