Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is a significant concept in geometry. With multiple real-world functions, you'd be amazed to find how relevant this figure can be. While you may wonder if it has no application in your life, we all should understand the concept to ace those tests in school.
To save your time and make this data readily accessible, here is an introductory insight into the properties of a linear pair of angles, with visualizations and examples to assist with your personal study sessions. We will also discuss few real-life and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are theories that exist to be relevant as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a easy definition in this unique point.
Definition
A linear pair of angles is the term given to two angles that are positioned on a straight line and the total of their measurement of angles is 180 degrees.
To put it simply, linear pairs of angles are two angles that are aligned 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 important to note that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This suggests that they always form on a straight line and are always supplementary angles.
It is essential to explain that, while the linear pair are at all times adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
Over the definition clarified, we will examine the two axioms seriously to fully grasp any example thrown at you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is approved without having to prove; it is considered clear and self-explanatory. A linear pair of angles has two axioms associated with them.
The first axiom establishes 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 among them. In other words, they make a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are a few figure examples with their corresponding answers.
Example One
In this example, we have two angles that are adjacent to each other. As you can notice in the figure, the adjacent angles form a linear pair due to the fact that the sum 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, producing four angles. Not every angles creates a linear pair, but each 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 a common point of three lines. Let's observe the axiom and properties of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations sum up to 180 degrees. As a consequence, we can conclude that this example has no linear pair until we expand one straight line.
Uses of Linear Pair of Angles
At the moment we have explored what linear pairs are and have observed some examples, let’s check how this concept can be used in geometry and the real-life scenario.
In Real-Life Scenarios
There are many implementations of linear pairs of angles in real life. One familiar example is architects, who use these axioms in their day-to-day job to determine if two lines are perpendicular and form a straight angle.
Construction and Building professionals also utilize expertise in this matter to make their work easier. They utilize linear pairs of angles to make sure that two close walls form a 90-degree angle with the ground.
Engineers also utilizes linear pairs of angles frequently. They do so by working out the tension in beams and trusses.
In Geometry
Linear pairs of angles additionally play a role in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles theorem. This concept 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 also replies 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 one another. Because of previously mentioned two rules, you are only required to determine the measurement of any one angle to figure out the measure of the rest.
The theorem of linear pairs is subsequently employed for more complex applications, such as figuring out the angles in polygons. It’s essential to grasp the fundamentals of linear pairs, so you are prepared for more progressive geometry.
As shown, linear pairs of angles are a somewhat simple theorem with some engaging applications. Later when you're out and about, see if you can notice some linear pairs! And, if you're attending a geometry class, bear in mind how linear pairs might be helpful in proofs.
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