Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for everyone who desires to learn arithmetic or any other subject that employs it. It's hard work, but we'll assure you get a handle on these concepts so you can attain the grade!
Don’t feel dispirited if you don’t recall or don’t have a good grasp on these theories, as this blog will help you understand all the basics. Moreover, we will help you understand the secret to learning faster and increasing your scores in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem states that when two straight lines meet, they create opposite angles, called vertical angles.
These opposite angles share a vertex. Additionally, the most essential point to keep in mind is that they are the same in measurement! This applies that regardless of where these straight lines cross, the angles opposite each other will always 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 possible to find the others utilizing 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 name this second line m.
After drawing these two lines, we will assume the angles formed by the intersecting lines l and m. To prevent confusion, we labeled 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 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 equals 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 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 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 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 the same measurement.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's talk specifically 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.
However, 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 sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose sum 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 meet the properties of vertically opposite angles.
There are many properties of vertically opposite angles. Still, odds are that you will only need these two to secure your exam.
Vertically opposite angles are always congruent. Consequently, 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 Situations?
You may wonder where you can find these concepts in the real world, and you'd be surprised to note that vertically opposite angles are quite common! You can discover them in many daily objects and scenarios.
For example, vertically opposite angles are made when two straight lines cross. Inside your room, the door attached to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and adjust the size of the angles. Track crossings are also a terrific example of vertically opposite angles.
Finally, vertically opposite angles are also present in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to notice your surroundings, as you will discover an example next to you.
PuttingEverything 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 explains that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line overlapping it and implementing the theorems 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 cannot be vertically opposite. However, 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 always 180°. Consequently, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are very common! You can discover them in many daily objects and situations, such as doors, windows, paintings, and trees.
Further Study
Search 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 concepts are ingrained in your mind.
Despite that, there is no problem if you require additional support. If you're struggling to comprehend vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our expert instructor can assist you comprehend the topic and nail your following test.
