Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is a crucial topic for anyone who desires to master math or another subject that uses it. It's tough work, but we'll assure you get a grip of these concepts so you can make the grade!
Don’t feel dispirited if you don’t recollect or don’t have a good grasp on these theories, as this blog will help you study all the basics. Moreover, we will help you understand the secret to learning quicker and increasing your scores in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines intersect, they make opposite angles, called vertical angles.
These opposite angles share a vertex. Furthermore, the most essential thing to remember is that they are the same in measurement! This applies that irrespective of where these straight lines cross, the angles converse each other will always share the same 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 moderately simple. First, let's draw a line and name it line l. After that, 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 labeled pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary due to the fact they share the same vertex but don’t share a side. Remember 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 notice that they are not linked at their vertex but close to each other. They have in common a side and a vertex, signifying they are supplementary angles, so the sum of both angles will be 180 degrees. This situation 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 sums up to equal the same, we can add these operations as follows:
∠A+∠C=∠B+∠C
By canceling out C on both sides of the equation, we will be left with:
∠A=∠B
So, we can conclude 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 discuss specifically regarding vertically opposite angles.
Definition
As we said earlier, 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 at no time next to each other. Adjacent angles are two angles that share 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 whose addition will equal 180°, which we just utilized in our proof of the vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are many properties of vertically opposite angles. Regardless, chances are that you will only need these two to nail your test.
Vertically opposite angles are always 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 Use Opposite Angles in Real-World Circumstances?
You may speculate where you can utilize these concepts in the real life, and you'd be stunned to notice that vertically opposite angles are fairly common! You can find them in various everyday things and situations.
For example, vertically opposite angles are made when two straight lines cross. Right in your room, the door attached to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and adjust the size of the angles. Track crossings are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also present in nature. If you watch a tree, the vertically opposite angles are created by the trunk and the branches.
Be sure to notice your surroundings, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have considered so far, vertically opposite angles are formed from two crossover lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem states that in the event of 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 overlapping it and using 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. However, they are complementary if the addition of these angles totals 90°. If the addition of both angles equals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are very common! You can discover them in several daily objects and scenarios, such as paintings, doors, windows, and trees.
Additional Study
Look 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 concepts are well-established in your mind.
Still, there is nothing humiliating if you require extra assistance. If you're struggling to understand vertical angles (or any other ideas of geometry), contemplate enrolling for a tutoring session with Grade Potential. One of our expert instructor can guide you grasp the material and ace your next examination.
