# Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an essential subject in geometry. With multiple real-life applications, you'd be amazed to figure out how relevant this figure can be. Although you might believe it has no relevant in your life, we all should understand the ideas to nail those examinations in school.

To save you time and create this info readily accessible, here is an preliminary insight into the properties of a linear pair of angles, with visualizations and examples to assist with your personal 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 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 positioned on a straight line and have the sum of their measurement of angles is 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 total of the angles in a linear pair will always create a straight angle equivalent

times to 180 degrees.

It is essential to bear in mind that linear pairs are always at adjacent angles. They share a common vertex and a common arm. This means that at all times make on a straight line and are at all times supplementary angles.

It is essential to make clear that, even though the linear pair are at all times adjacent angles, adjacent angles aren't always linear pairs.

### The Linear Pair Axiom

Through the definition simplified, we will explore the two axioms earnestly to completely understand any example given to you.

Initiate with definition of what an axiom is. It is a mathematical postulate or hypothesis that is accepted without having to prove; it is considered obvious and self-explanatory. A linear pair of angles has two axioms associated with them.

The first axiom states that if a ray is located on a line, the adjacent angles will form a straight angle, namely called a linear pair.

The second axiom establishes that if two angles makes a linear pair, then uncommon arms of both angles create a straight angle among them. In other words, they make a straight line.

## Examples of Linear Pairs of Angles

To envision these axioms better, here are some figure examples with their individual answers.

### Example One

As we can see in this example, we have two angles that are next to each other. As you can observe in the diagram, the adjacent angles form a linear pair due to the fact that the total of their measures equals 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

In this instance, we have two lines intersect, making four angles. Not every 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 instance shows convergence 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 sum up to 180 degrees. As a effect, we can come to the conclusion that this example has no linear pair until we stretch one straight line.

## Uses of Linear Pair of Angles

Now that we have explored what linear pairs are and have looked at some instances, let’s understand how this concept can be applied in geometry and the real world.

### In Real-Life Scenarios

There are many utilizations of linear pairs of angles in real life. One such example is architects, who utilize these axioms in their day-to-day job to identify if two lines are perpendicular and makes a straight angle.

Construction and Building professionals also employ expertise in this subject to make their job simpler. They employ linear pairs of angles to ensure that two adjacent walls form a 90-degree angle with the floor.

Engineers also uses linear pairs of angles frequently. They do so by working out the tension in beams and trusses.

### In Geometry

Linear pairs of angles additionally play a function in geometry proofs. A ordinary proof that employs 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 made are congruent.

The proof of vertical angles also depends on linear pairs of angles. While the adjacent angles are supplementary and add up to 180 degrees, the opposite vertical angles are always equal to each other. Because of above-mentioned two rules, you only need to figure out the measurement of any one angle to figure out the measure of the rest.

The theorem of linear pairs is also used for more sophisticated implementation, such as determining the angles in polygons. It’s essential to understand the basics of linear pairs, so you are ready for more advanced geometry.

As you can see, linear pairs of angles are a somewhat simple concept with several engaging applications. Later when you're out and about, take note if you can spot any linear pairs! And, if you're participating in a geometry class, bear in mind how linear pairs might be helpful in proofs.

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