Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a crucial topic for anyone who wants to master math or another subject that uses it. It's tough work, but we'll make sure you get a good grasp of these concepts so you can attain the grade!
Don’t feel disheartened if you don’t recall or don’t have a good grasp on these theories, as this blog will teach you all the basics. Furthermore, we will help you learn the secret to learning quicker and increasing your scores in mathematics and other prevailing subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines intersect, they make opposite angles, named 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 means that irrespective 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 find the others employing proportions.
Proving the Theorem
Proving this theorem is relatively simple. First, let's draw a line and name it line l. After that, we will pull another line that goes through line l at some point. We will assume 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 named pairs of vertically opposite angles. Accordingly, we named angle A, angle B, angle C, and angle D as follows:
We know that angles A and B are vertically contrary due to the fact they share the equivalent vertex but don’t share a side. Remember 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 adjacent to each other. They share a side and a vertex, signifying 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 sum up these operations as follows:
∠A+∠C=∠B+∠C
By removing C on both sides of the equation, we will be left with:
∠A=∠B
So, we can conclude that vertically opposite angles are congruent, as they have the same 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 mentioned, vertically opposite angles are two angles formed by the intersection of two straight lines. These angles opposite each other fulfill the vertical angle theorem.
However, vertically opposite angles are never adjacent. 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 case of complementary angles, the addition of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized to prove the vertical angle theorem.
These theories are applicable within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not fulfill the properties of vertically opposite angles.
There are several properties of vertically opposite angles. Still, chances are that you will only require these two to ace 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 at no time adjacent. They can share, at most, a vertex.
Where Can You Use Opposite Angles in Real-Life Scenario?
You might speculate where you can use these concepts in the real life, and you'd be surprised to note that vertically opposite angles are quite common! You can locate them in many everyday objects and scenarios.
For instance, vertically opposite angles are formed when two straight lines cross. Right in your room, the door installed 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 junctions are also a great example of vertically opposite angles.
In the end, 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 notice your environment, as you will find an example next to you.
PuttingEverything Together
So, to summarize what we have considered so far, vertically opposite angles are created from two overlapping lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem defines that when two intersecting straight lines, the angles formed are vertically opposite and congruent. This theorem can be tried out by depicting a straight line and another line intersecting it and applying the theorems of congruent angles to complete 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 addition of these angles equals 90°. If the addition of both angles totals 180°, they are considered supplementary.
The total of adjacent angles is always 180°. Thus, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are very common! You can locate them in many everyday objects and situations, such as windows, doors, paintings, and trees.
Additional Study
Look for a vertically opposite angles worksheet on the internet for examples and problems to practice. Math is not a spectator sport; keep applying until these theorems are ingrained in your brain.
However, there is no shame if you need further assistance. If you're struggling to comprehend vertical angles (or any other concept in geometry), think about signing up for a tutoring session with Grade Potential. One of our expert tutors can assist you understand the topic and ace your following examination.
