Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With several real-world applications, you'd be amazed to find how useful this figure can be. Although you might think it has no use in your life, we all should understand the concept to nail those exams in school.
To save your time and offer this info easy to access, here is an preliminary insight into the properties of a linear pair of angles, with visualizations and examples to help with your personal study sessions. We will also talk about few real-life and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are concepts that continue to be applicable as you go forward in geometry and more complicated theorems and proofs. We will answer this query with a straightforward definition in this unique point.
Definition
A linear pair of angles is the name designated 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 sit on the same line and together create a straight line. The sum of the angles in a linear pair will at all times make a straight angle equal to 180 degrees.
It is important to note that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This suggests that they always form on a straight line and are at all times supplementary angles.
It is important to explain that, although the linear pair are always adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
With the definition simplified, we will examine the two axioms earnestly to fully understand every example thrown at you.
Let’s start by defining what an axiom is. It is a mathematical postulate or assumption that is accepted without proof; it is deemed evident and self-explanatory. A linear pair of angles has two axioms connected with them.
The first axiom states that if a ray stands on a line, the adjacent angles will make a straight angle, also known as a linear pair.
The second axiom implies that if two angles create a linear pair, then uncommon arms of both angles create a straight angle between them. In other words, they make a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are a few diagram examples with their corresponding explanations.
Example One
Here in this instance, we have two angles that are adjacent to one another. As you can observe in the image, the adjacent angles form a linear pair due to the fact that 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 example, 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 case, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This example represents an intersection of three lines. Let's take note of the axiom and characteristics of linear pairs.
∠A 150 degrees
∠B: 50 degrees
∠C: 160 degrees
None of the angle combinations add up to 180 degrees. As a result, 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 gone through what linear pairs are and have looked at some instances, let's see how this theorem can be used in geometry and the real world.
In Real-Life Situations
There are many applications of linear pairs of angles in real life. One such case is architects, who utilize these axioms in their daily job to establish if two lines are perpendicular and creates a straight angle.
Construction and Building professionals also employ masters in this matter to make their work less complex. They employ linear pairs of angles to make sure that two adjacent walls make a 90-degree angle with the ground.
Engineers also uses linear pairs of angles frequently. They do so by calculating the weight on the beams and trusses.
In Geometry
Linear pairs of angles as well play a role in geometry proofs. A regular proof that employs linear pairs is the alternate interior angles theorem. This concept states that if two lines are parallel and intersected by a transversal line, the alternate interior angles made are congruent.
The proof of vertical angles additionally replies on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are at all times equal to each other. Because of previously mentioned two rules, you only need to determine the measurement of any one angle to determine the measurement of the rest.
The concept of linear pairs is further utilized for more complicated applications, such as working out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more progressive geometry.
As you can see, linear pairs of angles are a comparatively simple concept with several engaging implementations. Later when you're out and about, see if you can notice any linear pairs! And, if you're participating in a geometry class, be on the lookout for how linear pairs might be helpful in proofs.
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