Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential topic for anyone who desires to learn math or any related subject that utilizes it. It's tough work, but we'll make sure you get a grip of these theories so you can achieve 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 teach you all the essentials. Additionally, we will help you understand the tricks to learning faster and enhancing your grades in math and other common subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they form opposite angles, known as vertical angles.
These opposite angles share a vertex. Additionally, the most important thing to remember is that they also measure the same! This means that regardless of where these straight lines cross, the angles opposite each other will always share the exact value. These angles are called 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 straightforward. Primarily, let's pull a line and label it line l. Then, we will pull 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 created by the intersecting lines l and m. To prevent confusion, we named 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 because they share the equivalent vertex but don’t share a side. Remember that vertically opposite angles are also congruent, meaning that angle A equals angle B.
If you observe angles B and C, you will notice that they are not linked at their vertex but adjacent 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 say that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's talk specifically regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
However, vertically opposite angles are never next to each other. 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 case of complementary angles, the addition of two adjacent angles will equal 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 appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the properties of vertically opposite angles.
There are various properties of vertically opposite angles. But, chances are that you will only require these two to nail 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 never adjacent. They can share, at most, a vertex.
Where Can You Locate Opposite Angles in Real-Life Scenario?
You might wonder where you can find these concepts in the real life, and you'd be amazed to note that vertically opposite angles are fairly common! You can discover them in many everyday things and scenarios.
For example, vertically opposite angles are created when two straight lines overlap each other. 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 alter the size of the angles. Road intersections are also a wonderful example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you watch a tree, the vertically opposite angles are formed 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 discussed, vertically opposite angles are formed from two overlapping lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem defines that in the event of two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be tried out 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. 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 find them in several daily objects and circumstances, such as doors, windows, paintings, and trees.
Further Study
Search for a vertically opposite angles worksheet on the internet for examples and exercises to practice. Mathematics is not a onlooker sport; keep applying until these theorems are well-established in your head.
Still, there is no problem if you need additional help. If you're having a hard time to comprehend vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our skill teachers can assist you comprehend the material and nail your following examination.
