Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who wishes to learn arithmetic or any related subject that uses it. It's hard work, but we'll assure you get a grip of these concepts so you can attain the grade!
Don’t feel dispirited if you don’t remember or don’t have a good grasp on these theories, as this blog will help you study all the fundamentals. Additionally, we will help you understand the tricks to learning faster and improving your scores in math and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that at any time two straight lines intersect, 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 means that regardless of where these straight lines cross, the angles opposite each other will always share the equal 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 discover the others using proportions.
Proving the Theorem
Proving this theorem is somewhat easy. Primarily, let's draw 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 created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Therefore, we named angle A, angle B, angle C, and angle D as follows:
We are aware 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 the same as angle B.
If you observe angles B and C, you will note that they are not linked at their vertex but next 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 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 sums up to 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 conclude that vertically opposite angles are congruent, as they have the same measure.
Vertically Opposite Angles
Now that we know the theorem and how to prove it, let's discuss particularly regarding vertically opposite angles.
Definition
As we mentioned, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite one another satisfy the vertical angle theorem.
Despite that, vertically opposite angles are never next to each other. Adjacent angles are two angles that share 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 add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just used to prove the vertical angle theorem.
These concepts are applicable within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not meet the properties of vertically opposite angles.
There are several properties of vertically opposite angles. Regardless, chances are that you will only require these two to ace your examination.
Vertically opposite angles are always congruent. Hence, 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-Life Scenario?
You may speculate where you can use these theorems in the real world, and you'd be stunned to notice that vertically opposite angles are quite common! You can locate them in various daily things and scenarios.
For example, vertically opposite angles are formed when two straight lines overlap each other. Right in your room, the door connected to the door frame produces vertically opposite angles with the wall.
Open a pair of scissors to make two intersecting lines and modify 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 created 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 made from two overlapping lines. The two angles that are not next to each other have the same measure.
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 applying the theorems of congruent angles to complete measures.
Congruent angles refer to two angles that have identical measurements.
When two angles share a side and a vertex, they can’t be vertically opposite. Despite that, they are complementary if the addition of these angles totals 90°. If the addition of both angles equals 180°, they are assumed to be 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 quite common! You can find them in several everyday objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles practice questions online for examples and exercises to practice. Mathematics is not a onlooker sport; keep practicing until these concepts are rooted in your brain.
Despite that, there is no shame if you need extra assistance. If you're having a hard time to understand vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you understand the topic and nail your following examination.
