Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential topic for anyone who desires to learn math or any other subject that utilizes it. It's tough work, but we'll make sure you get a handle on these concepts so you can attain the grade!
Don’t feel disheartened if you don’t remember or don’t understand these concepts, as this blog will teach you all the essentials. Additionally, we will help you learn the secret to learning quicker and enhancing your scores in arithmetic and other prevailing subjects today.
The Theorem
The vertical angle theorem expresses that at any time two straight lines bisect, they make opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most essential point 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 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 feasible to find the others using proportions.
Proving the Theorem
Proving this theorem is relatively easy. Primarily, let's draw a line and label it line l. Then, we will draw another line that intersects 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. Thus, we named angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite because they share the equivalent vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A equals angle B.
If we look at angles B and C, you will notice that they are not joined at their vertex but close to one another. They share 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 manner:
∠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 eliminating 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 learned 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 made by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, 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 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 total 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are many characteristics of vertically opposite angles. Still, odds are that you will only need these two to ace your test.
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 Situations?
You may think where you can find these concepts in the real world, and you'd be stunned to observe that vertically opposite angles are quite common! You can discover them in various daily things and scenarios.
For example, vertically opposite angles are formed when two straight lines overlap each other. Inside your room, the door attached to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Road crossings are also a terrific example of vertically opposite angles.
Finally, 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 notice your surroundings, as you will discover an example next to you.
PuttingEverything Together
So, to summarize what we have talked about, vertically opposite angles are made from two intersecting lines. The two angles that are not next to each other 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 tested by drawing a straight line and another line intersecting it and applying the concepts of congruent angles to complete measures.
Congruent angles refer to two angles that have identical measurements.
When two angles share a side and a vertex, they cannot be vertically opposite. Nevertheless, they are complementary if the sum of these angles totals 90°. If the sum of both angles equals 180°, they are considered supplementary.
The total 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 many daily objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles worksheet on the internet for examples and problems to practice. Math is not a spectator sport; keep applying until these concepts are rooted in your mind.
Still, there is no shame if you need extra support. If you're having difficulty to comprehend vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our professional teachers can guide you comprehend the topic and ace your next examination.
