Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a important subject for anyone who desires to master mathematics or any related subject that employs it. It's tough work, but we'll make sure you get a grip of these concepts so you can make the grade!
Don’t feel dispirited if you don’t recollect or don’t have a good grasp on these concepts, as this blog will help you study all the fundamentals. Furthermore, we will help you understand the secret to learning quicker and improving your grades in arithmetic and other prevailing subjects today.
The Theorem
The vertical angle theorem states that when two straight lines intersect, they form opposite angles, known as vertical angles.
These opposite angles share a vertex. Additionally, the most essential thing 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 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 possible to find the others employing proportions.
Proving the Theorem
Proving this theorem is relatively simple. Primarily, let's draw 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 label the angles created by the intersecting lines l and m. To prevent confusion, we named pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically contrary 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 equals angle B.
If you see the angles B and C, you will note 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 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 be left with:
∠A=∠B
So, we can say that vertically opposite angles are congruent, as they have identical measurement.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's talk particularly 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 one another fulfill the vertical angle theorem.
However, vertically opposite angles are no way 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 whose addition will equal 180°, which we just used in our proof of the vertical angle theorem.
These concepts are appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the properties of vertically opposite angles.
There are many properties of vertically opposite angles. Regardless, chances are that you will only require these two to secure your test.
Vertically opposite angles are always congruent. Hence, 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 Situations?
You may wonder where you can use these concepts in the real life, and you'd be stunned to note that vertically opposite angles are very common! You can discover them in several daily things and circumstances.
For instance, vertically opposite angles are created when two straight lines cross. Right in 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 modify the size of the angles. Road intersections are also a great example of vertically opposite angles.
Eventually, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to observe your surroundings, as you will detect an example next to you.
Puttingit Together
So, to sum up what we have talked about, vertically opposite angles are formed from two overlapping lines. The two angles that are not next to each other have the same measure.
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 tried out by drawing a straight line and another line intersecting it and implementing the concepts of congruent angles to complete measures.
Congruent angles refer to two angles that measure the same.
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 sum of both angles equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are pretty common! You can discover them in many everyday objects and scenarios, such as doors, windows, paintings, and trees.
Additional Study
Search for a vertically opposite angles questionnaire on the internet for examples and exercises to practice. Mathematics is not a onlooker sport; keep applying until these theorems are well-established in your mind.
However, there is no shame if you need extra help. If you're having a hard time to grasp vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our expert teachers can guide you comprehend the material and nail your following examination.
