Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for anyone who desires to study mathematics or any related subject that uses it. It's tough work, but we'll make sure you get a grip of these theories so you can attain the grade!
Don’t feel dispirited if you don’t recall or don’t comprehend these theories, as this blog will help you understand all the basics. Moreover, we will help you learn the secret to learning faster and enhancing your grades in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they create opposite angles, called vertical angles.
These opposite angles share a vertex. Furthermore, the most essential thing to bear in mind is that they are the same in measurement! This means that regardless of where these straight lines cross, the angles opposite each other will constantly share the exact 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 employing proportions.
Proving the Theorem
Proving this theorem is somewhat easy. First, let's draw a line and call it line l. Then, we will draw another line that intersects line l at some point. We will call this second line m.
After drawing these two lines, we will label the angles created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically contrary because they share the equivalent vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but close to one another. They share a side and a vertex, signifying 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 sums up to equal the same, we can add these operations as follows:
∠A+∠C=∠B+∠C
By eliminating 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 measurement.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk particularly about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles created 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 adjacent. Adjacent angles are two angles that share 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 the 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 utilized in our proof of the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not meet the properties of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, odds are that you will only require these two to ace your examination.
Vertically opposite angles are always congruent. Hence, 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 Use Opposite Angles in Real-World Circumstances?
You might think where you can utilize these concepts in the real life, and you'd be stunned to note that vertically opposite angles are fairly common! You can discover them in many daily things and situations.
For example, vertically opposite angles are created when two straight lines overlap each other. Right in 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. Road junctions are also a wonderful example of vertically opposite angles.
Eventually, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are formed 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 sum up what we have considered so far, vertically opposite angles are formed from two overlapping lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem explains that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line overlapping it and implementing the theorems 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 cannot be vertically opposite. Nevertheless, they are complementary if the addition of these angles totals 90°. If the sum of both angles totals 180°, they are assumed to be 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 quite common! You can locate them in several daily objects and circumstances, such as windows, doors, paintings, and trees.
Further Study
Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Mathematics is not a spectator sport; keep applying until these theorems are ingrained in your mind.
Still, there is nothing humiliating if you require further support. If you're having difficulty to understand vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert instructor can help you grasp the material and nail your following examination.
