Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential subject for everyone who desires to master math or another subject that uses it. It's hard work, but we'll ensure you get a handle on these theories so you can make the grade!
Don’t feel disheartened if you don’t remember or don’t have a good grasp on these theories, as this blog will help you understand all the fundamentals. Furthermore, we will help you learn the tricks to learning quicker and increasing your grades in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that at any time two straight lines bisect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most important thing to bear 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 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 moderately simple. First, let's pull a line and call it line l. After that, we will draw 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 formed 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 know that angles A and B are vertically contrary because 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 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 one another. They share a side and a vertex, therefore they are supplementary angles, so the sum 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 additions 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 say that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's discuss particularly 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 each other satisfy 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 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 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 relevant 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. Still, chances are that you will only need these two to ace your test.
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 Locate Opposite Angles in Real-World Scenario?
You might wonder where you can use these concepts in the real life, and you'd be surprised to notice that vertically opposite angles are fairly common! You can discover them in various daily things and scenarios.
For instance, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door attached to the door frame creates 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 great 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 observe your surroundings, as you will find an example next to you.
Puttingit Together
So, to sum up what we have considered so far, vertically opposite angles are formed from two crossover lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem defines that whenever two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting it and using the concepts of congruent angles to complete measures.
Congruent angles means two angles that measure the same.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the sum of these angles totals 90°. If the addition of both angles totals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are pretty common! You can discover them in many daily objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles practice questions on the internet for examples and exercises to practice. Math is not a onlooker sport; keep applying until these theorems are rooted in your head.
However, there is no problem if you need extra help. If you're struggling to comprehend vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our skill tutors can assist you comprehend the material and nail your next examination.
