Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important subject in geometry. With multiple real-life functions, you'd be astonished to figure out how relevant this figure can be. Although you may wonder if it has no relevant in your life, we all must grasp the ideas to nail those exams in school.
To save your time and offer this information easy to access, here is an preliminary insight into the characteristics of a linear pair of angles, with images and examples to guide with your private 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 useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a simple definition in this single point.
Definition
A linear pair of angles is the term given to two angles that are situated on a straight line and the total of their angles measure 180 degrees.
To put it simply, linear pairs of angles are two angles that sit on the same line and together form a straight line. The total of the angles in a linear pair will always create a straight angle equal to 180 degrees.
It is crucial to bear in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This means that at all times form on a straight line and are always supplementary angles.
It is crucial to explain that, although the linear pair are at all times adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
With the precise explanation, we will explore the two axioms critical to completely comprehend every example provided to you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is acknowledged without proof; it is deemed clear and self-evident. A linear pair of angles has two axioms linked with them.
The first axiom implies that if a ray is located on a line, the adjacent angles will create a straight angle, making them a linear pair.
The second axiom establishes that if two angles produces a linear pair, then uncommon arms of both angles create a straight angle among them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are a few figure examples with their corresponding explanations.
Example One
Here in this instance, we have two angles that are next to one another. As you can observe in the diagram, the adjacent angles form a linear pair since the total 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 possess two lines intersect, creating four angles. Not every angles creates 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 example 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 totals sum up to 180 degrees. As a consequence, we can conclude that this example has no linear pair unless we stretch one straight line.
Applications of Linear Pair of Angles
At the moment we have gone through what linear pairs are and have looked at some instances, let's see how this theorem can be used in geometry and the real-life scenario.
In Real-World Situations
There are multiple uses of linear pairs of angles in real life. One familiar example is architects, who apply these axioms in their daily work to determine if two lines are perpendicular and creates a straight angle.
Construction and Building professionals also use expertise in this subject to make their work less complex. They employ linear pairs of angles to assure that two close walls create a 90-degree angle with the floor.
Engineers also uses linear pairs of angles regularly. They do so by calculating the tension in beams and trusses.
In Geometry
Linear pairs of angles as well perform a function in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles concept. This concept states that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.
The proof of vertical angles as well depends on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equal to one another. Because of above-mentioned two rules, you are only required to figure out the measure of one angle to figure out the measure of the rest.
The theorem of linear pairs is further utilized for more sophisticated implementation, such as measuring the angles in polygons. It’s important to grasp the fundamentals of linear pairs, so you are prepared for more complex geometry.
As shown, linear pairs of angles are a relatively easy concept with some engaging implementations. Later when you're out and about, see if you can spot some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be useful in proofs.
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