Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential subject in geometry. With several real-world uses, you'd be surprised to find how relevant this figure can be. While you may wonder if it has no use in your life, we all should grasp the concept to ace those exams in school.
To save your time and create this information easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with images and examples to help with your personal study sessions. We will also discuss some real-life and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are theories that remain to be relevant as you move ahead with geometry and more complicated theorems and proofs. We will answer this query with a simple explanation in this single point.
Definition
A linear pair of angles is the term 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 simply, linear pairs of angles are two angles that sit on the same line and together create a straight line. The sum of the angles in a linear pair will at all times create a straight angle equal to 180 degrees.
It is important to keep in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This means that at all times form on a straight line and are always supplementary angles.
It is important to explain that, although the linear pair are constantly 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 understand every example given to you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is acknowledged without having to prove; it is considered obvious and self-explanatory. 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 implies that if two angles create a linear pair, then uncommon arms of both angles create a straight angle among them. This is also known as a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are some drawn examples with their individual answers.
Example One
As we can see in this instance, we have two angles that are adjacent to each other. As you can observe in the image, the adjacent angles form a linear pair due to the fact that the total of their measurement 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 have two lines intersect, making four angles. Not all angles makes 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 instance, 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 look at the axiom and characteristics of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle totals add up to 180 degrees. As a effect, we can conclude that this figure has no linear pair unless we expand a straight line.
Uses of Linear Pair of Angles
Now that we have explored what linear pairs are and have looked at some instances, let’s understand how this concept can be applied in geometry and the real-life scenario.
In Real-World Situations
There are multiple utilizations of linear pairs of angles in real life. One such example is architects, who apply these axioms in their daily job to check if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also utilize masters in this field to make their work simpler. They utilize linear pairs of angles to make sure that two adjacent walls create a 90-degree angle with the floor.
Engineers also utilizes linear pairs of angles frequently. They do so by working out the pressure on the beams and trusses.
In Geometry
Linear pairs of angles also perform a role in geometry proofs. A regular proof that utilizes linear pairs is the alternate interior angles theorem. This concept expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.
The proof of vertical angles also replies on linear pairs of angles. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to each other. Because of above-mentioned two rules, you only need to figure out the measurement of one angle to work out the measure of the rest.
The theorem of linear pairs is further employed for more complex uses, such as working out the angles in polygons. It’s critical to grasp the fundamentals of linear pairs, so you are ready for more progressive geometry.
As demonstrated, linear pairs of angles are a somewhat easy theorem with some interesting applications. Next time you're out and about, take note if you can see some linear pairs! And, if you're attending a geometry class, bear in mind how linear pairs may be helpful in proofs.
Improve Your Geometry Skills through Grade Potential
Geometry is entertaining and beneficial, maily if you are interested in the field of architecture or construction.
Still, if you're having difficulty understanding linear pairs of angles (or any concept in geometry), consider signing up for a tutoring session with Grade Potential. One of our expert teachers can help you comprehend the material and nail your next exam.
