Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential subject for anyone who desires to study mathematics or any related subject that uses it. It's tough work, but we'll assure you get a handle on these theories so you can attain the grade!
Don’t feel disheartened if you don’t remember or don’t have a good grasp on these theories, as this blog will help you study all the fundamentals. Moreover, we will help you learn the tricks to learning quicker and improving your grades in arithmetic and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that at any time two straight lines meet, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Furthermore, the most important point to keep in mind is that they also measure the same! This means that regardless of where these straight lines cross, the angles converse 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 feasible to find the others using proportions.
Proving the Theorem
Proving this theorem is somewhat straightforward. Primarily, let's draw a line and call it line l. Then, we will draw another line that goes through line l at some point. We will call 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 named pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically contrary due to the fact 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 identical angle B.
If you see the angles B and C, you will note that they are not connected at their vertex but adjacent to one another. They have in common a side and a vertex, meaning they are supplementary angles, so the sum 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 way:
∠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 the same measure.
Vertically Opposite Angles
Now that we have studied about the theorem and how to prove it, let's discuss specifically about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles made by the intersection of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
However, vertically opposite angles are no way adjacent. 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 the case of complementary angles, the sum of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized to prove the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles since supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are various characteristics of vertically opposite angles. Still, odds are that you will only require these two to ace your test.
Vertically opposite angles are always congruent. Hence, 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-Life Situations?
You may wonder where you can find these theorems in the real world, and you'd be stunned to observe that vertically opposite angles are fairly common! You can find them in several everyday objects and scenarios.
For instance, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door installed to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Track intersections are also a terrific example of vertically opposite angles.
Eventually, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to watch your surroundings, as you will detect an example next to you.
Puttingit Together
So, to sum up what we have discussed, vertically opposite angles are formed from two crossover lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem defines that whenever 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 theorems 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. Despite that, they are complementary if the addition of these angles equals 90°. If the addition of both angles equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Consequently, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are pretty common! You can discover them in several everyday objects and circumstances, such as doors, windows, paintings, and trees.
Further Study
Look for a vertically opposite angles questionnaire on the internet for examples and problems to practice. Mathematics is not a onlooker sport; keep applying until these concepts are ingrained in your mind.
However, there is no problem if you need additional assistance. If you're having a hard time to comprehend vertical angles (or any other ideas of geometry), think about signing up for a tutoring session with Grade Potential. One of our expert teachers can assist you grasp the topic and nail your next test.
