Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential concept in geometry. With so many real-life applications, you'd be astonished to figure out how applicable this figure can be. While you may believe it has no application in your life, we all should understand the ideas to nail those exams in school.
To save you time and make this info easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with visualizations and examples to guide with your personal study sessions. We will also discuss few real-world and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that remain to be relevant as you progress in geometry and more complex theorems and proofs. We will answer this question with a easy explanation in this unique point.
Definition
A linear pair of angles is the term designated to two angles that are positioned on a straight line and the total of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that are adjacent on the same line and pair up to create a straight line. The total of the angles in a linear pair will always produce a straight angle equivalent
times to 180 degrees.
It is essential to bear in mind that linear pairs are always at adjacent angles. They share a common apex and a common arm. This suggests that at all times make on a straight line and are at all times supplementary angles.
It is essential to explain that, while the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
With the precise explanation, we will study the two axioms critical to fully understand every example given to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is considered evident and self-explanatory. A linear pair of angles has two axioms associated with them.
The first axiom implies that if a ray is located on a line, the adjacent angles will make a straight angle, also known as a linear pair.
The second axiom implies that if two angles create a linear pair, then uncommon arms of both angles produces a straight angle between them. In other words, they make a straight line.
Examples of Linear Pairs of Angles
To envision these axioms better, here are some drawn examples with their respective explanations.
Example One
Here in this example, we have two angles that are next to each other. As you can notice in the diagram, the adjacent angles form a linear pair since the sum of their measures is equivalent to 180 degrees. They are also supplementary angles, since they share a side and a common vertex.
Angle A: 75 degrees
Angle B: 105 degrees
Sum of Angles A and B: 75 + 105 = 180
Example Two
In this example, we have two lines intersect, making four angles. Not every angles makes a linear pair, but respective angle and the one next to it form a linear pair.
∠A 30 degrees
∠B: 150 degrees
∠C: 30 degrees
∠D: 150 degrees
In this example, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This case shows convergence of three lines. Let's take note of the axiom and characteristics of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations add up to 180 degrees. As a result, we can come to the conclusion that this diagram has no linear pair unless we extend a straight line.
Implementations of Linear Pair of Angles
Now that we have learned what linear pairs are and have observed some instances, let's see how this concept can be utilized in geometry and the real-life scenario.
In Real-World Scenarios
There are several utilizations of linear pairs of angles in real-world. One common case is architects, who use these axioms in their day-to-day work to check if two lines are perpendicular and creates a straight angle.
Construction and Building professionals also use expertise in this matter to make their work simpler. They use linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the ground.
Engineers also uses linear pairs of angles frequently. They do so by working out the weight on the beams and trusses.
In Geometry
Linear pairs of angles additionally perform a function in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles theorem. This concept explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.
The proof of vertical angles additionally replies on linear pairs of angles. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to each other. Because of previously mentioned two rules, you are only required to determine the measure of any one angle to figure out the measure of the rest.
The concept of linear pairs is subsequently employed for more complicated applications, such as determining the angles in polygons. It’s important to grasp the fundamentals of linear pairs, so you are prepared for more advanced geometry.
As demonstrated, linear pairs of angles are a somewhat simple concept with several fascinating uses. Later when you're out and about, take note if you can spot some linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs may be useful in proofs.
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