Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who wants to study arithmetic or any related subject that uses it. It's tough work, but we'll ensure you get a handle on these theories so you can make the grade!
Don’t feel dispirited if you don’t recollect or don’t have a good grasp on these concepts, as this blog will help you study all the fundamentals. Moreover, we will help you understand the tricks to learning faster and increasing your grades in math and other common subjects today.
The Theorem
The vertical angle theorem expresses that whenever two straight lines bisect, they make opposite angles, called vertical angles.
These opposite angles share a vertex. Moreover, the most crucial thing to bear in mind is that they also measure the same! This means that irrespective of where these straight lines cross, the angles opposite each other will consistently share the equal 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 employing proportions.
Proving the Theorem
Proving this theorem is somewhat straightforward. First, let's pull a line and name it line l. After that, we will draw 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 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 contrary 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 the same as 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, signifying they are supplementary angles, so the total of both angles will be 180 degrees. This instance 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 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 be left 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 studied about the theorem and how to prove it, let's talk particularly about 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 fulfill 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 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 utilized to prove the vertical angle theorem.
These concepts are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are various characteristics of vertically opposite angles. Regardless, chances are that you will only require 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 at no time adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-World Scenario?
You might wonder where you can utilize these theorems in the real world, and you'd be amazed to note that vertically opposite angles are very common! You can discover them in several everyday things and circumstances.
For instance, vertically opposite angles are created when two straight lines cross. Inside your room, the door connected to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Road junctions are also a great example of vertically opposite angles.
In the end, vertically opposite angles are also discovered in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to watch your surroundings, as you will find an example next to you.
Puttingit Together
So, to sum up what we have considered so far, vertically opposite angles are formed from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem defines that whenever two intersecting straight lines, the angles created 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 complete 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 sum of these angles equals 90°. If the addition of both angles totals 180°, they are considered supplementary.
The total of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will at all time equal 180°.
Vertically opposite angles are quite common! You can find them in various everyday objects and scenarios, such as windows, doors, paintings, and trees.
Further Study
Look for a vertically opposite angles questionnaire online for examples and exercises to practice. Math is not a spectator sport; keep applying until these concepts are rooted in your mind.
However, there is nothing humiliating if you need extra support. If you're having a hard time to grasp vertical angles (or any other ideas of geometry), think about enrolling for a tutoring session with Grade Potential. One of our professional instructor can guide you grasp the material and ace your next test.
