Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for anyone who wishes to learn mathematics or any related subject that uses 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 dispirited if you don’t recollect or don’t comprehend these concepts, as this blog will help you study all the essentials. Additionally, we will teach you the secret to learning faster and enhancing your scores in math 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. Additionally, the most crucial thing to remember is that they are the same in measurement! This means that regardless of where these straight lines cross, the angles converse each other will consistently share the same 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 find the others employing proportions.
Proving the Theorem
Proving this theorem is somewhat straightforward. Primarily, 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 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 named 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 note that they are not connected at their vertex but adjacent to each other. They share a side and a vertex, therefore they are supplementary angles, so the sum 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 way:
∠B+∠C=180 and ∠A+∠C=180
Since both sums up to 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 have studied about the theorem and how to prove it, let's discuss particularly about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
However, vertically opposite angles are at no time adjacent. 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 the case of complementary angles, the sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used in our proof of the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are several properties of vertically opposite angles. But, chances are that you will only need these two to ace your examination.
Vertically opposite angles are always congruent. Therefore, 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 Use Opposite Angles in Real-Life Situations?
You may think where you can find these concepts in the real life, and you'd be stunned to observe that vertically opposite angles are fairly common! You can discover them in various everyday objects and circumstances.
For example, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door connected to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and modify the size of the angles. Track intersections are also a great example of vertically opposite angles.
In the end, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to watch your environment, as you will detect an example next to you.
Puttingit Together
So, to summarize what we have discussed, vertically opposite angles are formed from two crossover lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem defines that when 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 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 can’t be vertically opposite. However, they are complementary if the sum of these angles equals 90°. If the sum of both angles totals 180°, they are deemed supplementary.
The sum of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are very common! You can discover them in several daily objects and scenarios, such as doors, windows, paintings, and trees.
Additional Study
Look for a vertically opposite angles practice questions online for examples and exercises to practice. Math is not a spectator sport; keep practicing until these theorems are ingrained in your mind.
Still, there is no problem if you require additional support. If you're having a hard time to understand vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional teachers can assist you understand the topic and nail your next examination.
