Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for everyone who wants to master math or another subject that utilizes it. It's hard work, but we'll assure you get a grip of these theories 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 understand all the basics. Additionally, we will teach you the tricks to learning faster and increasing your scores in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they form opposite angles, called vertical angles.
These opposite angles share a vertex. Moreover, the most essential 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 always 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 somewhat straightforward. Primarily, let's draw a line and call it line l. Then, we will draw another line that intersects line l at some point. We will call this second line m.
After drawing these two lines, we will label the angles created by the intersecting lines l and m. To avoid confusion, we labeled 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 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 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 close to each other. They have in common a side and a vertex, therefore 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 add 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 conclude 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 specifically regarding vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, 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 equal 90°. Supplementary angles are adjacent angles which will add up to 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 several characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to secure 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 Find Opposite Angles in Real-World Circumstances?
You may think where you can find these theorems in the real life, and you'd be surprised to notice that vertically opposite angles are very common! You can discover them in various everyday objects 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 produce two intersecting lines and alter the size of the angles. Track junctions are also a terrific example of vertically opposite angles.
In the end, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to notice your environment, as you will discover an example next to you.
PuttingEverything Together
So, to summarize what we have discussed, vertically opposite angles are formed from two overlapping lines. The two angles that are not adjacent 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 tested by drawing a straight line and another line intersecting it and implementing the theorems of congruent angles to finish measures.
Congruent angles refer to two angles that have identical measurements.
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 addition of both angles equals 180°, they are deemed supplementary.
The total of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are pretty 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 on the internet for examples and exercises to practice. Math is not a spectator sport; keep practicing until these theorems are ingrained in your brain.
However, there is nothing humiliating if you need further help. If you're struggling to understand vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our skill teachers can assist you comprehend the topic and ace your next test.
