Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a crucial topic for anyone who wishes to study math or another subject that employs it. It's hard work, but we'll assure you get a grip of these theories so you can make the grade!
Don’t feel disheartened if you don’t recollect or don’t comprehend these theories, as this blog will help you study all the essentials. Moreover, we will help you understand the tricks to learning faster and enhancing your grades in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Moreover, the most essential thing to bear in mind is that they are the same in measurement! This applies that irrespective of where these straight lines cross, the angles converse each other will consistently share the equal 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 using proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. First, let's draw a line and call it line l. Then, we will pull another line that intersects line l at some point. We will name 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. Thus, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically opposite reason being that they share the equivalent vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A equals angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but close to each other. They share a side and a vertex, signifying they are supplementary angles, so the total 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 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 end with:
∠A=∠B
So, we can say that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's discuss specifically regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
Still, vertically opposite angles are never 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 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 concepts are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Regardless, chances are that you will only require these two to ace your examination.
Vertically opposite angles are always congruent. Hence, 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-World Circumstances?
You might speculate where you can use these theorems in the real world, and you'd be surprised 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 created when two straight lines overlap each other. Back of your room, the door installed to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and alter the size of the angles. Road crossings are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to notice your surroundings, as you will find an example next to you.
PuttingEverything 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 explains that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line overlapping it and implementing the theorems 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. However, they are complementary if the addition of these angles totals 90°. If the sum of both angles equals 180°, they are considered supplementary.
The sum of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will always 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.
Additional Study
Look for a vertically opposite angles questionnaire on the internet for examples and sums to practice. Math is not a spectator sport; keep applying until these theorems are ingrained in your brain.
Still, there is nothing humiliating if you need additional support. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you understand the material and ace your next test.
