Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is a significant concept in geometry. With several real-life applications, you'd be surprised to figure out how relevant this figure can be. Although you may wonder if it has no application in your life, we all should learn the concept to ace those examinations in school.
To save your time and create this information easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to guide with your private study sessions. We will also talk about few real-life and geometric uses.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that exist to be useful as you move ahead with geometry and more sophisticated theorems and proofs. We will answer this query with a simple explanation in this unique point.
Definition
A linear pair of angles is the term provided to two angles that are situated on a straight line and have the sum of their measurement of angles is 180 degrees.
To put it simply, linear pairs of angles are two angles that are aligned on the same line and together create a straight line. The sum of the angles in a linear pair will always produce a straight angle equal to 180 degrees.
It is crucial to note that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This implies that at all times form on a straight line and are at all times supplementary angles.
It is crucial to clarify that, even though the linear pair are at all times adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
Through the definition simplified, we will examine the two axioms earnestly to completely comprehend any example provided to you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is approved without proof; it is believed clear and self-explanatory. A linear pair of angles has two axioms associated 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 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 figure examples with their individual explanations.
Example One
As we can see in this instance, we have two angles that are adjacent to one another. As you can notice in the diagram, 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 example, we have two lines intersect, making four angles. Not all angles makes a linear pair, but each angle and the one adjacent 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 an intersection 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 figure has no linear pair until we expand one straight line.
Implementations of Linear Pair of Angles
At the moment we have gone through what linear pairs are and have looked at some examples, let’s check how this theorem can be utilized in geometry and the real-life scenario.
In Real-Life Situations
There are multiple implementations of linear pairs of angles in real life. One common case is architects, who apply these axioms in their daily job to establish if two lines are perpendicular and makes a straight angle.
Builders and construction professionals also use expertise in this subject to make their work easier. They utilize linear pairs of angles to assure that two adjacent walls make a 90-degree angle with the floor.
Engineers also uses linear pairs of angles frequently. They do so by calculating the pressure on the beams and trusses.
In Geometry
Linear pairs of angles also perform a role in geometry proofs. A common proof that employs linear pairs is the alternate interior angles concept. This theorem explains 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 depends on linear pairs of angles. Although the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equal to one another. Because of these two rules, you only need to determine the measure of any one angle to figure out the measurement of the rest.
The theorem of linear pairs is further utilized for more complex implementation, such as figuring out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more complex geometry.
As demonstrated, linear pairs of angles are a somewhat easy concept with some fascinating applications. Later when you're out and about, see if you can spot some linear pairs! And, if you're taking a geometry class, take notes on how linear pairs may be useful in proofs.
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