Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential subject for everyone who wants to learn mathematics or any other subject that uses it. It's tough work, but we'll assure you get a grip of these theories so you can make the grade!
Don’t feel disheartened if you don’t remember or don’t comprehend these theories, as this blog will help you understand all the fundamentals. Furthermore, we will help you understand the secret to learning quicker and enhancing your grades in math and other common subjects today.
The Theorem
The vertical angle theorem states that when two straight lines bisect, they create opposite angles, called vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial thing to remember is that they are the same in measurement! This applies that irrespective of where these straight lines cross, the angles opposite each other will always share the equal 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 discover the others using proportions.
Proving the Theorem
Proving this theorem is moderately simple. First, let's pull a line and call it line l. After that, we will pull another line that intersects line l at some point. We will assume this second line m.
After drawing these two lines, we will assume the angles formed by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite reason being that they share the same vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A is identical angle B.
If you observe angles B and C, you will note that they are not linked at their vertex but close to each other. They share a side and a vertex, signifying they are supplementary angles, so the total of both angles will be 180 degrees. This scenario 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 add 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 conclude 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 explicitly regarding 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 one another fulfill the vertical angle theorem.
Despite that, 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 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 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 due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are various properties of vertically opposite angles. But, odds are that you will only require these two to ace your exam.
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 never adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-Life Scenario?
You might think where you can use these concepts in the real life, and you'd be amazed to observe that vertically opposite angles are fairly common! You can find them in various everyday objects and scenarios.
For instance, vertically opposite angles are created when two straight lines cross. Right in your room, the door installed 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. Track intersections are also a terrific 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 made by the trunk and the branches.
Be sure to watch your surroundings, as you will find an example next to you.
PuttingEverything Together
So, to sum up what we have talked about, vertically opposite angles are made from two crossover lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem explains that whenever two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line intersecting it and applying the theorems of congruent angles to complete measures.
Congruent angles means two angles that have identical measurements.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the sum of these angles totals 90°. If the addition of both angles equals 180°, they are deemed supplementary.
The total of adjacent angles is consistently 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 various daily objects and circumstances, such as windows, doors, paintings, and trees.
Further Study
Search for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a spectator sport; keep practicing until these theorems are rooted in your head.
However, there is no shame if you need extra support. If you're having difficulty to understand vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our professional tutors can assist you comprehend the material and nail your following examination.
