Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential subject in geometry. With so many real-world applications, you'd be astonished to find how relevant this figure can be. Although you might think it has no use in your life, we all must understand the ideas to nail those examinations in school.
To save you time and offer this info easily accessible, here is an preliminary insight into the characteristics of a linear pair of angles, with images and examples to assist with your personal study sessions. We will also talk about few real-world and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are concepts that remain to be applicable as you progress in geometry and more complicated theorems and proofs. We will answer this question with a easy definition in this unique point.
Explanation
A linear pair of angles is the name provided to two angles that are located on a straight line and the total of their measurement of angles is 180 degrees.
To put it easily, linear pairs of angles are two angles that are aligned on the same line and together form a straight line. The total of the angles in a linear pair will always create a straight angle equivalent
times to 180 degrees.
It is crucial to keep in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This suggests that at all times form on a straight line and are at all times supplementary angles.
It is essential to make clear that, while the linear pair are at all times adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
With the definition clarified, we will examine the two axioms critical to fully comprehend every example provided to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is accepted without having to prove; it is deemed obvious and self-explanatory. A linear pair of angles has two axioms associated with them.
The first axiom implies 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 produces a linear pair, then uncommon arms of both angles produces a straight angle among them. In other words, they make a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are some diagram examples with their respective explanations.
Example One
In this example, we have two angles that are adjacent to one another. As you can notice in the figure, the adjacent angles form a linear pair because the total of their measurement is equivalent to 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, creating four angles. Not all angles form a linear pair, but respective 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 case, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This case represents a common point of three lines. Let's take note of 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.
Uses of Linear Pair of Angles
Now that we have learned what linear pairs are and have looked at some examples, let’s understand how this concept can be implemented in geometry and the real-life scenario.
In Real-World Scenarios
There are multiple applications of linear pairs of angles in real life. One familiar example is architects, who utilize these axioms in their day-to-day job to establish if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also use expertise in this field to make their job easier. They use linear pairs of angles to make sure that two adjacent walls form a 90-degree angle with the floor.
Engineers also apply linear pairs of angles frequently. They do so by working out the tension in beams and trusses.
In Geometry
Linear pairs of angles as well play a role in geometry proofs. A common proof that uses linear pairs is the alternate interior angles concept. This concept expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.
The proof of vertical angles additionally depends 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 each other. Because of these two rules, you are only required to know the measurement of any one angle to figure out the measurement of the rest.
The theorem of linear pairs is subsequently used for more complicated applications, such as figuring out the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared for more complex geometry.
As you can see, linear pairs of angles are a comparatively easy concept with few engaging uses. Later when you're out and about, take note if you can notice some linear pairs! And, if you're taking a geometry class, bear in mind how linear pairs might be useful in proofs.
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