Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With multiple real-life uses, you'd be surprised to find how useful this figure can be. Even though you may believe it has no use in your life, we all must grasp the ideas to nail those exams in school.
To save your time and make this information readily accessible, here is an preliminary insight into the properties of a linear pair of angles, with diagrams and examples to assist with your personal study sessions. We will also discuss few real-life and geometric uses.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are concepts that remain to be useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a easy definition in this unique point.
Explanation
A linear pair of angles is the term 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 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 important to bear in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This suggests that at all times make on a straight line and are at all times supplementary angles.
It is crucial to make clear that, although the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
Over the definition clarified, we will examine the two axioms critical to fully understand any example given to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is approved without proof; it is deemed evident 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 make a straight angle, also known as a linear pair.
The second axiom states that if two angles create a linear pair, then uncommon arms of both angles makes a straight angle among them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To envision these axioms better, here are some drawn examples with their corresponding answers.
Example One
In this example, we have two angles that are neighboring each other. As you can observe in the diagram, the adjacent angles form a linear pair since the total of their measures 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, creating four angles. Not all angles creates a linear pair, but each angle and the one close to it makes 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 example shows convergence 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 sum up to 180 degrees. As a effect, we can conclude that this example has no linear pair unless we stretch one straight line.
Uses of Linear Pair of Angles
At the moment we have explored what linear pairs are and have looked at some cases, let’s understand how this theorem can be implemented in geometry and the real-life scenario.
In Real-World Situations
There are many applications of linear pairs of angles in real-world. One common example is architects, who use these axioms in their daily work to establish if two lines are perpendicular and makes a straight angle.
Builders and construction professionals also utilize experts in this matter to make their work easier. They employ linear pairs of angles to assure that two adjacent walls create a 90-degree angle with the floor.
Engineers also utilizes linear pairs of angles frequently. They do so by figuring 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 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 made are congruent.
The proof of vertical angles also replies 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 equivalent to one another. Because of these two rules, you are only required to figure out the measurement of one angle to determine the measurement of the rest.
The theorem of linear pairs is also used for more complicated uses, such as measuring the angles in polygons. It’s important to grasp the fundamentals of linear pairs, so you are prepared for more advanced geometry.
As demonstrated, linear pairs of angles are a relatively easy concept with several interesting uses. Later when you're out and about, take note if you can spot 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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