Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who wishes to learn arithmetic or another subject that uses it. It's hard work, but we'll make sure you get a handle on these concepts so you can make 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 understand all the basics. Moreover, we will help you learn the secret to learning faster and improving your grades in math and other common subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines bisect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most crucial thing to keep in mind is that they also measure the same! This refers that irrespective of where these straight lines cross, the angles opposite each other will always 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 find the others utilizing proportions.
Proving the Theorem
Proving this theorem is relatively simple. First, let's pull a line and name it line l. Then, we will draw another line that intersects line l at some point. We will assume 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 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 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 is the same as angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but next to each other. They share a side and a vertex, meaning 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 way:
∠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 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 the same measurement.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's talk explicitly regarding vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
Despite that, 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 equal 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.
These theories are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are various characteristics of vertically opposite angles. But, chances are that you will only need these two to ace your examination.
Vertically opposite angles are always 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 Find Opposite Angles in Real-World Situations?
You may speculate where you can use these theorems in the real life, and you'd be surprised to note that vertically opposite angles are very common! You can discover them in various daily things and circumstances.
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 creates vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and modify the size of the angles. Track crossings are also a terrific example of vertically opposite angles.
Eventually, 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 surroundings, as you will discover an example next to you.
Puttingit All Together
So, to summarize what we have discussed, vertically opposite angles are made from two intersecting lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem defines that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line overlapping it and using the concepts of congruent angles to finish measures.
Congruent angles refer to two angles that measure the same.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are quite common! You can locate them in many everyday objects and scenarios, such as doors, windows, paintings, and trees.
Additional Study
Search for a vertically opposite angles questionnaire on the internet for examples and problems to practice. Mathematics is not a spectator sport; keep applying until these concepts are well-established in your mind.
However, there is no problem if you require extra support. If you're struggling to grasp vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill instructor can guide you grasp the topic and ace your following test.
