Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With so many real-life applications, you'd be surprised to discover how useful this figure can be. Although you may wonder if it has no relevant in your life, we all must learn the ideas to nail those examinations in school.
To save your time and create this info easy to access, here is an introductory insight into the characteristics 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 relevant as you progress in geometry and more complicated theorems and proofs. We will answer this question with a easy explanation in this single point.
Definition
A linear pair of angles is the name given to two angles that are situated 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 are aligned 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 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 apex and a common arm. This means that they always make on a straight line and are always supplementary angles.
It is crucial to make clear 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 study the two axioms earnestly to completely comprehend every example given to you.
First, let's define what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; it is considered clear 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 make a straight angle, namely called a linear pair.
The second axiom establishes that if two angles create a linear pair, then uncommon arms of both angles create a straight angle between them. This is also known as a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are a few diagram examples with their corresponding answers.
Example One
In this instance, we have two angles that are adjacent to one another. As you can see in the image, the adjacent angles form a linear pair since the total of their measurement is equivalent to 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, creating four angles. Not all 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 instance represents an intersection of three lines. Let's look at 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 diagram has no linear pair until we extend a straight line.
Implementations of Linear Pair of Angles
At the moment we have gone through what linear pairs are and have observed some cases, let’s check how this concept can be utilized 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 example is architects, who utilize these axioms in their daily job to determine if two lines are perpendicular and form a straight angle.
Builders and construction professionals also utilize experts in this field to make their work less complex. They employ linear pairs of angles to ensure that two adjacent walls create a 90-degree angle with the ground.
Engineers also utilizes linear pairs of angles frequently. They do so by calculating the weight on the beams and trusses.
In Geometry
Linear pairs of angles also perform a role in geometry proofs. A regular proof that uses linear pairs is the alternate interior angles theorem. This theorem expresses 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. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equal to one another. Because of previously mentioned two rules, you only need to determine the measure of one angle to work out the measure of the rest.
The theorem of linear pairs is also employed for more sophisticated implementation, such as working out the angles in polygons. It’s critical to grasp the basics of linear pairs, so you are prepared for more complex geometry.
As shown, linear pairs of angles are a comparatively simple concept with several engaging uses. Next time you're out and about, see if you can notice some linear pairs! And, if you're attending a geometry class, take notes on how linear pairs may be helpful in proofs.
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