Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a crucial subject for anyone who wants to master mathematics or any other subject that uses it. It's hard work, but we'll make sure you get a grip of these theories so you can attain the grade!
Don’t feel discouraged if you don’t recollect or don’t understand these concepts, as this blog will teach you all the fundamentals. Additionally, we will teach you the tricks to learning faster and increasing your grades in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines intersect, they form opposite angles, called vertical angles.
These opposite angles share a vertex. Furthermore, the most important 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 same 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 feasible to work out the others using 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 assume 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 labeled pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary due to the fact 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 each other. They share a side and a vertex, therefore they are supplementary angles, so the total 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 manner:
∠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 the same measurement.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's discuss specifically about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
Despite that, vertically opposite angles are never next to each other. Adjacent angles are two angles that share 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 case of complementary angles, the sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized 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 meet the properties of vertically opposite angles.
There are several characteristics of vertically opposite angles. Still, chances are that you will only need these two to ace your examination.
Vertically opposite angles are at all time congruent. Therefore, 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-Life Circumstances?
You might think where you can find these theorems in the real world, and you'd be surprised to notice that vertically opposite angles are very common! You can locate them in many everyday objects and situations.
For instance, vertically opposite angles are made when two straight lines cross. Right in your room, the door connected to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and adjust the size of the angles. Road intersections are also a wonderful example of vertically opposite angles.
In the end, vertically opposite angles are also present in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to observe your environment, as you will discover an example next to you.
Puttingit All Together
So, to sum up what we have discussed, vertically opposite angles are made from two crossover lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem explains that whenever two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line intersecting it and implementing 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. However, they are complementary if the addition of these angles equals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.
The total of adjacent angles is consistently 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 find them in several everyday objects and situations, such as doors, windows, paintings, and trees.
Further Study
Search for a vertically opposite angles practice questions online for examples and exercises to practice. Mathematics is not a spectator sport; keep practicing until these theorems are rooted in your brain.
Despite that, there is no problem if you require further help. If you're having a hard time to understand vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our skill tutors can help you understand the material and nail your following examination.
