Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a important topic for everyone who wishes to learn mathematics or any other subject that uses it. It's tough work, but we'll ensure you get a grip of these theories so you can make the grade!
Don’t feel discouraged if you don’t remember or don’t have a good grasp on these concepts, as this blog will help you study all the essentials. Furthermore, we will teach you the tricks to learning faster and increasing your grades in math and other common subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines bisect, they make opposite angles, called vertical angles.
These opposite angles share a vertex. Furthermore, the most essential thing to keep in mind is that they also measure the same! This means that regardless of where these straight lines cross, the angles opposite each other will consistently share the exact 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 possible to discover the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively easy. First, let's pull a line and label it line l. After that, we will draw another line that goes through line l at some point. We will name this second line m.
After drawing these two lines, we will assume the angles created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically opposite because they share the same 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 we look at angles B and C, you will notice that they are not joined at their vertex but close to each other. They have in common a side and a vertex, therefore they are supplementary angles, so the total of both angles will be 180 degrees. This case 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 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 the same measure.
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 stated, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Despite that, 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 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 addition will equal 180°, which we just used to prove 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 properties of vertically opposite angles.
There are several characteristics of vertically opposite angles. Still, chances are that you will only need these two to nail your exam.
Vertically opposite angles are always 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 Use Opposite Angles in Real-Life Scenario?
You may wonder where you can utilize these theorems in the real world, and you'd be surprised to notice that vertically opposite angles are fairly common! You can locate them in several everyday objects and situations.
For example, vertically opposite angles are made when two straight lines overlap each other. Back of 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 adjust the size of the angles. Track intersections are also a terrific example of vertically opposite angles.
Finally, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to observe your surroundings, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have discussed, vertically opposite angles are created from two crossover lines. The two angles that are not next to each other have identical measurements.
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 drawing a straight line and another line overlapping it and implementing 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 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 considered supplementary.
The total of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are quite common! You can find them in several daily objects and circumstances, such as doors, windows, paintings, and trees.
Further Study
Look for a vertically opposite angles worksheet 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 no problem if you require extra support. If you're struggling to grasp vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our skill instructor can assist you understand the material and nail your next test.
