Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With several real-world uses, you'd be astonished to find how applicable this figure can be. Even though you might wonder if it has no relevant in your life, we all must learn the concept to nail those examinations in school.
To save your time and create this info easily accessible, here is an preliminary insight into the properties of a linear pair of angles, with images and examples to guide with your personal study sessions. We will also talk about some real-world and geometric applications.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are ideas that continue to be relevant as you go forward in geometry and more complicated theorems and proofs. We will answer this question with a straightforward definition in this unique point.
Explanation
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 angles measure 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 total of the angles in a linear pair will always produce a straight angle equivalent
times 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 implies that at all times make on a straight line and are always supplementary angles.
It is important to clarify that, although the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
Over the definition clarified, we will study the two axioms seriously to completely understand any example provided to you.
First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is approved without having to prove; it is considered clear and self-explanatory. A linear pair of angles has two axioms linked 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 establishes that if two angles makes a linear pair, then uncommon arms of both angles makes a straight angle between them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are some drawn examples with their respective answers.
Example One
As we can see in this example, we have two angles that are neighboring each other. As you can notice in the image, the adjacent angles form a linear pair due to the fact that the total of their measures is equivalent to 180 degrees. They are also supplementary angles, since 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, creating four angles. Not all angles makes a linear pair, but each 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 convergence 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 add up to 180 degrees. As a consequence, we can conclude that this figure has no linear pair until we extend a straight line.
Applications of Linear Pair of Angles
At the moment we have learned what linear pairs are and have looked at some cases, let’s check how this theorem can be used in geometry and the real-life scenario.
In Real-World Situations
There are multiple applications of linear pairs of angles in real life. One familiar case is architects, who apply these axioms in their daily work to establish if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also employ masters in this matter to make their work less complex. They employ linear pairs of angles to ensure that two close walls make a 90-degree angle with the floor.
Engineers also apply linear pairs of angles frequently. They do so by figuring out the tension in beams and trusses.
In Geometry
Linear pairs of angles additionally play a function in geometry proofs. A ordinary proof that utilizes linear pairs is the alternate interior angles theorem. This theorem states that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.
The proof of vertical angles also 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 above-mentioned two rules, you only need to determine the measure of any one angle to work out the measure of the rest.
The theorem of linear pairs is also utilized for more complicated implementation, such as working out the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are ready for more advanced geometry.
As demonstrated, linear pairs of angles are a somewhat easy theorem with some interesting applications. Next time you're out and about, take note if you can see some 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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