Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who wants to learn math or any related subject that uses it. It's hard work, but we'll make sure you get a good grasp of these theories so you can make the grade!
Don’t feel discouraged if you don’t recollect or don’t comprehend these theories, as this blog will help you study all the fundamentals. Furthermore, we will help you learn the tricks to learning quicker and enhancing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines intersect, they create opposite angles, called vertical angles.
These opposite angles share a vertex. Moreover, the most essential point to keep in mind is that they also measure the same! This applies that regardless of where these straight lines cross, the angles converse each other will constantly 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 relatively easy. Primarily, let's pull a line and label it line l. Then, we will pull another line that intersects line l at some point. We will assume this second line m.
After drawing these two lines, we will name the angles formed by the intersecting lines l and m. To avoid confusion, we labeled 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 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 equals angle B.
If we look at angles B and C, you will note that they are not joined at their vertex but next 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 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 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 measure.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's talk explicitly about vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles made by the convergence of two straight lines. These angles opposite each other satisfy the vertical angle theorem.
However, vertically opposite angles are no way 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 equal 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used to prove the vertical angle theorem.
These concepts are relevant 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 many properties of vertically opposite angles. Still, odds are that you will only need these two to nail your exam.
Vertically opposite angles are at all time 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-World Situations?
You may speculate where you can find these theorems in the real world, and you'd be surprised to note that vertically opposite angles are fairly common! You can locate them in several everyday things and scenarios.
For instance, vertically opposite angles are made when two straight lines cross. Back of your room, the door connected to the door frame creates vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and modify the size of the angles. Track crossings are also a great example of vertically opposite angles.
Finally, 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 observe your environment, as you will discover an example next to you.
PuttingEverything Together
So, to sum up what we have discussed, vertically opposite angles are made from two overlapping lines. The two angles that are not next to each other have the same measure.
The vertical angle theorem defines that in the event of two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be proven by depicting a straight line and another line overlapping it and implementing 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 considered supplementary.
The total of adjacent angles is consistently 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 find them in several daily objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles worksheet online for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these theorems are well-established in your head.
However, there is no problem if you require extra help. If you're having difficulty to understand vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our expert teachers can help you understand the topic and ace your following test.
