Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a important topic for anyone who wants to study math or another subject that employs it. It's hard work, but we'll ensure you get a grip of these theories so you can make the grade!
Don’t feel disheartened if you don’t recall or don’t understand these concepts, as this blog will help you study all the basics. Furthermore, we will help you learn the secret to learning faster and improving your grades in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines intersect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most important point to remember 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 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 possible to work out the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively 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 avoid 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 due to the fact 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 equals angle B.
If you observe angles B and C, you will notice that they are not linked at their vertex but close to each other. They share a side and a vertex, therefore they are supplementary angles, so the total of both angles will be 180 degrees. This situation 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 conclude 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 discuss explicitly regarding vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
However, vertically opposite angles are no way next to each other. Adjacent angles are two angles that have 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 add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized in our proof of the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are various properties of vertically opposite angles. Regardless, chances are that you will only need these two to nail your exam.
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 never adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-Life Situations?
You might wonder where you can utilize these theorems in the real life, and you'd be amazed to notice that vertically opposite angles are quite common! You can find them in many everyday things and scenarios.
For instance, vertically opposite angles are made when two straight lines overlap each other. Right in your room, the door attached to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Track crossings are also a great example of vertically opposite angles.
Finally, 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 detect an example next to you.
Puttingit All Together
So, to sum up what we have considered so far, vertically opposite angles are made from two crossover lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem defines that in the event of 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 implementing the theorems of congruent angles to finish measures.
Congruent angles means two angles that have identical measurements.
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 sum of both angles totals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are pretty common! You can discover them in various daily objects and situations, such as doors, windows, paintings, and trees.
Additional Study
Look for a vertically opposite angles questionnaire online for examples and sums to practice. Mathematics is not a spectator sport; keep applying until these theorems are ingrained in your head.
However, there is nothing humiliating if you need extra assistance. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our skill teachers can help you grasp the material and ace your following test.
