Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial subject for everyone who wants to study mathematics or any other subject that employs it. It's hard work, but we'll ensure you get a good grasp of these concepts so you can make the grade!
Don’t feel disheartened if you don’t recollect or don’t comprehend these theories, as this blog will teach you all the essentials. Moreover, we will teach you the secret to learning faster and enhancing your scores in math and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial point to keep 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 constantly 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 possible to work out the others employing proportions.
Proving the Theorem
Proving this theorem is moderately easy. Primarily, 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 call this second line m.
After drawing these two lines, we will name the angles formed by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically opposite due to the fact they share the equivalent vertex but don’t share a side. Bear in mind 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 note that they are not connected at their vertex but close to one another. They share a side and a vertex, signifying they are supplementary angles, so the sum 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 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 be left 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 studied about the theorem and how to prove it, let's talk particularly about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
However, vertically opposite angles are at no time adjacent. 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 equal 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 since supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several properties of vertically opposite angles. Regardless, chances are that you will only need these two to ace your test.
Vertically opposite angles are at all time 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 Use Opposite Angles in Real-World Scenario?
You may think where you can utilize these theorems in the real life, and you'd be stunned to note that vertically opposite angles are fairly common! You can find them in many everyday objects and scenarios.
For instance, vertically opposite angles are formed when two straight lines overlap each other. Inside your room, the door installed 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 intersections are also a terrific example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to notice your surroundings, as you will detect an example next to you.
Puttingit Together
So, to summarize what we have talked about, vertically opposite angles are formed from two overlapping 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 formed are vertically opposite and congruent. This theorem can be tried out by drawing 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 measure the same.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the addition of these angles equals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.
The sum of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are quite common! You can locate them in several everyday objects and situations, such as paintings, doors, windows, and trees.
Further Study
Search for a vertically opposite angles practice questions online for examples and exercises to practice. Math is not a onlooker sport; keep applying until these concepts are well-established in your brain.
Still, there is no shame if you require extra help. If you're having difficulty to grasp vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our professional teachers can assist you grasp the material and ace your following examination.
