Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial subject for anyone who desires to study arithmetic or another subject that employs it. It's tough work, but we'll make sure you get a grip of these concepts so you can achieve the grade!
Don’t feel dispirited if you don’t recall or don’t understand these theories, as this blog will help you understand all the basics. Furthermore, we will help you understand the tricks to learning faster and improving your scores in arithmetic and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines bisect, they make opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most important thing to keep in mind is that they are the same in measurement! This refers that regardless of where these straight lines cross, the angles opposite each other will consistently share the exact 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 discover the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. Primarily, let's pull a line and label it line l. After that, we will draw 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 formed by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite reason being that 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 equals 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 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 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 measure.
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 said earlier, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
Despite that, vertically opposite angles are never 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 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 in our proof of the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several characteristics of vertically opposite angles. Regardless, odds are that you will only require these two to secure your test.
Vertically opposite angles are at all time congruent. Hence, 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 Use Opposite Angles in Real-World Situations?
You might speculate where you can utilize these concepts in the real life, and you'd be amazed to note that vertically opposite angles are fairly common! You can locate them in various everyday things and scenarios.
For instance, vertically opposite angles are created 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 make two intersecting lines and adjust the size of the angles. Track intersections are also a great example of vertically opposite angles.
In the end, 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 observe your environment, as you will discover an example next to you.
Puttingit All Together
So, to sum up what we have talked about, vertically opposite angles are made from two crossover lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem states that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line overlapping it and using the theorems of congruent angles to finish measures.
Congruent angles refer to two angles that have identical measurements.
When two angles share a side and a vertex, they can’t be vertically opposite. Nevertheless, they are complementary if the addition of these angles equals 90°. If the sum of both angles equals 180°, they are deemed supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are pretty common! You can find them in several everyday objects and situations, such as paintings, doors, windows, and trees.
Further Study
Look for a vertically opposite angles questionnaire online for examples and sums to practice. Math is not a spectator sport; keep applying until these theorems are ingrained in your mind.
Still, there is no problem if you need additional assistance. If you're struggling to understand vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our professional instructor can help you comprehend the topic and nail your next test.
