Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is a important subject for anyone who wishes to study arithmetic or any other subject that utilizes it. It's hard work, but we'll make sure you get a good grasp 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 fundamentals. Furthermore, we will teach you the secret to learning quicker and improving your scores in math and other popular subjects today.
The Theorem
The vertical angle theorem states that at any time two straight lines bisect, they make opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial thing to remember is that they also measure the same! This refers that irrespective of where these straight lines cross, the angles converse each other will always share the same 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 using proportions.
Proving the Theorem
Proving this theorem is moderately straightforward. First, let's draw a line and label it line l. Then, we will draw another line that goes through line l at some point. We will assume 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 named angle A, angle B, angle C, and angle D as follows:
We understand that angles A and B are vertically opposite 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 close 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 situation 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 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 the same measure.
Vertically Opposite Angles
Now that we have learned about the theorem and how to prove it, let's discuss explicitly 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.
Still, 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 sum 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 appropriate within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not meet the characteristics of vertically opposite angles.
There are several properties of vertically opposite angles. Still, odds are that you will only need these two to secure 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 at no time adjacent. They can share, at most, a vertex.
Where Can You Find Opposite Angles in Real-Life Circumstances?
You might think where you can utilize these theorems in the real world, and you'd be amazed to note that vertically opposite angles are quite common! You can find them in many everyday things and scenarios.
For example, vertically opposite angles are made when two straight lines overlap each other. Inside your room, the door installed to the door frame makes vertically opposite angles with the wall.
Open a pair of scissors to produce two intersecting lines and modify the size of the angles. Track intersections 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 surroundings, as you will discover an example next to you.
PuttingEverything Together
So, to summarize what we have talked about, vertically opposite angles are made from two intersecting lines. The two angles that are not adjacent have the same measure.
The vertical angle theorem states that whenever 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 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. Nevertheless, they are complementary if the sum of these angles equals 90°. If the sum of both angles totals 180°, they are deemed supplementary.
The sum of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will always equal 180°.
Vertically opposite angles are very common! You can find them in various everyday objects and situations, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles practice questions online for examples and problems to practice. Math is not a spectator sport; keep applying until these concepts are well-established in your brain.
Still, there is no shame if you need further assistance. If you're having difficulty to grasp vertical angles (or any other ideas of geometry), contemplate signing up for a tutoring session with Grade Potential. One of our skill teachers can help you grasp the topic and ace your next test.
