Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential subject for anyone who wishes to study arithmetic or another subject that employs it. It's tough work, but we'll assure you get a handle on these theories so you can attain the grade!
Don’t feel disheartened if you don’t recall or don’t understand these theories, as this blog will help you study all the fundamentals. Furthermore, we will help you understand the secret to learning quicker and improving your grades in math and other popular subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines bisect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Moreover, the most crucial point to keep in mind is that they are the same in measurement! This means that irrespective of where these straight lines cross, the angles converse each other will consistently 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 employing proportions.
Proving the Theorem
Proving this theorem is relatively easy. Primarily, let's pull a line and label it line l. After that, we will draw another line that goes through line l at some point. We will call this second line m.
After drawing these two lines, we will name the angles created 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 understand that angles A and B are vertically opposite 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 observe angles B and C, you will note that they are not linked at their vertex but adjacent 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 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 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 end 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 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 satisfy the vertical angle theorem.
However, vertically opposite angles are at no time adjacent. 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 addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized in our proof of the vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. But, chances are that you will only need these two to nail your exam.
Vertically opposite angles are always congruent. Consequently, 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 Find Opposite Angles in Real-Life Circumstances?
You may wonder where you can utilize these concepts in the real life, and you'd be amazed to notice that vertically opposite angles are fairly common! You can locate them in various everyday things and situations.
For instance, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door installed 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. Track intersections are also a wonderful example of vertically opposite angles.
In the end, vertically opposite angles are also present in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to observe your surroundings, as you will detect an example next to you.
Puttingit Together
So, to sum up what we have discussed, vertically opposite angles are created from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem explains that whenever two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by depicting a straight line and another line intersecting it and using the concepts of congruent angles to finish 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 sum of these angles totals 90°. If the addition of both angles totals 180°, they are deemed supplementary.
The total of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are quite common! You can find them in several daily objects and circumstances, such as paintings, doors, windows, and trees.
Further Study
Look for a vertically opposite angles practice questions online for examples and sums to practice. Math is not a onlooker sport; keep practicing until these theorems are rooted in your head.
Still, there is nothing humiliating if you require additional support. If you're struggling to understand vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our professional tutors can help you comprehend the material and ace your following test.
