Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a important subject for anyone who desires to study mathematics or any other subject that employs it. It's tough work, but we'll make sure you get a good grasp of these theories so you can achieve the grade!
Don’t feel disheartened if you don’t remember or don’t understand these concepts, as this blog will teach you all the essentials. Additionally, we will teach you the tricks to learning quicker and improving your grades in arithmetic and other common subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most important thing to remember is that they are the same in measurement! 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 feasible to work out the others utilizing proportions.
Proving the Theorem
Proving this theorem is somewhat straightforward. First, let's draw a line and call it line l. Then, we will draw another line that goes through line l at some point. We will call this second line m.
After drawing these two lines, we will assume the angles formed by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite due to the fact they share the equivalent vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A equals angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but adjacent 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 instance 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 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 explicitly 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 one another fulfill the vertical angle theorem.
Still, 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 sum of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized in our proof of the vertical angle theorem.
These theories are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, odds are that you will only require these two to ace your test.
Vertically opposite angles are at all time 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 Locate Opposite Angles in Real-World Scenario?
You may speculate where you can use these theorems in the real life, and you'd be amazed to observe that vertically opposite angles are fairly common! You can discover them in several everyday objects and circumstances.
For instance, vertically opposite angles are created 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 make 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 discovered in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to watch your environment, as you will detect an example next to you.
PuttingEverything Together
So, to summarize 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 explains that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and applying 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 can’t be vertically opposite. Despite that, they are complementary if the sum of these angles equals 90°. If the sum of both angles equals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are very common! You can find them in various everyday objects and scenarios, such as windows, doors, paintings, and trees.
Additional Study
Look for a vertically opposite angles worksheet on the internet for examples and sums to practice. Mathematics 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 having a hard time to grasp vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our professional instructor can guide you understand the topic and nail your next test.
