Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for anyone who wants to learn math or any other subject that employs it. It's hard work, but we'll assure you get a handle on these theories so you can make the grade!
Don’t feel disheartened if you don’t remember or don’t comprehend these theories, as this blog will teach you all the essentials. Additionally, we will help you learn the tricks to learning faster and improving your scores in mathematics and other prevailing subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines intersect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most important point to bear in mind is that they are the same in measurement! This applies that irrespective of where these straight lines cross, the angles opposite each other will consistently share the equal 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 relatively easy. First, let's pull a line and name it line l. Then, we will draw another line that goes through line l at some point. We will name this second line m.
After drawing these two lines, we will name the angles formed by the intersecting lines l and m. To prevent confusion, we labeled pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically contrary due to the fact they share the same vertex but don’t share a side. If you recall that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If you observe angles B and C, you will note that they are not linked at their vertex but adjacent 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 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 add these operations as follows:
∠A+∠C=∠B+∠C
By eliminating 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 measurement.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk specifically about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles made by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, vertically opposite angles are no way adjacent. Adjacent angles are two angles that have 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 add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.
These concepts are appropriate within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are various properties of vertically opposite angles. Still, odds are that you will only require these two to secure your examination.
Vertically opposite angles are at all time 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 Locate Opposite Angles in Real-Life Circumstances?
You might speculate where you can find these concepts in the real world, and you'd be amazed to note that vertically opposite angles are very common! You can discover them in various daily things and scenarios.
For example, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door attached to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and alter the size of the angles. Road intersections are also a terrific example of vertically opposite angles.
Eventually, 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 observe your environment, as you will find an example next to you.
Puttingit All Together
So, to sum up what we have discussed, vertically opposite angles are created from two crossover lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem states that in the event of two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line overlapping it and using the theorems of congruent angles to finish 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 totals 90°. If the sum of both angles equals 180°, they are assumed to be supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can locate them in several daily objects and circumstances, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles practice questions online for examples and problems to practice. Mathematics is not a spectator sport; keep practicing until these concepts are well-established in your mind.
Still, there is no shame if you require extra assistance. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our professional instructor can assist you grasp the material and nail your next examination.
