Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial subject for everyone who wishes to learn math or any other subject that employs it. It's tough work, but we'll make sure you get a grip of these concepts so you can attain the grade!
Don’t feel disheartened if you don’t recollect or don’t have a good grasp on these theories, as this blog will help you study all the essentials. Moreover, we will help you learn the tricks to learning quicker and enhancing your grades in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem states that when two straight lines intersect, they create opposite angles, called 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 irrespective of where these straight lines cross, the angles opposite each other will consistently share the same value. These angles are referred as congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to discover the others employing proportions.
Proving the Theorem
Proving this theorem is moderately simple. Primarily, let's draw a line and label it line l. Then, we will draw another line that goes through line l at some point. We will assume this second line m.
After drawing these two lines, we will label the angles formed by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically contrary reason being that 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 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 each other. They share a side and a vertex, signifying 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 eliminating 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 identical measure.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's talk particularly about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles made by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
Still, vertically opposite angles are at no time next to each other. Adjacent angles are two angles that share 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 addition will equal 180°, which we just used to prove the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are many properties of vertically opposite angles. But, chances are that you will only require these two to nail your examination.
Vertically opposite angles are at all time 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-Life Scenario?
You might speculate where you can utilize these theorems in the real world, and you'd be amazed to note that vertically opposite angles are very common! You can locate them in many daily things and circumstances.
For instance, vertically opposite angles are formed when two straight lines overlap each other. Right in your room, the door connected to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Track junctions are also a terrific example of vertically opposite angles.
Finally, 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 watch your surroundings, as you will detect an example next to you.
PuttingEverything Together
So, to sum up what we have talked about, vertically opposite angles are formed from two overlapping lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem explains that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line intersecting it and using the concepts 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 equals 90°. If the addition of both angles equals 180°, they are deemed supplementary.
The total 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 pretty common! You can find them in various everyday objects and circumstances, such as windows, doors, paintings, 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 spectator sport; keep practicing until these theorems are well-established in your mind.
However, there is no problem if you require additional assistance. If you're having difficulty to grasp vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our professional teachers can guide you comprehend the topic and ace your next test.
