Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a important subject for anyone who desires to master math or another subject that utilizes it. It's hard work, but we'll ensure you get a handle on these concepts so you can achieve the grade!
Don’t feel dispirited if you don’t remember or don’t comprehend these concepts, as this blog will help you understand all the essentials. Additionally, we will help you learn the secret to learning faster and increasing your scores in arithmetic and other common subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines intersect, they create opposite angles, named vertical angles.
These opposite angles share a vertex. Moreover, the most important thing to remember is that they also measure the same! This applies that irrespective of where these straight lines cross, the angles opposite each other will constantly 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 feasible to work out the others utilizing proportions.
Proving the Theorem
Proving this theorem is moderately simple. First, let's pull a line and call it line l. Then, we will draw another line that intersects 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 labeled pairs of vertically opposite angles. Thus, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically opposite because 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 we look at angles B and C, you will note that they are not joined at their vertex but close to one another. They have in common a side and a vertex, therefore 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 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 conclude 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 discuss specifically regarding 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 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 addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used in our proof of the vertical angle theorem.
These theories are appropriate within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are various characteristics of vertically opposite angles. Regardless, odds are that you will only require these two to secure your test.
Vertically opposite angles are at all time congruent. Consequently, if angles A and B are vertically opposite, they will measure the same.
Vertically opposite angles are at no time adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-World Situations?
You might think where you can use these concepts in the real life, and you'd be amazed to observe that vertically opposite angles are quite common! You can discover them in several everyday things and scenarios.
For instance, vertically opposite angles are created when two straight lines overlap each other. Back of your room, the door attached to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and alter the size of the angles. Road junctions are also a great example of vertically opposite angles.
Finally, vertically opposite angles are also found in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to watch your environment, as you will find an example next to you.
Puttingit All Together
So, to sum up what we have considered so far, vertically opposite angles are formed from two intersecting lines. The two angles that are not next to each other have identical measurements.
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 tried out by drawing a straight line and another line overlapping it and using the concepts of congruent angles to finish measures.
Congruent angles means two angles that measure the same.
When two angles share a side and a vertex, they can’t be vertically opposite. However, they are complementary if the sum of these angles totals 90°. If the addition of both angles totals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are pretty common! You can locate them in various everyday objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles practice questions on the internet for examples and problems to practice. Math is not a onlooker sport; keep practicing until these theorems are rooted in your mind.
Still, there is no shame if you need further assistance. If you're having difficulty to comprehend vertical angles (or any other ideas of 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 following test.
