Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a important topic for everyone who desires to study mathematics or another subject that employs it. It's tough work, but we'll make sure you get a good grasp of these concepts so you can make the grade!
Don’t feel discouraged if you don’t remember or don’t understand these theories, as this blog will help you understand all the fundamentals. Additionally, we will help you understand the secret to learning faster and increasing your scores in math and other popular subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines meet, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most crucial point to bear in mind is that they also measure the same! This means that irrespective of where these straight lines cross, the angles converse each other will constantly share the equal value. These angles are called congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to work out the others using proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. Primarily, let's draw a line and call it line l. After that, 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 name the angles created by the intersecting lines l and m. To prevent confusion, we named 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 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 equals angle B.
If we look at angles B and C, you will note that they are not joined at their vertex but adjacent to one another. They have in common a side and a vertex, therefore they are supplementary angles, so the sum 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 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 identical measurement.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's talk explicitly about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, vertically opposite angles are no way next to each other. 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 addition 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 appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are various characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to nail your test.
Vertically opposite angles are at all time congruent. Therefore, 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-World Situations?
You may wonder where you can find these theorems in the real life, and you'd be stunned to notice that vertically opposite angles are fairly common! You can find them in several daily objects and scenarios.
For instance, vertically opposite angles are created when two straight lines cross. Inside your room, the door attached to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Road intersections are also a terrific example of vertically opposite angles.
In the end, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to observe your environment, as you will find an example next to you.
Puttingit Together
So, to sum up what we have considered so far, vertically opposite angles are made from two overlapping lines. The two angles that are not adjacent have the same measure.
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 applying the theorems of congruent angles to complete 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. However, they are complementary if the addition of these angles totals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can locate them in several daily objects and circumstances, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles practice questions on the internet for examples and sums to practice. Math is not a spectator sport; keep practicing until these concepts are well-established in your brain.
Despite that, there is nothing humiliating if you require additional support. If you're having a hard time to understand vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill tutors can assist you comprehend the topic and nail your following test.
