Vertical Angles: Theorem, Proof, Vertically Opposite Angles
Understanding vertical angles is a important topic for anyone who desires to learn arithmetic or another subject that employs it. It's hard work, but we'll ensure you get a handle on these concepts so you can achieve the grade!
Don’t feel disheartened if you don’t remember or don’t have a good grasp on these concepts, as this blog will help you understand all the basics. Additionally, we will teach you the tricks to learning quicker and enhancing your grades in mathematics and other prevailing subjects today.
The Theorem
The vertical angle theorem expresses that whenever two straight lines meet, they make opposite angles, known as vertical angles.
These opposite angles share a vertex. Furthermore, the most crucial point to keep in mind is that they are the same in measurement! This means that irrespective of where these straight lines cross, the angles opposite 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 find the others utilizing proportions.
Proving the Theorem
Proving this theorem is somewhat simple. Primarily, let's draw a line and label 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 created by the intersecting lines l and m. To avoid confusion, we labeled pairs of vertically opposite angles. Therefore, we label angle A, angle B, angle C, and angle D as follows:
We understand 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 see the angles B and C, you will notice that they are not connected at their vertex but adjacent to one another. They share a side and a vertex, therefore 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 be left 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 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 one another satisfy the vertical angle theorem.
Despite that, 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 the case of complementary angles, the addition of two adjacent angles will equal 90°. Supplementary angles are adjacent angles whose sum will equal 180°, which we just used in our proof of the vertical angle theorem.
These theories are relevant within the vertical angle theorem and vertically opposite angles due to this reason supplementary and complementary angles do not fulfill the characteristics of vertically opposite angles.
There are various characteristics of vertically opposite angles. Regardless, chances are that you will only require these two to nail your examination.
Vertically opposite angles are always 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 Find Opposite Angles in Real-Life Circumstances?
You may think where you can find these theorems in the real world, and you'd be stunned to notice that vertically opposite angles are very common! You can discover them in various everyday things and scenarios.
For example, vertically opposite angles are created 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 make two intersecting lines and modify the size of the angles. Road crossings are also a wonderful example of vertically opposite angles.
Eventually, vertically opposite angles are also discovered 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 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 adjacent have identical measurements.
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 drawing a straight line and another line overlapping it and applying the theorems of congruent angles to finish measures.
Congruent angles means two angles that have identical measurements.
When two angles share a side and a vertex, they cannot be vertically opposite. Despite that, they are complementary if the addition of these angles totals 90°. If the sum of both angles totals 180°, they are assumed to be supplementary.
The total of adjacent angles is always 180°. Therefore, if angles B and C are adjacent angles, they will at all time add up to 180°.
Vertically opposite angles are very common! You can discover them in several daily objects and scenarios, such as paintings, doors, windows, and trees.
Additional Study
Search for a vertically opposite angles worksheet online for examples and problems to practice. Math is not a onlooker sport; keep practicing until these theorems are ingrained in your mind.
Still, there is no shame if you require additional assistance. If you're struggling to comprehend vertical angles (or any other concept in geometry), contemplate signing up for a tutoring session with Grade Potential. One of our expert instructor can guide you comprehend the topic and nail your next test.
