Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important subject in geometry. With multiple real-world functions, you'd be surprised to discover how useful this figure can be. While you might wonder if it has no use in your life, we all should grasp the ideas to nail those examinations in school.
To save your time and create this info easy to access, here is an introductory insight into the properties of a linear pair of angles, with images and examples to guide with your private study sessions. We will also talk about some real-life and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that continue to be applicable as you move ahead with geometry and more complex theorems and proofs. We will answer this question with a simple definition in this unique point.
Definition
A linear pair of angles is the name provided to two angles that are situated on a straight line and the total of their angles measure 180 degrees.
To put it simply, linear pairs of angles are two angles that sit on the same line and together form a straight line. The sum of the angles in a linear pair will at all times make a straight angle equal to 180 degrees.
It is important to bear in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This implies that at all times make on a straight line and are always supplementary angles.
It is essential to make clear that, while the linear pair are always adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
Over the definition simplified, we will examine the two axioms earnestly to fully grasp any example provided to you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is acknowledged without proof; it is deemed evident and self-evident. A linear pair of angles has two axioms linked with them.
The first axiom establishes that if a ray stands on a line, the adjacent angles will form a straight angle, namely called a linear pair.
The second axiom states that if two angles makes a linear pair, then uncommon arms of both angles makes a straight angle among them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are a few diagram examples with their individual explanations.
Example One
Here in this example, we have two angles that are adjacent to each other. As you can see in the figure, the adjacent angles form a linear pair due to the fact that the sum of their measurement is equivalent to 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 example, we possess two lines intersect, creating 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 case, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This example represents convergence of three lines. Let's observe the axiom and characteristics of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations sum up to 180 degrees. As a effect, we can conclude that this diagram has no linear pair until we extend one straight line.
Implementations of Linear Pair of Angles
At the moment we have learned what linear pairs are and have looked at some cases, let's see how this concept can be used in geometry and the real world.
In Real-Life Situations
There are multiple implementations of linear pairs of angles in real-world. One such case is architects, who apply these axioms in their daily job to establish if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also employ experts in this subject to make their job less complex. They use linear pairs of angles to ensure that two adjacent walls make a 90-degree angle with the floor.
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 additionally perform a function in geometry proofs. A common proof that employs 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 created are congruent.
The proof of vertical angles as well replies on linear pairs of angles. While the adjacent angles are supplementary and add 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 measure of any one angle to determine the measurement of the rest.
The concept of linear pairs is further employed for more complex applications, such as measuring the angles in polygons. It’s important to grasp the basics of linear pairs, so you are ready for more progressive geometry.
As you can see, linear pairs of angles are a relatively simple theorem with some interesting implementations. Next time you're out and about, observe if you can spot any linear pairs! And, if you're participating in a geometry class, bear in mind how linear pairs may be useful in proofs.
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