Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential concept in geometry. With several real-world uses, you'd be amazed to figure out how relevant this figure can be. Although you may wonder if it has no application in your life, we all must learn the concept to ace those tests in school.
To save your time and make this information readily accessible, here is an preliminary insight into the properties 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 continue to be relevant as you go forward in geometry and more complex theorems and proofs. We will answer this query with a straightforward definition in this single point.
Definition
A linear pair of angles is the name provided to two angles that are positioned on a straight line and have the sum of their measurement of angles is 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 total of the angles in a linear pair will at all times make a straight angle equivalent
times to 180 degrees.
It is crucial to keep in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This implies that they always make on a straight line and are at all times supplementary angles.
It is important to clarify that, even though the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
With the definition clarified, we will examine the two axioms critical to fully comprehend every example provided to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is approved without proof; it is considered obvious 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 create a straight angle, namely called a linear pair.
The second axiom implies that if two angles produces a linear pair, then uncommon arms of both angles makes a straight angle among them. This is also known as a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are a few diagram examples with their corresponding answers.
Example One
As we can see in this instance, we have two angles that are next to one another. As you can notice in the image, the adjacent angles form a linear pair since the sum of their measurement is equivalent to 180 degrees. They are also supplementary angles, as 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 instance, we possess two lines intersect, creating four angles. Not every angles form a linear pair, but each angle and the one adjacent to it form a linear pair.
∠A 30 degrees
∠B: 150 degrees
∠C: 30 degrees
∠D: 150 degrees
In this case, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This case represents a common point 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 come to the conclusion that this diagram has no linear pair unless we stretch a straight line.
Uses of Linear Pair of Angles
At the moment we have learned what linear pairs are and have observed some instances, let’s check how this theorem can be used in geometry and the real-life scenario.
In Real-World Situations
There are many utilizations of linear pairs of angles in real life. One common example is architects, who use these axioms in their day-to-day job to determine if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also employ masters in this field to make their work easier. They employ linear pairs of angles to ensure that two adjacent walls form a 90-degree angle with the ground.
Engineers also apply linear pairs of angles frequently. They do so by figuring out the tension in beams and trusses.
In Geometry
Linear pairs of angles also perform a function in geometry proofs. A regular proof that employs linear pairs is the alternate interior angles concept. This theorem states that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.
The proof of vertical angles additionally 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 equivalent to each other. Because of these two rules, you only need to know the measurement of one angle to work out the measurement of the rest.
The theorem of linear pairs is subsequently used for more sophisticated uses, such as figuring out the angles in polygons. It’s essential to understand the basics of linear pairs, so you are prepared for more progressive geometry.
As shown, linear pairs of angles are a relatively simple concept with several engaging uses. Next time you're out and about, observe if you can see any linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be helpful in proofs.
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