Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is a significant subject in geometry. With multiple real-life functions, you'd be amazed to find how applicable this figure can be. Even though you might wonder if it has no use in your life, we all must grasp the ideas to nail those exams in school.
To save you time and make this data easy to access, here is an preliminary insight into the properties of a linear pair of angles, with images and examples to guide with your private study sessions. We will also talk about few real-life and geometric uses.
What Is a Linear Pair of Angles?
Linearity, angles, and intersections are theories that continue to be useful as you go forward in geometry and more complex theorems and proofs. We will answer this question with a easy definition in this unique point.
Definition
A linear pair of angles is the term provided to two angles that are located 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 together create a straight line. The sum of the angles in a linear pair will at all times produce a straight angle equivalent
times to 180 degrees.
It is important to bear in mind that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This suggests that at all times form on a straight line and are always supplementary angles.
It is essential to clarify that, although the linear pair are at all times adjacent angles, adjacent angles not at all times linear pairs.
The Linear Pair Axiom
With the definition clarified, we will study the two axioms earnestly to completely comprehend any example thrown at you.
Initiate with definition of what an axiom is. It is a mathematical postulate or hypothesis that is approved without having to prove; it is believed clear 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 create a straight angle, also known as a linear pair.
The second axiom implies that if two angles makes a linear pair, then uncommon arms of both angles makes a straight angle among them. In other words, they make a straight line.
Examples of Linear Pairs of Angles
To visualize these axioms better, here are a few diagram examples with their corresponding explanations.
Example One
In this example, we have two angles that are adjacent to each other. As you can notice in the figure, the adjacent angles form a linear pair since 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 close to it makes 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 shows a common point 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 combinations add up to 180 degrees. As a result, we can conclude that this diagram has no linear pair until we expand one straight line.
Implementations of Linear Pair of Angles
Now that we have explored what linear pairs are and have observed some cases, let’s understand how this concept can be utilized in geometry and the real-life scenario.
In Real-World Situations
There are multiple uses of linear pairs of angles in real-world. One familiar case is architects, who use these axioms in their day-to-day work to determine if two lines are perpendicular and form a straight angle.
Construction and Building professionals also utilize experts in this field to make their job easier. They use linear pairs of angles to make sure that two adjacent walls make a 90-degree angle with the floor.
Engineers also uses linear pairs of angles frequently. They do so by working 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 theorem. This concept 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 as well replies on linear pairs of angles. Although the adjacent angles are supplementary and sum up to 180 degrees, the opposite vertical angles are always equivalent to one another. Because of previously mentioned two rules, you are only required to know the measurement of any one angle to figure out the measurement of the rest.
The theorem of linear pairs is further used for more complicated uses, such as figuring out the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared for more complex geometry.
As shown, linear pairs of angles are a relatively easy theorem with some fascinating implementations. Next time you're out and about, take note if you can spot some linear pairs! And, if you're participating in a geometry class, take notes on how linear pairs may be useful in proofs.
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