Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for anyone who wishes to learn mathematics or another subject that uses 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 disheartened if you don’t recall or don’t have a good grasp on these theories, as this blog will help you understand all the basics. Additionally, we will teach you the secret to learning quicker and improving your scores in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that at any time two straight lines bisect, they make opposite angles, called vertical angles.
These opposite angles share a vertex. Moreover, the most crucial point to remember is that they also measure the same! This applies that regardless of where these straight lines cross, the angles converse each other will always 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 possible to discover the others utilizing proportions.
Proving the Theorem
Proving this theorem is moderately easy. Primarily, 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 name this second line m.
After drawing these two lines, we will name the angles created by the intersecting lines l and m. To avoid 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 reason being that 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 adjacent 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 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 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 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 specifically about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
However, 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 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 to prove the vertical angle theorem.
These concepts are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several characteristics of vertically opposite angles. But, odds are that you will only require these two to secure your test.
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 never adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-World Scenario?
You might speculate where you can use these theorems in the real world, and you'd be stunned to note that vertically opposite angles are very common! You can find them in various everyday objects and circumstances.
For instance, vertically opposite angles are created when two straight lines cross. Right in your room, the door attached to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and adjust the size of the angles. Road crossings are also a wonderful example of vertically opposite angles.
In the end, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to watch your environment, as you will find an example next to you.
Puttingit All Together
So, to sum up what we have considered so far, vertically opposite angles are formed from two crossover lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem explains that when two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and using the concepts 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 can’t be vertically opposite. Nevertheless, they are complementary if the sum of these angles totals 90°. If the addition of both angles totals 180°, they are deemed supplementary.
The total of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will at all time 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.
Further Study
Search for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a onlooker sport; keep practicing until these theorems are ingrained in your head.
Still, there is no problem if you require additional help. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our professional teachers can help you comprehend the topic and nail your next examination.
