Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential topic for everyone who wishes to learn mathematics or any related subject that uses it. It's hard work, but we'll ensure you get a handle on these theories so you can achieve the grade!
Don’t feel disheartened if you don’t recollect or don’t understand these concepts, as this blog will teach you all the fundamentals. Additionally, we will help you learn the tricks to learning quicker and increasing your grades in math and other prevailing subjects today.
The Theorem
The vertical angle theorem expresses that at any time two straight lines intersect, they form opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most essential thing to keep in mind is that they are the same in measurement! This means that irrespective of where these straight lines cross, the angles opposite each other will consistently share the equal 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 simple. Primarily, let's draw a line and name it line l. After that, we will draw another line that goes through line l at some point. We will name this second line m.
After drawing these two lines, we will label the angles formed by the intersecting lines l and m. To prevent confusion, we named 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 opposite because they share the same vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A equals angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but next to one another. They have in common a side and a vertex, therefore 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 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 eliminating 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 have studied about the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
However, vertically opposite angles are at no time next to each other. Adjacent angles are two angles that have 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 add up to 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 theories are applicable within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are various characteristics of vertically opposite angles. But, odds are that you will only need these two to secure your exam.
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 Locate Opposite Angles in Real-World Circumstances?
You may wonder where you can find these theorems in the real life, and you'd be amazed to note that vertically opposite angles are quite common! You can discover them in various everyday things and scenarios.
For example, vertically opposite angles are made 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 create two intersecting lines and adjust the size of the angles. Road crossings are also a great example of vertically opposite angles.
In the end, vertically opposite angles are also discovered in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to watch your environment, 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 made from two overlapping lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem states that in the event of 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 overlapping it and using the theorems 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. However, they are complementary if the addition 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 equal 180°.
Vertically opposite angles are very common! You can discover them in various everyday objects and scenarios, such as windows, doors, paintings, and trees.
Additional Study
Look for a vertically opposite angles worksheet online for examples and exercises to practice. Mathematics is not a spectator sport; keep practicing until these theorems are well-established in your mind.
Still, there is nothing humiliating if you require further assistance. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our expert teachers can assist you comprehend the topic and nail your next examination.
