Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is an essential subject for anyone who desires to master arithmetic or any related subject that uses it. It's tough work, but we'll ensure you get a grip of these concepts so you can achieve the grade!
Don’t feel discouraged if you don’t recall or don’t have a good grasp on these theories, as this blog will help you study all the basics. Additionally, we will teach you the tricks to learning faster and increasing your scores in mathematics and other prevailing subjects today.
The Theorem
The vertical angle theorem states that whenever two straight lines bisect, they form opposite angles, named vertical angles.
These opposite angles share a vertex. Additionally, the most crucial thing to remember is that they also measure the same! This applies that regardless of where these straight lines cross, the angles opposite each other will constantly share the exact 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 feasible to work out the others utilizing proportions.
Proving the Theorem
Proving this theorem is somewhat easy. First, let's draw a line and call it line l. Then, we will draw another line that intersects line l at some point. We will assume this second line m.
After drawing these two lines, we will label the angles created by the intersecting lines l and m. To prevent confusion, we labeled 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 same vertex but don’t share a side. Bear in mind that vertically opposite angles are also congruent, meaning that angle A is identical 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 have in common 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 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 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 measurement.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's discuss explicitly about vertically opposite angles.
Definition
As we said earlier, vertically opposite angles are two angles created by the intersection of two straight lines. These angles opposite one another fulfill the vertical angle theorem.
Still, 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 addition 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 due to this reason supplementary and complementary angles do not satisfy the characteristics of vertically opposite angles.
There are many properties of vertically opposite angles. Regardless, odds are that you will only need these two to ace your exam.
Vertically opposite angles are at all time congruent. Therefore, 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-World Scenario?
You may speculate where you can find these concepts in the real life, and you'd be stunned to observe that vertically opposite angles are fairly common! You can find them in various everyday things and circumstances.
For example, vertically opposite angles are formed when two straight lines overlap each other. Inside your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and adjust the size of the angles. Track crossings are also a great example of vertically opposite angles.
In the end, vertically opposite angles are also discovered in nature. If you watch a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to notice your environment, as you will discover an example next to you.
PuttingEverything Together
So, to summarize what we have discussed, vertically opposite angles are made from two overlapping lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem states that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tested by drawing a straight line and another line overlapping it and using the concepts of congruent angles to complete measures.
Congruent angles refer to two angles that measure the same.
When two angles share a side and a vertex, they can’t be vertically opposite. Despite that, they are complementary if the sum 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°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are quite common! You can discover them in several daily objects and situations, such as doors, windows, paintings, and trees.
Additional Study
Look for a vertically opposite angles practice questions on the internet for examples and sums to practice. Math is not a onlooker sport; keep practicing until these concepts are rooted in your head.
Still, there is no shame if you need additional help. If you're having difficulty to comprehend vertical angles (or any other concept in geometry), think about enrolling for a tutoring session with Grade Potential. One of our skill tutors can guide you comprehend the material and nail your next examination.
