Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential topic for anyone who wants to study math or any other subject that utilizes it. It's tough work, but we'll ensure you get a handle on these concepts so you can attain the grade!
Don’t feel disheartened if you don’t remember or don’t understand these theories, as this blog will teach you all the fundamentals. Moreover, we will teach you the secret to learning faster and increasing your scores in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that whenever two straight lines bisect, they make opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial thing to bear in mind is that they are the same in measurement! This applies that regardless of where these straight lines cross, the angles converse each other will always share the same 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 utilizing proportions.
Proving the Theorem
Proving this theorem is relatively simple. First, let's pull a line and call it line l. Then, we will pull another line that intersects line l at some point. We will name this second line m.
After drawing these two lines, we will name the angles formed by the intersecting lines l and m. To prevent 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 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, signifying 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 manner:
∠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 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 have learned about the theorem and how to prove it, let's talk specifically about vertically opposite angles.
Definition
As we said earlier, 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 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 total 90°. Supplementary angles are adjacent angles whose addition will equal 180°, which we just utilized in our proof of the vertical angle theorem.
These concepts are relevant within the vertical angle theorem and vertically opposite angles because supplementary and complementary angles do not satisfy the properties of vertically opposite angles.
There are many properties of vertically opposite angles. But, odds are that you will only require these two to nail your test.
Vertically opposite angles are at all time congruent. Therefore, 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 wonder where you can find these theorems in the real life, and you'd be amazed to note that vertically opposite angles are quite common! You can discover them in several 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 makes vertically opposite angles with the wall.
Open a pair of scissors to create two intersecting lines and alter the size of the angles. Road crossings are also a great example of vertically opposite angles.
Eventually, 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 environment, as you will discover 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 the same measure.
The vertical angle theorem explains 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 intersecting it and implementing the concepts of congruent angles to finish measures.
Congruent angles refer to 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 equals 180°, they are assumed to be supplementary.
The sum of adjacent angles is consistently 180°. Thus, if angles B and C are adjacent angles, they will always add up to 180°.
Vertically opposite angles are very common! You can discover them in various daily objects and scenarios, such as paintings, doors, windows, and trees.
Further Study
Search for a vertically opposite angles questionnaire online for examples and sums to practice. Math is not a spectator sport; keep applying until these concepts are ingrained in your mind.
Despite that, there is no problem if you require further assistance. If you're struggling to grasp vertical angles (or any other concept in geometry), consider enrolling for a tutoring session with Grade Potential. One of our professional tutors can assist you grasp the topic and nail your next test.
