Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial topic for anyone who wants to learn math or any related subject that employs it. It's hard work, but we'll ensure you get a handle on these concepts so you can achieve the grade!
Don’t feel dispirited if you don’t recollect or don’t understand these theories, as this blog will help you understand all the essentials. Additionally, we will help you learn the tricks to learning faster and increasing your scores in arithmetic and other common subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines intersect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most important point to remember 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 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 possible to discover the others employing proportions.
Proving the Theorem
Proving this theorem is relatively easy. Primarily, let's pull a line and label 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 created by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Therefore, we label 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 equivalent vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A is the same as angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but close to one another. They have in common a side and a vertex, signifying 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 way:
∠B+∠C=180 and ∠A+∠C=180
Since both additions equal the same, we can add these operations as follows:
∠A+∠C=∠B+∠C
By canceling out C on both sides of the equation, we will end with:
∠A=∠B
So, we can conclude that vertically opposite angles are congruent, as they have the same measurement.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's discuss specifically about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, vertically opposite angles are no way next to each other. 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 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 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 fulfill the properties of vertically opposite angles.
There are various properties of vertically opposite angles. But, chances are that you will only require these two to nail 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 at no time adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-World Situations?
You may think where you can find these concepts in the real life, and you'd be amazed to notice that vertically opposite angles are fairly common! You can find them in several everyday things and situations.
For instance, vertically opposite angles are created when two straight lines cross. Inside your room, the door attached to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Track junctions are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to observe your environment, as you will detect an example next to you.
PuttingEverything Together
So, to summarize what we have talked about, vertically opposite angles are created from two overlapping lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem states that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line overlapping it and using the concepts of congruent angles to finish measures.
Congruent angles means two angles that measure the same.
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 addition of both angles totals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are very common! You can locate them in several daily objects and scenarios, such as paintings, doors, windows, and trees.
Further Study
Search for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a spectator sport; keep practicing until these concepts are rooted in your head.
Despite that, there is no problem if you require additional help. 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 expert instructor can guide you grasp the material and nail your following test.
