Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential subject in geometry. With so many real-life uses, you'd be astonished to figure out how useful this figure can be. Although you might wonder if it has no relevant in your life, we all should learn the concept to nail those exams 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 visualizations and examples to guide with your private study sessions. We will also discuss some 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 move ahead with geometry and more complex theorems and proofs. We will answer this question with a straightforward definition in this unique point.
Definition
A linear pair of angles is the name given to two angles that are positioned on a straight line and the total of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that sit on the same line and pair up to create a straight line. The total of the angles in a linear pair will at all times produce a straight angle equal to 180 degrees.
It is crucial to note that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This suggests that they always create on a straight line and are always supplementary angles.
It is important to make clear that, while the linear pair are always adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
Through the definition simplified, we will study the two axioms seriously to completely understand any example provided to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is approved without having to prove; it is deemed obvious and self-evident. A linear pair of angles has two axioms connected with them.
The first axiom establishes that if a ray stands on a line, the adjacent angles will form a straight angle, also known as a linear pair.
The second axiom implies that if two angles produces a linear pair, then uncommon arms of both angles makes a straight angle among them. This is also known as a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are a few drawn examples with their corresponding answers.
Example One
As we can see in this example, we have two angles that are adjacent to each other. As you can see in the diagram, the adjacent angles form a linear pair because the total of their measures 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
In this instance, we have two lines intersect, producing four angles. Not every 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 example, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This example represents a common point of three lines. Let's observe the axiom and properties of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations add up to 180 degrees. As a effect, we can come to the conclusion that this image has no linear pair until we extend one straight line.
Applications of Linear Pair of Angles
At the moment we have explored what linear pairs are and have observed some instances, let’s check how this theorem can be applied in geometry and the real-life scenario.
In Real-World Scenarios
There are several implementations of linear pairs of angles in real life. One common case is architects, who use these axioms in their day-to-day work to identify if two lines are perpendicular and makes a straight angle.
Builders and construction professionals also employ masters in this subject to make their work simpler. They use linear pairs of angles to assure that two close walls form a 90-degree angle with the ground.
Engineers also uses linear pairs of angles frequently. They do so by working out the weight on the beams and trusses.
In Geometry
Linear pairs of angles as well perform a function in geometry proofs. A common proof that utilizes 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 made are congruent.
The proof of vertical angles also replies on linear pairs of angles. Although the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equivalent to one another. Because of previously mentioned two rules, you only need to determine the measure of any one angle to work out the measurement of the rest.
The theorem of linear pairs is further employed for more complex uses, such as measuring the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are prepared for more complex geometry.
As demonstrated, linear pairs of angles are a relatively simple concept with few interesting applications. Later when you're out and about, take note if you can spot any linear pairs! And, if you're participating in a geometry class, take notes on how linear pairs might be useful in proofs.
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