Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an important concept in geometry. With multiple real-world uses, you'd be astonished to find how useful this figure can be. Although you may wonder if it has no relevant in your life, we all must understand the concept to ace those examinations in school.
To save you time and make this information easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with visualizations and examples to help with your private study sessions. We will also discuss some real-life and geometric uses.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are concepts that exist to be useful as you go forward in geometry and more complex 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 provided to two angles that are located on a straight line and the total 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 pair up to form a straight line. The total of the angles in a linear pair will at all times create a straight angle equivalent
times to 180 degrees.
It is essential to note that linear pairs are always at adjacent angles. They share a common apex and a common arm. This implies that at all times create on a straight line and are always supplementary angles.
It is essential to clarify that, while the linear pair are always adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
Through the definition simplified, we will explore the two axioms critical to completely comprehend every example provided to you.
Initiate with definition of 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 associated with them.
The first axiom establishes that if a ray is located on a line, the adjacent angles will create a straight angle, namely called a linear pair.
The second axiom states that if two angles makes a linear pair, then uncommon arms of both angles create a straight angle among them. This is commonly called a straight line.
Examples of Linear Pairs of Angles
To imagine these axioms better, here are some figure examples with their respective answers.
Example One
In this example, we have two angles that are adjacent to each other. As you can observe in the image, the adjacent angles form a linear pair due to the fact that the sum of their measures 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
Here, we have two lines intersect, making four angles. Not every 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 instance, the linear pairs are:
∠A and ∠B
∠B and ∠C
∠C and ∠D
∠D and ∠A
Example Three
This instance shows an intersection 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 result, we can conclude that this example has no linear pair until we extend a straight line.
Implementations of Linear Pair of Angles
At the moment we have explored what linear pairs are and have observed some cases, let’s check how this concept can be implemented in geometry and the real-life scenario.
In Real-World Situations
There are many implementations of linear pairs of angles in real life. One common example is architects, who utilize these axioms in their daily job to check if two lines are perpendicular and creates a straight angle.
Builders and construction professionals also use experts in this matter to make their work simpler. They employ linear pairs of angles to ensure that two adjacent walls make a 90-degree angle with the ground.
Engineers also apply linear pairs of angles frequently. They do so by figuring out the weight on the beams and trusses.
In Geometry
Linear pairs of angles as well play a role in geometry proofs. A common proof that utilizes linear pairs is the alternate interior angles concept. This concept 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 as well replies on linear pairs of angles. While the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are at all times equivalent to each other. Because of previously mentioned two rules, you are only required to figure out the measure of any one angle to determine the measure of the rest.
The concept of linear pairs is subsequently utilized for more complicated uses, such as determining the angles in polygons. It’s critical to understand the basics of linear pairs, so you are ready for more complex geometry.
As you can see, linear pairs of angles are a somewhat 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 attending a geometry class, take notes on how linear pairs may be helpful in proofs.
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