Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for anyone who desires to learn mathematics or any related subject that uses it. It's hard work, but we'll make sure you get a handle on these theories so you can achieve the grade!
Don’t feel discouraged if you don’t remember or don’t understand these concepts, as this blog will teach you all the fundamentals. Furthermore, we will help you learn the secret to learning faster and improving your scores in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem expresses that when two straight lines meet, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most important point to keep in mind is that they also measure the same! This means that irrespective of where these straight lines cross, the angles converse 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 work out the others using proportions.
Proving the Theorem
Proving this theorem is somewhat simple. First, let's draw a line and call it line l. Then, we will pull another line that intersects line l at some point. We will name this second line m.
After drawing these two lines, we will assume the angles created by the intersecting lines l and m. To prevent 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 opposite 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 is the same as angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but next to each other. 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 situation 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 sums up to equal the same, we can sum up 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 measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
Despite that, vertically opposite angles are at no time next to each other. 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 equal 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized in our proof of the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. But, chances are that you will only need these two to ace your examination.
Vertically opposite angles are always 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 Locate Opposite Angles in Real-Life Circumstances?
You may wonder where you can find these theorems in the real world, and you'd be stunned to notice that vertically opposite angles are fairly common! You can discover them in many daily things and scenarios.
For instance, vertically opposite angles are made when two straight lines overlap each other. Right in 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 great example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered 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 discover an example next to you.
Puttingit Together
So, to summarize what we have discussed, vertically opposite angles are formed from two intersecting lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem defines that when two intersecting straight lines, the angles made 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 refer to 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 addition of these angles totals 90°. If the sum 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 equal 180°.
Vertically opposite angles are very common! You can locate them in several daily objects and situations, such as paintings, doors, windows, and trees.
Further Study
Look for a vertically opposite angles practice questions online for examples and sums to practice. Mathematics is not a spectator sport; keep practicing until these concepts are ingrained in your brain.
Despite that, there is nothing humiliating if you need additional help. If you're having a hard time to understand vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill tutors can help you understand the topic and ace your next examination.
