Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for everyone who desires to master math or any other subject that employs it. It's tough work, but we'll assure you get a handle on these theories so you can achieve 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 essentials. Furthermore, we will help you learn the secret to learning quicker and increasing your grades in math and other common subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they form opposite angles, called vertical angles.
These opposite angles share a vertex. Additionally, the most important thing to remember is that they also measure the same! This refers that regardless of where these straight lines cross, the angles converse each other will consistently share the same 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 using proportions.
Proving the Theorem
Proving this theorem is moderately easy. First, let's draw 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 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 know that angles A and B are vertically opposite due to the fact they share the same vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If we look at angles B and C, you will note that they are not joined at their vertex but next 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 sums up to equal the same, we can sum up 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 identical measure.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's discuss 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 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 at no time share a side. When angles share a side, these adjacent angles could be complementary or supplementary.
In case of complementary angles, the sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles since 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 require these two to ace your examination.
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 never adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-Life Scenario?
You may think where you can find these concepts in the real life, and you'd be surprised to observe that vertically opposite angles are quite common! You can find them in various everyday things and scenarios.
For example, vertically opposite angles are created when two straight lines overlap each other. Right in your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Road crossings are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to watch your environment, as you will discover an example next to you.
Puttingit Together
So, to summarize what we have talked about, vertically opposite angles are created from two crossover lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem defines that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line overlapping it and applying the concepts 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 can’t be vertically opposite. Despite that, they are complementary if the sum of these angles equals 90°. If the addition of both angles equals 180°, they are considered supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can locate them in many everyday objects and circumstances, such as windows, doors, paintings, and trees.
Additional Study
Search for a vertically opposite angles practice questions on the internet for examples and problems to practice. Math is not a onlooker sport; keep applying until these concepts are rooted in your mind.
However, there is nothing humiliating if you need further support. If you're struggling to comprehend vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our expert instructor can guide you comprehend the material and ace your next examination.
