Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for everyone who wishes to learn math or any related subject that uses it. It's tough work, but we'll make sure you get a handle on these concepts so you can achieve the grade!
Don’t feel discouraged if you don’t recall or don’t comprehend these concepts, as this blog will help you study all the essentials. Moreover, we will teach you the tricks to learning quicker and enhancing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines intersect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Moreover, the most important point to bear in mind is that they also measure the same! This means that regardless of where these straight lines cross, the angles opposite each other will always share the exact value. These angles are referred as congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is feasible to find the others employing proportions.
Proving the Theorem
Proving this theorem is relatively straightforward. Primarily, let's draw a line and name it line l. After that, we will draw another line that goes through line l at some point. We will call 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 named pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite 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 identical angle B.
If you see the angles B and C, you will notice that they are not connected at their vertex but close to one another. They have in common a side and a vertex, meaning 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 way:
∠B+∠C=180 and ∠A+∠C=180
Since both sums up to equal the same, we can add these operations as follows:
∠A+∠C=∠B+∠C
By canceling out 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 explicitly regarding 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 no way next to each other. Adjacent angles are two angles that have 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 the case of complementary angles, the sum of two adjacent angles will total 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used in our proof of the vertical angle theorem.
These concepts are relevant within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, chances are that you will only require these two to nail your exam.
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 Locate Opposite Angles in Real-Life Scenario?
You may speculate where you can utilize these theorems in the real life, and you'd be surprised to observe that vertically opposite angles are quite common! You can locate them in many daily objects and situations.
For instance, vertically opposite angles are created when two straight lines overlap each other. Back of your room, the door installed to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and adjust the size of the angles. Road junctions are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to notice your surroundings, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have considered so far, vertically opposite angles are made from two intersecting lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem states that when two intersecting straight lines, the angles created are vertically opposite and congruent. This theorem can be proven by depicting a straight line and another line intersecting it and applying the concepts 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 cannot be vertically opposite. However, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are quite common! You can discover them in several daily objects and scenarios, such as paintings, doors, windows, and trees.
Further Study
Search for a vertically opposite angles worksheet online for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these theorems are rooted in your head.
Despite that, there is nothing humiliating if you need extra help. If you're struggling to grasp vertical angles (or any other ideas of geometry), consider enrolling for a tutoring session with Grade Potential. One of our professional tutors can help you comprehend the topic and ace your next test.
