Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for everyone who desires to master mathematics or another subject that utilizes it. It's tough work, but we'll assure you get a good grasp of these concepts so you can make the grade!
Don’t feel discouraged if you don’t recall or don’t comprehend these theories, as this blog will help you understand all the fundamentals. Furthermore, we will help you understand the tricks to learning faster and enhancing your grades in mathematics and other prevailing subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines meet, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most essential point to keep in mind 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 exact value. These angles are called congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to discover the others using proportions.
Proving the Theorem
Proving this theorem is moderately straightforward. First, let's pull a line and label it line l. Then, we will draw another line that intersects line l at some point. We will assume this second line m.
After drawing these two lines, we will label the angles formed by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Accordingly, we label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite because 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 is the same as angle B.
If you see the angles B and C, you will note that they are not connected at their vertex but close to one another. They share a side and a vertex, signifying 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 sum up these operations as follows:
∠A+∠C=∠B+∠C
By removing C on both sides of the equation, we will end with:
∠A=∠B
So, we can conclude that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's discuss explicitly about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
Despite that, vertically opposite angles are at no time 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 case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used to prove 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 properties of vertically opposite angles.
There are several characteristics of vertically opposite angles. But, odds are that you will only need 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 never adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-Life Circumstances?
You may wonder where you can find these concepts in the real world, and you'd be stunned to notice that vertically opposite angles are quite common! You can discover them in many daily things and situations.
For instance, vertically opposite angles are formed when two straight lines overlap each other. Back of your room, the door connected to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Road intersections are also a terrific example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to watch your surroundings, as you will discover an example next to you.
Puttingit All Together
So, to summarize what we have discussed, vertically opposite angles are formed from two intersecting lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem states that whenever two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by drawing a straight line and another line intersecting it and implementing the theorems 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 addition of these angles equals 90°. If the addition of both angles equals 180°, they are deemed supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are pretty common! You can find them in several daily objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles worksheet on the internet for examples and sums to practice. Math is not a onlooker sport; keep applying until these concepts are well-established in your brain.
Still, there is no shame if you require further help. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill tutors can assist you grasp the material and ace your next examination.
