Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who wants to study mathematics or any related subject that employs it. It's hard work, but we'll make sure you get a handle on these concepts so you can attain the grade!
Don’t feel disheartened 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 teach you the secret to learning quicker and increasing your grades in arithmetic and other common subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines meet, they create opposite angles, called vertical angles.
These opposite angles share a vertex. Moreover, the most essential thing to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles converse each other will consistently share the equal 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 feasible to find the others using proportions.
Proving the Theorem
Proving this theorem is relatively easy. First, let's draw a line and name it line l. Then, we will pull another line that goes through line l at some point. We will call this second line m.
After drawing these two lines, we will name 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 understand that angles A and B are vertically contrary 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 identical angle B.
If you see the angles B and C, you will note that they are not connected at their vertex but close to each other. They have in common a side and a vertex, signifying they are supplementary angles, so the sum of both angles will be 180 degrees. This scenario 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 canceling out 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 measurement.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Still, vertically opposite angles are never 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 total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just used in our proof of the vertical angle theorem.
These theories are relevant within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are various characteristics of vertically opposite angles. Still, odds are that you will only need these two to nail your examination.
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 Locate Opposite Angles in Real-Life Situations?
You might wonder where you can find these concepts in the real life, and you'd be amazed to observe that vertically opposite angles are fairly common! You can discover them in many everyday things and situations.
For example, vertically opposite angles are formed when two straight lines cross. Back of your room, the door attached to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Track intersections are also a terrific example of vertically opposite angles.
Eventually, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to observe your environment, as you will find an example next to you.
PuttingEverything Together
So, to summarize what we have talked about, vertically opposite angles are made from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem states that in the event of two intersecting straight lines, the angles formed 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 means two angles that have identical measurements.
When two angles share a side and a vertex, they cannot be vertically opposite. However, 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 add up to 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 practice questions online for examples and problems to practice. Math is not a spectator sport; keep practicing until these concepts are ingrained in your mind.
However, there is no problem if you require additional assistance. If you're having a hard time to grasp vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our professional teachers can assist you understand the material and nail your following test.
