Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is an essential subject in geometry. With several real-world applications, you'd be surprised to find how applicable this figure can be. While you might wonder if it has no use in your life, we all should learn the concept to nail those exams in school.
To save your time and offer this info easily accessible, here is an preliminary insight into the characteristics of a linear pair of angles, with visualizations and examples to guide with your private study sessions. We will also talk about few real-world and geometric uses.
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 sophisticated theorems and proofs. We will answer this query with a straightforward explanation in this unique point.
Definition
A linear pair of angles is the term given to two angles that are located on a straight line and have the sum of their angles measure 180 degrees.
To put it easily, linear pairs of angles are two angles that are adjacent 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 equal to 180 degrees.
It is crucial to note that linear pairs are at all times at adjacent angles. They share a common vertex and a common arm. This means that they always make on a straight line and are always supplementary angles.
It is important to explain that, although the linear pair are constantly adjacent angles, adjacent angles never constantly linear pairs.
The Linear Pair Axiom
Over the precise explanation, we will study the two axioms earnestly to fully grasp any example thrown at you.
First, let's define what an axiom is. It is a mathematical postulate or assumption that is accepted without having to prove; it is deemed clear and self-explanatory. A linear pair of angles has two axioms linked with them.
The first axiom implies that if a ray stands on a line, the adjacent angles will make a straight angle, making them a linear pair.
The second axiom states 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 imagine these axioms better, here are a few figure examples with their respective answers.
Example One
As we can see in this example, we have two angles that are next 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 measurement 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, making four angles. Not every angles form a linear pair, but respective angle and the one next 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 case represents convergence of three lines. Let's observe 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 effect, we can come to the conclusion that this figure has no linear pair unless we stretch one straight line.
Applications of Linear Pair of Angles
Now that we have learned what linear pairs are and have observed some examples, let's see how this theorem can be applied in geometry and the real world.
In Real-Life Scenarios
There are multiple applications of linear pairs of angles in real life. One familiar example is architects, who use these axioms in their day-to-day work to check if two lines are perpendicular and form a straight angle.
Construction and Building professionals also use masters in this subject to make their job simpler. They use linear pairs of angles to make sure that two close walls make a 90-degree angle with the floor.
Engineers also utilizes linear pairs of angles frequently. They do so by figuring out the tension in beams and trusses.
In Geometry
Linear pairs of angles additionally perform a role in geometry proofs. A common proof that uses linear pairs is the alternate interior angles theorem. This theorem explains that if two lines are parallel and intersected by a transversal line, the alternate interior angles formed are congruent.
The proof of vertical angles additionally replies on linear pairs of angles. Even though the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equivalent to one another. Because of these two rules, you are only required to know the measurement of one angle to figure out the measure of the rest.
The theorem of linear pairs is also utilized for more complex applications, such as working out the angles in polygons. It’s important to understand the basics of linear pairs, so you are prepared for more complex geometry.
As you can see, linear pairs of angles are a relatively easy theorem with some fascinating uses. Later when you're out and about, take note if you can spot any linear pairs! And, if you're taking a geometry class, be on the lookout for how linear pairs may be helpful in proofs.
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