Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With so many real-world applications, you'd be surprised to find how relevant this figure can be. Although you might believe it has no relevant in your life, we all should grasp the ideas to nail those tests in school.
To save your time and offer this info readily accessible, here is an preliminary insight into the characteristics of a linear pair of angles, with images and examples to assist with your private study sessions. We will also discuss few real-life and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that exist to be relevant as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a straightforward explanation in this single point.
Explanation
A linear pair of angles is the name given 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 easily, 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 always produce a straight angle equal to 180 degrees.
It is essential to note that linear pairs are always at adjacent angles. They share a common vertex 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, while the linear pair are at all times adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
With the precise explanation, we will study the two axioms earnestly to completely grasp any example provided to you.
Initiate with definition of what an axiom is. It is a mathematical postulate or assumption that is approved without having to prove; it is believed clear 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 states that if two angles produces a linear pair, then uncommon arms of both angles produces a straight angle between 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 respective explanations.
Example One
As we can see in this example, we have two angles that are next to one another. As you can see in the image, the adjacent angles form a linear pair due to the fact that 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
In this instance, we possess two lines intersect, making four angles. Not every angles makes 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 instance, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This case represents 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 combinations sum up to 180 degrees. As a consequence, we can come to the conclusion that this diagram has no linear pair until we stretch 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 understand how this concept can be implemented in geometry and the real world.
In Real-World Situations
There are multiple utilizations of linear pairs of angles in real life. One familiar example is architects, who apply these axioms in their daily work to identify if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also use experts in this field to make their job easier. They utilize linear pairs of angles to make sure that two adjacent 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 perform a role in geometry proofs. A common 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 formed are congruent.
The proof of vertical angles as well replies on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equal to each other. Because of previously mentioned two rules, you are only required to know the measurement of one angle to determine the measure of the rest.
The concept of linear pairs is subsequently employed for more complex implementation, such as determining the angles in polygons. It’s essential to understand the basics of linear pairs, so you are ready for more complex geometry.
As demonstrated, linear pairs of angles are a relatively simple theorem with some interesting implementations. Next time you're out and about, see if you can notice any linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs might be helpful in proofs.
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