Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential concept in geometry. With multiple real-life applications, you'd be amazed to figure out how useful this figure can be. While you may believe it has no relevant in your life, we all should understand the ideas to nail those exams in school.
To save you time and make this data easily accessible, here is an preliminary insight into the characteristics of a linear pair of angles, with images and examples to help with your personal study sessions. We will also talk about some real-world and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that continue to be useful as you go forward in geometry and more sophisticated 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 provided 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 adjacent 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 equal to 180 degrees.
It is important to note that linear pairs are always at adjacent angles. They share a common apex and a common arm. This means that they always make on a straight line and are at all times supplementary angles.
It is essential to clarify that, even though the linear pair are constantly adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
With the definition clarified, we will study the two axioms seriously to fully comprehend any 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 deemed obvious and self-evident. A linear pair of angles has two axioms linked with them.
The first axiom establishes 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 create a linear pair, then uncommon arms of both angles makes a straight angle between them. This is also known as a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are some diagram examples with their individual explanations.
Example One
In this instance, we have two angles that are neighboring each other. As you can see in the image, the adjacent angles form a linear pair since the sum of their measures equals 180 degrees. They are also supplementary angles, because 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, producing four angles. Not all angles creates a linear pair, but each 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 instance presents an intersection 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 totals sum up to 180 degrees. As a consequence, we can conclude that this image has no linear pair unless we extend a straight line.
Implementations of Linear Pair of Angles
Now that we have learned what linear pairs are and have looked at some examples, let's see how this theorem can be utilized in geometry and the real world.
In Real-Life Situations
There are multiple applications of linear pairs of angles in real life. One familiar case is architects, who use these axioms in their day-to-day work to determine if two lines are perpendicular and creates a straight angle.
Construction and Building professionals also use expertise in this field to make their work easier. They use linear pairs of angles to assure that two adjacent walls create a 90-degree angle with the ground.
Engineers also uses linear pairs of angles frequently. They do so by figuring out the weight on the beams and trusses.
In Geometry
Linear pairs of angles as well play a role 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 made 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 always equivalent to one another. Because of previously mentioned two rules, you only need to figure out the measurement of one angle to figure out the measurement of the rest.
The theorem of linear pairs is subsequently utilized for more complicated applications, such as working out the angles in polygons. It’s essential to grasp the fundamentals of linear pairs, so you are ready for more advanced geometry.
As you can see, linear pairs of angles are a somewhat easy concept with several interesting uses. Next time you're out and about, see if you can see some linear pairs! And, if you're participating in a geometry class, bear in mind how linear pairs might be useful in proofs.
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