Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a crucial subject for everyone who wants to study mathematics or any related subject that utilizes it. It's tough work, but we'll assure you get a grip of these theories so you can achieve the grade!
Don’t feel dispirited if you don’t recall or don’t understand these theories, as this blog will teach you all the basics. Moreover, we will teach you the secret to learning faster and improving your grades in arithmetic and other common subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines bisect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Additionally, the most crucial thing to keep in mind is that they are the same in measurement! This refers that irrespective of where these straight lines cross, the angles opposite each other will constantly share the same 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 using proportions.
Proving the Theorem
Proving this theorem is somewhat simple. First, let's draw a line and call it line l. After that, 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 label angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically opposite due to the fact they share the same vertex but don’t share a side. Bear in mind 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 notice that they are not joined at their vertex but adjacent to each other. They have in common 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 additions equal the same, we can sum up these operations as follows:
∠A+∠C=∠B+∠C
By removing C on both sides of the equation, we will end 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 talk particularly regarding vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Despite that, vertically opposite angles are no way 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 sum of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just used in our proof of the vertical angle theorem.
These concepts are relevant within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are several properties of vertically opposite angles. Still, chances are that you will only require these two to nail your test.
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 Use Opposite Angles in Real-World Situations?
You might wonder where you can use these concepts in the real life, and you'd be surprised to observe that vertically opposite angles are quite common! You can locate them in several everyday objects and scenarios.
For example, vertically opposite angles are made when two straight lines overlap each other. Right in your room, the door attached to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and alter the size of the angles. Track crossings are also a great example of vertically opposite angles.
Eventually, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to notice your surroundings, as you will find an example next to you.
Puttingit Together
So, to summarize what we have considered so far, vertically opposite angles are created from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem defines that whenever two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line overlapping it and applying the theorems of congruent angles to complete measures.
Congruent angles means 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 sum of these angles totals 90°. If the sum of both angles equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are quite common! You can locate them in many daily objects and scenarios, such as doors, windows, paintings, and trees.
Further Study
Look for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a spectator sport; keep practicing until these concepts are well-established in your head.
However, there is nothing humiliating if you need further support. If you're struggling to understand vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our professional tutors can guide you grasp the material and nail your next test.
