Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a important topic for everyone who desires to master arithmetic or another subject that uses it. It's hard work, but we'll assure you get a grip of these concepts so you can make the grade!
Don’t feel dispirited if you don’t remember or don’t have a good grasp on these concepts, as this blog will help you study all the fundamentals. Moreover, we will help you understand the secret to learning quicker and increasing your grades in math and other prevailing subjects today.
The Theorem
The vertical angle theorem states that when two straight lines meet, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most important point to bear in mind is that they also measure the same! This refers that regardless of where these straight lines cross, the angles opposite each other will consistently share the same value. These angles are known 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 relatively straightforward. Primarily, let's pull a line and call it line l. After that, we will pull another line that goes through line l at some point. We will name 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 label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically contrary due to the fact 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 is the same as angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but close 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 scenario 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 sums up to equal the same, we can add 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 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 particularly about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Despite that, vertically opposite angles are no way 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 whose sum 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 since supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several properties of vertically opposite angles. Regardless, chances are that you will only require these two to ace your test.
Vertically opposite angles are at all time 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 Use Opposite Angles in Real-Life Situations?
You might think where you can find these theorems in the real life, and you'd be amazed to note that vertically opposite angles are fairly common! You can find them in several daily things and situations.
For instance, vertically opposite angles are formed when two straight lines overlap each other. Inside your room, the door installed to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and modify the size of the angles. Track junctions are also a wonderful example of vertically opposite angles.
Eventually, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to notice your environment, as you will find an example next to you.
PuttingEverything Together
So, to summarize what we have discussed, vertically opposite angles are created from two intersecting lines. The two angles that are not next to each other 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 tried out by drawing a straight line and another line overlapping it and implementing the theorems of congruent angles to finish measures.
Congruent angles refer to two angles that measure the same.
When two angles share a side and a vertex, they cannot be vertically opposite. However, they are complementary if the sum of these angles totals 90°. If the sum of both angles totals 180°, they are deemed supplementary.
The total of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are quite common! You can locate them in various daily objects and circumstances, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles questionnaire on the internet for examples and exercises to practice. Math is not a spectator sport; keep applying until these concepts are rooted in your mind.
However, there is no problem if you need extra assistance. If you're having a hard time to grasp vertical angles (or any other ideas of geometry), think about signing up for a tutoring session with Grade Potential. One of our skill teachers can help you understand the topic and ace your next examination.
