Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is a significant subject in geometry. With several real-life uses, you'd be surprised to figure out how relevant this figure can be. Although you might think it has no use in your life, we all must learn the concept to nail those exams in school.
To save your time and make this data easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to help with your private 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 ideas that exist to be useful as you go forward in geometry and more sophisticated theorems and proofs. We will answer this question with a straightforward explanation in this unique point.
Definition
A linear pair of angles is the term given to two angles that are situated on a straight line and have the sum of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that are aligned on the same line and together create a straight line. The total of the angles in a linear pair will always produce a straight angle equivalent
times to 180 degrees.
It is crucial to keep in mind that linear pairs are always at adjacent angles. They share a common apex and a common arm. This implies that at all times create on a straight line and are always supplementary angles.
It is important to clarify that, even though the linear pair are always adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
With the definition clarified, we will examine the two axioms earnestly to fully grasp any example thrown at you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is accepted without having to prove; it is deemed clear and self-explanatory. A linear pair of angles has two axioms linked with them.
The first axiom states that if a ray is located on a line, the adjacent angles will make a straight angle, namely called a linear pair.
The second axiom states that if two angles create a linear pair, then uncommon arms of both angles create 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 some 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 notice in the figure, 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, 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
Here, we possess two lines intersect, making 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 example, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This case presents an intersection of three lines. Let's look at 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 result, we can come to the conclusion that this example has no linear pair unless we extend a straight line.
Applications of Linear Pair of Angles
At the moment we have explored what linear pairs are and have observed some cases, let's see how this concept can be applied in geometry and the real world.
In Real-Life Situations
There are multiple utilizations of linear pairs of angles in real life. One common case is architects, who use these axioms in their day-to-day job to establish if two lines are perpendicular and creates a straight angle.
Construction and Building professionals also employ experts in this subject to make their job less complex. They use linear pairs of angles to make sure that two adjacent walls make a 90-degree angle with the floor.
Engineers also utilizes linear pairs of angles regularly. They do so by working out the pressure on the beams and trusses.
In Geometry
Linear pairs of angles as well perform a function in geometry proofs. A common proof that uses 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 also depends on linear pairs of angles. Even though the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are at all times equal to one another. Because of previously mentioned two rules, you are only required to determine the measurement of one angle to figure out the measure of the rest.
The theorem of linear pairs is subsequently employed for more complicated implementation, such as working out the angles in polygons. It’s important to understand the fundamentals of linear pairs, so you are prepared for more complex geometry.
As shown, linear pairs of angles are a relatively simple concept with some engaging implementations. Next time you're out and about, observe if you can see any linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be useful in proofs.
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