October 04, 2022

Linear Pair of AnglesDefinition, Axiom, Examples

The linear pair of angles is an important concept in geometry. With so many real-world functions, you'd be amazed to figure out how applicable this figure can be. While you might think it has no relevant in your life, we all should learn the ideas to nail those exams in school.

To save your time and offer this data easy to access, here is an introductory insight into the characteristics of a linear pair of angles, with images and examples to assist with your private study sessions. We will also discuss few real-life and geometric uses.

What Is a Linear Pair of Angles?

Linearity, angles, and intersections are concepts that continue to be useful as you progress in geometry and more complex theorems and proofs. We will answer this question with a straightforward explanation in this single point.


A linear pair of angles is the name provided to two angles that are located on a straight line and have the sum of their measurement of angles is 180 degrees.

To put it simply, linear pairs of angles are two angles that are adjacent on the same line and together form 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 keep in mind that linear pairs are always at adjacent angles. They share a common apex and a common arm. This implies that they always make on a straight line and are always 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

With the definition simplified, we will explore the two axioms earnestly to completely grasp every example given to you.

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

The first axiom implies that if a ray stands on a line, the adjacent angles will create a straight angle, also known as a linear pair.

The second axiom establishes that if two angles produces 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 visualize these axioms better, here are some drawn examples with their respective answers.

Example One

As we can see in this instance, 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 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

In this example, we possess two lines intersect, creating four angles. Not every angles creates a linear pair, but each 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 shows convergence 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 effect, we can conclude that this example has no linear pair unless we stretch a straight line.

Uses of Linear Pair of Angles

Now that we have gone through what linear pairs are and have looked at some examples, let’s check how this concept can be applied in geometry and the real-life scenario.

In Real-World Scenarios

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

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

Engineers also apply linear pairs of angles frequently. They do so by figuring out the pressure on the beams and trusses.

In Geometry

Linear pairs of angles also perform a function in geometry proofs. A common proof that utilizes linear pairs is the alternate interior angles concept. 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 replies on linear pairs of angles. Although 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 these two rules, you are only required to determine the measure of any one angle to determine the measurement of the rest.

The theorem of linear pairs is further used for more sophisticated implementation, such as figuring out the angles in polygons. It’s critical to grasp the fundamentals of linear pairs, so you are prepared for more progressive geometry.

As shown, linear pairs of angles are a relatively simple theorem with some fascinating uses. Next time you're out and about, see if you can notice any linear pairs! And, if you're attending a geometry class, be on the lookout for how linear pairs may be useful in proofs.

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