Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial topic for anyone who wishes to study arithmetic or any related subject that employs it. It's hard work, but we'll ensure you get a good grasp of these concepts so you can attain the grade!
Don’t feel dispirited if you don’t remember or don’t have a good grasp on these theories, as this blog will help you study all the basics. Furthermore, we will help you learn the secret to learning faster and improving your scores in math and other prevailing subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines meet, they form opposite angles, known as vertical angles.
These opposite angles share a vertex. Additionally, the most essential 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 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 somewhat straightforward. Primarily, let's draw a line and label it line l. Then, we will pull another line that intersects line l at some point. We will name 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 named 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 same vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If you observe angles B and C, you will note that they are not linked at their vertex but close 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 instance 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 sum up these operations as follows:
∠A+∠C=∠B+∠C
By canceling out 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 learned about the theorem and how to prove it, let's discuss explicitly about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Still, vertically opposite angles are never 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 case of complementary angles, the sum of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used in our proof of the vertical angle theorem.
These concepts are relevant within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Regardless, chances are that you will only need these two to nail your examination.
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-World Scenario?
You might speculate where you can find these concepts in the real world, and you'd be amazed to note that vertically opposite angles are very common! You can discover them in several everyday things and situations.
For example, vertically opposite angles are formed when two straight lines cross. Back of your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and adjust the size of the angles. Track intersections are also a wonderful example of vertically opposite angles.
Eventually, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to watch your environment, as you will detect an example next to you.
PuttingEverything Together
So, to sum up what we have considered so far, 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 defines that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by depicting a straight line and another line intersecting it and implementing 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 can’t be vertically opposite. Despite that, they are complementary if the sum 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°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are pretty common! You can locate them in various daily objects and situations, such as windows, doors, paintings, and trees.
Additional Study
Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are ingrained in your mind.
Still, there is nothing humiliating if you need further help. If you're having difficulty to understand vertical angles (or any other ideas of geometry), think about signing up for a tutoring session with Grade Potential. One of our expert tutors can guide you understand the material and ace your following test.
