Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a important subject for everyone who wishes to study math or another subject that employs it. It's tough work, but we'll make sure you get a grip of these theories so you can make the grade!
Don’t feel discouraged if you don’t recall or don’t have a good grasp on these theories, as this blog will help you study all the essentials. Moreover, we will teach you the secret to learning quicker and improving your grades in mathematics and other common 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. Additionally, the most important point to keep in mind is that they also measure the same! This applies that regardless of where these straight lines cross, the angles converse each other will always share the equal 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 find the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. First, let's pull a line and call it line l. Then, we will pull another line that intersects 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 avoid confusion, we labeled pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:
We know 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 identical angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but adjacent to each other. They share 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 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 removing 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 measure.
Vertically Opposite Angles
Now that we have learned 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 formed by the intersection of two straight lines. These angles opposite one another fulfill 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 never 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 total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.
These concepts are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are various characteristics of vertically opposite angles. Regardless, chances are that you will only need these two to ace your examination.
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-World Situations?
You might wonder where you can utilize these theorems in the real world, and you'd be surprised to notice that vertically opposite angles are very common! You can locate them in various daily things and situations.
For instance, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Track crossings are also a terrific example of vertically opposite angles.
Eventually, 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 notice your surroundings, as you will detect an example next to you.
Puttingit All Together
So, to summarize what we have discussed, vertically opposite angles are made from two intersecting 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 proven by depicting a straight line and another line overlapping it and applying the theorems of congruent angles to complete 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 totals 90°. If the addition of both angles equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are quite common! You can locate them in several everyday objects and situations, such as doors, windows, paintings, and trees.
Additional Study
Look for a vertically opposite angles practice questions on the internet for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are rooted in your brain.
Still, there is no shame if you need extra help. 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 tutors can help you comprehend the topic and ace your next examination.
