Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important subject in geometry. With multiple real-life applications, you'd be amazed to find how useful this figure can be. Even though you might wonder if it has no use in your life, we all must understand the concept to nail those tests in school.
To save you time and create this information easily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to help with your personal 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 remain to be relevant as you move ahead with geometry and more complex theorems and proofs. We will answer this question with a easy explanation in this single point.
Explanation
A linear pair of angles is the term designated to two angles that are situated 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 sit on the same line and together form a straight line. The total of the angles in a linear pair will always make a straight angle equivalent
times to 180 degrees.
It is essential to keep in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This suggests that they always make on a straight line and are always supplementary angles.
It is essential to make clear that, although the linear pair are at all times adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
Over the definition simplified, we will examine the two axioms critical to fully comprehend any example thrown at 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 evident and self-explanatory. A linear pair of angles has two axioms associated with them.
The first axiom implies that if a ray stands on a line, the adjacent angles will form a straight angle, also known as a linear pair.
The second axiom states that if two angles produces a linear pair, then uncommon arms of both angles produces 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 diagram examples with their corresponding answers.
Example One
Here in this instance, we have two angles that are adjacent to one another. As you can see in the diagram, the adjacent angles form a linear pair because the sum of their measurement equals 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
Here, we have two lines intersect, producing four angles. Not all angles creates a linear pair, but respective angle and the one next 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 case 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 combinations add up to 180 degrees. As a effect, we can conclude that this example has no linear pair unless we expand a straight line.
Uses of Linear Pair of Angles
Now that we have gone through what linear pairs are and have observed some examples, let's see how this concept can be utilized in geometry and the real world.
In Real-Life Scenarios
There are many applications of linear pairs of angles in real life. One such case is architects, who utilize these axioms in their day-to-day job to identify if two lines are perpendicular and makes a straight angle.
Construction and Building professionals also use experts in this field to make their job simpler. They utilize linear pairs of angles to make sure that two close walls create a 90-degree angle with the ground.
Engineers also utilizes linear pairs of angles frequently. They do so by figuring out the tension in beams and trusses.
In Geometry
Linear pairs of angles also play a function in geometry proofs. A common proof that uses linear pairs is the alternate interior angles concept. This concept explains 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 depends on linear pairs of angles. Even though the adjacent angles are supplementary and sum 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 measurement of one angle to determine the measurement of the rest.
The theorem of linear pairs is subsequently utilized for more complicated implementation, such as determining the angles in polygons. It’s essential to grasp the basics of linear pairs, so you are ready for more complex geometry.
As shown, linear pairs of angles are a relatively easy theorem with some engaging applications. Later when you're out and about, observe if you can see any linear pairs! And, if you're participating in a geometry class, take notes on how linear pairs might be useful in proofs.
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