Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential subject for everyone who desires to learn mathematics or another subject that uses it. It's tough work, but we'll ensure you get a grip of these theories so you can make the grade!
Don’t feel disheartened if you don’t recall or don’t comprehend these concepts, as this blog will help you understand all the basics. Moreover, we will teach you the tricks 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 form opposite angles, known as vertical angles.
These opposite angles share a vertex. Moreover, the most important thing to keep in mind is that they are the same in measurement! This refers that irrespective of where these straight lines cross, the angles opposite each other will consistently share the same 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 possible to work out the others employing proportions.
Proving the Theorem
Proving this theorem is somewhat simple. Primarily, let's draw a line and label it line l. After that, we will draw another line that goes through 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 named pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically opposite reason being that 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 you observe angles B and C, you will note that they are not linked at their vertex but adjacent to each other. They share a side and a vertex, therefore 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 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 identical measurement.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk specifically about vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles created by the convergence of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Still, 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 case of complementary angles, the sum of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used 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 properties of vertically opposite angles.
There are several properties of vertically opposite angles. But, chances are that you will only require these two to nail 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 Find Opposite Angles in Real-World Circumstances?
You may speculate where you can find these concepts in the real world, and you'd be amazed to note that vertically opposite angles are fairly common! You can locate them in various everyday objects and scenarios.
For instance, vertically opposite angles are made when two straight lines cross. 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 adjust the size of the angles. Road junctions are also a wonderful example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered in nature. If you look at a tree, the vertically opposite angles are made by the trunk and the branches.
Be sure to notice your environment, as you will find an example next to you.
Puttingit All Together
So, to sum up what we have considered so far, vertically opposite angles are made from two overlapping lines. The two angles that are not adjacent have identical measurements.
The vertical angle theorem states that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line overlapping it and implementing the concepts of congruent angles to finish measures.
Congruent angles means 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 equals 90°. If the addition of both angles equals 180°, they are deemed supplementary.
The total 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 many daily objects and scenarios, such as doors, windows, paintings, and trees.
Further Study
Search for a vertically opposite angles questionnaire on the internet for examples and problems to practice. Math is not a onlooker sport; keep applying until these theorems are ingrained in your mind.
Still, there is no problem if you need additional support. If you're having difficulty to comprehend vertical angles (or any other concept in geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our skill teachers can guide you comprehend the topic and ace your next examination.
