Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential topic for anyone who desires to learn math or any other subject that employs it. It's tough work, but we'll assure you get a handle on these concepts so you can attain the grade!
Don’t feel disheartened if you don’t recall or don’t have a good grasp on these theories, as this blog will help you understand all the fundamentals. Moreover, we will teach you the secret to learning faster and increasing your grades in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines intersect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most important thing to remember is that they are the same in measurement! This applies 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 find the others using proportions.
Proving the Theorem
Proving this theorem is moderately easy. First, let's draw a line and name it line l. After that, we will pull 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 avoid confusion, we named pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:
We are aware 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 the same as 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 share a side and a vertex, signifying 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 additions equal the same, we can sum up 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 have learned about the theorem and how to prove it, let's discuss explicitly regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other 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 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 to prove the vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, chances are that you will only need these two to secure your exam.
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 at no time adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-World Circumstances?
You might wonder where you can use these theorems in the real world, and you'd be amazed to note that vertically opposite angles are very common! You can find them in several daily things and circumstances.
For example, vertically opposite angles are formed when two straight lines overlap each other. Inside your room, the door installed to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Track intersections are also a terrific 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 formed by the trunk and the branches.
Be sure to notice your environment, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have discussed, vertically opposite angles are made from two crossover lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem states that whenever 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 using the concepts of congruent angles to finish 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 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 find them in various everyday objects and circumstances, such as windows, doors, paintings, and trees.
Further Study
Look for a vertically opposite angles worksheet on the internet for examples and exercises to practice. Math is not a onlooker sport; keep practicing until these concepts are ingrained in your head.
Still, there is nothing humiliating if you need further assistance. 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 tutors can guide you grasp the material and ace your next examination.
