Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Learning vertical angles is an essential subject for anyone who wishes to learn arithmetic or any other subject that uses it. It's tough work, but we'll ensure you get a good grasp of these theories so you can achieve the grade!
Don’t feel dispirited if you don’t recollect or don’t comprehend these concepts, as this blog will teach you all the fundamentals. Additionally, we will help you understand the tricks to learning faster and enhancing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines intersect, they create opposite angles, known as vertical angles.
These opposite angles share a vertex. Additionally, the most important 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 consistently share the equal value. These angles are referred as congruent angles.
Vertically opposite angles are congruent, so if you have a value for one angle, then it is possible to find the others utilizing proportions.
Proving the Theorem
Proving this theorem is somewhat easy. First, let's draw a line and label 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 label angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary because they share the same 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 you see the angles B and C, you will note that they are not connected at their vertex but next to each other. They have in common a side and a vertex, meaning they are supplementary angles, so the sum 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 way:
∠B+∠C=180 and ∠A+∠C=180
Since both additions equal the same, we can sum up 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 identical measurement.
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 said earlier, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
Despite that, vertically opposite angles are no way next to each other. 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 add up to 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just utilized to prove the vertical angle theorem.
These theories are appropriate 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 various properties of vertically opposite angles. But, odds are that you will only require these two to ace your test.
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 Use Opposite Angles in Real-Life Circumstances?
You might speculate where you can find these concepts in the real life, and you'd be surprised to observe that vertically opposite angles are very common! You can locate them in various daily things and circumstances.
For example, vertically opposite angles are made when two straight lines overlap each other. Right in your room, the door attached 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 intersections are also a wonderful example of vertically opposite angles.
Finally, vertically opposite angles are also present in nature. If you look at a tree, the vertically opposite angles are formed by the trunk and the branches.
Be sure to notice your surroundings, as you will detect an example next to you.
PuttingEverything Together
So, to summarize what we have considered so far, vertically opposite angles are formed from two overlapping lines. The two angles that are not next to each other have identical measurements.
The vertical angle theorem states that when two intersecting straight lines, the angles made are vertically opposite and congruent. This theorem can be proven by drawing a straight line and another line overlapping it and applying the theorems 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 sum of both angles totals 180°, they are assumed to be supplementary.
The total of adjacent angles is consistently 180°. Therefore, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can locate them in many daily objects and situations, such as windows, doors, paintings, and trees.
Further Study
Search for a vertically opposite angles worksheet online for examples and sums to practice. Math is not a spectator sport; keep applying until these theorems are well-established in your mind.
Despite that, there is no shame if you require extra support. If you're having difficulty to understand vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our skill tutors can guide you comprehend the material and nail your following test.
