Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for everyone who desires to learn mathematics or another subject that employs it. It's tough work, but we'll make sure you get a grip of these theories so you can attain the grade!
Don’t feel dispirited if you don’t remember or don’t understand these concepts, as this blog will help you study all the essentials. Furthermore, we will teach you the secret to learning quicker and increasing your grades in mathematics and other popular subjects today.
The Theorem
The vertical angle theorem stipulates that at any time two straight lines meet, they form opposite angles, named 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 irrespective of where these straight lines cross, the angles opposite each other will consistently share the equal value. These angles are known as 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 using proportions.
Proving the Theorem
Proving this theorem is moderately straightforward. Primarily, 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 name this second line m.
After drawing these two lines, we will label the angles formed by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Thus, we label angle A, angle B, angle C, and angle D as follows:
We are aware that angles A and B are vertically opposite because 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 adjacent to one another. They share a side and a vertex, signifying 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 canceling out 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 have learned about the theorem and how to prove it, let's discuss particularly regarding vertically opposite angles.
Definition
As we stated, vertically opposite angles are two angles formed by the convergence of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, vertically opposite angles are at no time adjacent. 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 addition will equal 180°, which we just utilized in our proof of the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are many properties of vertically opposite angles. But, odds 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 at no time adjacent. They can share, at most, a vertex.
Where Can You Use Opposite Angles in Real-World Scenario?
You may speculate where you can find these concepts in the real life, and you'd be surprised to observe that vertically opposite angles are quite common! You can discover them in various everyday objects and circumstances.
For example, 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. 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 detect an example next to you.
Puttingit All Together
So, to sum up what we have discussed, 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 formed are vertically opposite and congruent. This theorem can be tested by depicting a straight line and another line overlapping it and implementing 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 cannot be vertically opposite. Nevertheless, they are complementary if the sum of these angles totals 90°. If the sum of both angles equals 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 at all time equal 180°.
Vertically opposite angles are quite common! You can locate them in several daily objects and situations, such as paintings, doors, windows, 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 concepts are ingrained in your brain.
Still, there is no problem if you need additional support. If you're struggling to understand vertical angles (or any other concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our skill teachers can help you comprehend the topic and ace your next examination.
