Linear Pair of AnglesDefinition, Axiom, Examples
The linear pair of angles is a significant concept in geometry. With multiple real-life applications, you'd be surprised to figure out how useful this figure can be. Even though you may think it has no use in your life, we all must understand the concept to nail those examinations in school.
To save your time and create this data readily accessible, here is an introductory insight into the characteristics of a linear pair of angles, with diagrams and examples to assist with your personal 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 theories that remain to be applicable as you move ahead with geometry and more sophisticated theorems and proofs. We will answer this query with a easy explanation in this unique point.
Definition
A linear pair of angles is the term designated 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 sit on the same line and pair up to form a straight line. The sum of the angles in a linear pair will always produce a straight angle equivalent
times to 180 degrees.
It is essential to keep in mind that linear pairs are at all times at adjacent angles. They share a common apex and a common arm. This implies that they always form on a straight line and are at all times supplementary angles.
It is important to explain that, even though the linear pair are constantly adjacent angles, adjacent angles aren't always linear pairs.
The Linear Pair Axiom
Through the definition simplified, we will study the two axioms seriously to fully understand every example given to you.
Let’s start by defining what an axiom is. It is a mathematical postulate or hypothesis that is accepted without proof; 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, making them a linear pair.
The second axiom implies that if two angles makes 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 visualize these axioms better, here are a few drawn examples with their individual answers.
Example One
Here in this example, we have two angles that are neighboring one another. As you can see in the figure, the adjacent angles form a linear pair because the sum of their measures is equivalent to 180 degrees. They are also supplementary angles, as 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, creating four angles. Not all angles makes a linear pair, but each angle and the one adjacent 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 presents convergence 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 totals add up to 180 degrees. As a consequence, we can conclude that this image has no linear pair unless we expand one straight line.
Uses of Linear Pair of Angles
At the moment we have explored what linear pairs are and have looked at some examples, let’s check how this theorem can be used in geometry and the real-life scenario.
In Real-World Scenarios
There are several uses of linear pairs of angles in real life. One familiar case is architects, who apply these axioms in their daily work to identify if two lines are perpendicular and makes a straight angle.
Construction and Building professionals also use expertise in this subject to make their job easier. They utilize linear pairs of angles to assure that two close walls form a 90-degree angle with the ground.
Engineers also utilizes linear pairs of angles regularly. They do so by working out the tension in beams and trusses.
In Geometry
Linear pairs of angles as well play a function in geometry proofs. A common proof that utilizes linear pairs is the alternate interior angles theorem. This concept expresses that if two lines are parallel and intersected by a transversal line, the alternate interior angles created are congruent.
The proof of vertical angles additionally depends 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 equal to one another. Because of above-mentioned two rules, you only need to know the measure of one angle to work out the measure of the rest.
The theorem of linear pairs is further employed for more complex applications, such as working out the angles in polygons. It’s critical to understand the basics of linear pairs, so you are prepared for more complex geometry.
As shown, linear pairs of angles are a somewhat simple theorem with few interesting implementations. Next time you're out and about, observe if you can spot some linear pairs! And, if you're attending a geometry class, bear in mind how linear pairs may be useful in proofs.
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