Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With several real-world functions, you'd be surprised to figure out how relevant this figure can be. Although you may think it has no application in your life, we all should grasp the concept to nail those exams in school.
To save you time and offer this data easily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with visualizations and examples to guide with your private study sessions. We will also discuss some real-life and geometric uses.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that exist to be relevant as you move ahead with geometry and more complicated theorems and proofs. We will answer this query with a simple definition in this single point.
Explanation
A linear pair of angles is the term provided to two angles that are situated on a straight line and have the sum of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that are aligned on the same line and together form a straight line. The sum of the angles in a linear pair will always make a straight angle equivalent
times to 180 degrees.
It is important to bear in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This implies that at all times form on a straight line and are at all times supplementary angles.
It is important to explain that, while the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
Through the definition clarified, we will examine the two axioms critical to fully comprehend every example given to you.
First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is accepted without having to prove; it is deemed clear and self-evident. 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 form a straight angle, making them a linear pair.
The second axiom states that if two angles create a linear pair, then uncommon arms of both angles create a straight angle between them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are some drawn examples with their corresponding answers.
Example One
In this example, we have two angles that are adjacent to one another. As you can see in the figure, the adjacent angles form a linear pair since the total of their measures equals 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
Here, we possess two lines intersect, making four angles. Not all angles makes a linear pair, but each angle and the one close to it makes a linear pair.
∠A 30 degrees
∠B: 150 degrees
∠C: 30 degrees
∠D: 150 degrees
In this instance, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This example presents an intersection of three lines. Let's look at 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 image has no linear pair unless we extend one straight line.
Uses of Linear Pair of Angles
Now that we have explored what linear pairs are and have observed some instances, let’s understand how this concept can be applied in geometry and the real-life scenario.
In Real-World Scenarios
There are multiple utilizations of linear pairs of angles in real-world. One familiar case is architects, who use these axioms in their daily job to establish if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also use experts in this matter to make their job simpler. They use linear pairs of angles to ensure that two adjacent walls create a 90-degree angle with the floor.
Engineers also uses linear pairs of angles regularly. They do so by figuring out the weight on the beams and trusses.
In Geometry
Linear pairs of angles also perform a role in geometry proofs. A regular proof that employs 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 created are congruent.
The proof of vertical angles as well depends on linear pairs of angles. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are at all times equal to each other. Because of above-mentioned two rules, you only need to figure out the measure of any one angle to figure out the measurement of the rest.
The theorem of linear pairs is also used for more complicated implementation, such as determining the angles in polygons. It’s important to grasp the basics of linear pairs, so you are ready for more complex geometry.
As shown, linear pairs of angles are a somewhat simple theorem with some interesting applications. Later when you're out and about, take note if you can see some linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs may be helpful in proofs.
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