Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Studying vertical angles is an essential subject for anyone who wants to learn arithmetic or another subject that employs it. It's hard work, but we'll ensure you get a grip of these theories so you can attain the grade!
Don’t feel disheartened if you don’t recall or don’t understand these theories, as this blog will help you understand all the essentials. Moreover, we will teach you the tricks to learning quicker and enhancing your grades in mathematics and other common subjects today.
The Theorem
The vertical angle theorem stipulates that when two straight lines meet, they form opposite angles, called vertical angles.
These opposite angles share a vertex. Additionally, the most important thing to remember is that they also measure the same! This refers that irrespective of where these straight lines cross, the angles opposite each other will constantly 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 feasible to discover the others employing proportions.
Proving the Theorem
Proving this theorem is somewhat easy. First, let's draw a line and name it line l. After that, 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 name the angles created by the intersecting lines l and m. To avoid confusion, we named pairs of vertically opposite angles. Accordingly, we label 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 equivalent 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 we look at angles B and C, you will note that they are not joined at their vertex but close 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 instance 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 additions equal the same, we can sum up these operations as follows:
∠A+∠C=∠B+∠C
By eliminating C on both sides of the equation, we will end with:
∠A=∠B
So, we can say 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 talk explicitly 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.
Still, vertically opposite angles are no way 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 the case of complementary angles, the addition of two adjacent angles will add up to 90°. Supplementary angles are adjacent angles which will add up to equal 180°, which we just utilized to prove the vertical angle theorem.
These concepts 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 characteristics of vertically opposite angles. Regardless, odds are that you will only need these two to nail your exam.
Vertically opposite angles are always congruent. Consequently, 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 Locate Opposite Angles in Real-Life Situations?
You may think where you can utilize these theorems in the real life, and you'd be amazed to notice that vertically opposite angles are quite common! You can locate them in various daily things and circumstances.
For example, vertically opposite angles are formed when two straight lines overlap each other. Back of your room, the door installed 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. Track intersections are also a terrific 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 made by the trunk and the branches.
Be sure to watch your surroundings, as you will discover an example next to you.
Puttingit All Together
So, to sum up what we have discussed, vertically opposite angles are created from two crossover lines. The two angles that are not next to each other have the same measure.
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 drawing a straight line and another line overlapping it and applying the theorems of congruent angles to finish measures.
Congruent angles refer to two angles that have identical measurements.
When two angles share a side and a vertex, they can’t be vertically opposite. Despite that, they are complementary if the addition of these angles equals 90°. If the sum of both angles totals 180°, they are considered supplementary.
The sum of adjacent angles is always 180°. Consequently, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are quite common! You can locate them in many daily objects and circumstances, such as paintings, doors, windows, and trees.
Additional Study
Look for a vertically opposite angles questionnaire online for examples and sums to practice. Math is not a spectator sport; keep practicing until these theorems are rooted in your mind.
Despite that, there is no problem if you require further support. If you're struggling to comprehend vertical angles (or any other ideas of geometry), consider signing up for a tutoring session with Grade Potential. One of our professional teachers can assist you understand the topic and ace your next test.
