A linear pair of angles is a pair of adjacent angles formed when two lines intersect. They share a common arm (or ray), and their non-common arms are on the same line, making a straight line.
Let’s learn what is a Linear Pair of Angle in geometry, including its definition, properties, axioms and examples.
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Linear Pair of Angles Definition
A linear pair of angles is formed when two adjacent angles share a common arm and their non-common arms form opposite rays, creating a straight line.
In other words, the sum of the measures of two linear pair angles is always 180°.
Linear Pair Explanation with Example
In the example given below, there is a straight line AB on which a ray OC intersect AB at O forming two angles namely angle AOC and angle BOC.
If we join these both angle we find that they have common vertex O and a common arm OC and they combine to form a straight line AB.
We know that the angle on one side of a straight line is 180° i.e. a straight angle. Hence, the angle AOC and the angle BOC are called a Linear Pair of angles.
Linear Pair of Angles Properties
These are some of the most important properties of a linear pair of angles:
- Sum of Measures: The two angles in a linear pair always combine to form a total angle measure of 180°.
- Adjacency: All linear pairs of angles are adjacent, meaning they share a common arm and a common vertex. However, it’s important to note that while all linear pairs are adjacent, not all adjacent angles necessarily form a linear pair.
- Common Vertex and Arm: Linear pairs of angles share both a common vertex (point of intersection) and a common arm (the side they share).
- Formation on a Straight Line: They are always formed on a straight line, contributing to the definition of a linear pair.
- Equivalent to Straight Angle Parts: Linear pairs of angles can be conceptualized as constituting two parts of either a 180-degree angle or a straight angle. This characteristic underscores their role in forming straight lines.
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Linear Pair Postulate
The postulate of linear pair says that,
When a ray is positioned on a line, the total measure of two adjacent angles formed is always 180°.
- Conversely, if the sum of two adjacent angles equals 180°, the non-common arms of these angles will form a straight line.
- Both of these postulates are collectively grouped under the term “linear pair axiom.”
Linear Pair Axioms
There are two axioms which are related to Linear Pair of angles.
Let’s learn about each of them in detail.
Linear Pair Axiom 1
This axiom of linear pair states that if a ray is positioned on a line, then the total measure of the two adjacent angles formed by the ray and the line is always 180°.
In simpler terms, when you have a straight line and place a ray on it, the sum of the two angles on either side of the ray will always equal 180°.
- This fundamental property is crucial in understanding the relationship between angles in a linear pair.
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In a straight line
with a ray placed on it, forming two angles ∠ABC and ∠CBD - According to this axiom, if a ray is on a line, the sum of the two adjacent angles is 180°. If m(∠ABC) + m(∠CBD) = 180°, then these angles form a linear pair.
- For instance, if m(∠ABC) = 80°, then m(∠CBD) would be 180°- 80°= 100°. This satisfies the axiom, confirming that ∠ ABC and ∠CBD form a linear pair.
Linear Pair Axiom 2
This axiom of linear pair states that if the sum of the measures of two adjacent angles is 180°, then the non-common arms of these angles form a straight line.
In other words, when you have two angles whose measures add up to 180°, you can conclude that the arms of these angles create a straight line.
- This axiom provides a way to identify the presence of a linear pair based on the sum of angle measures, reinforcing the concept that a linear pair of angles forms a straight line.
- If you know that m(∠ABC) + m(∠CBD) = 180°, then you can conclude that the non-common arms of these angles form a straight line.
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For example, if m(∠ABC) = 60° and m(∠CBD) = 120°, their sum is (60° + 120°)= 180°. By Axiom 2, this implies that the arms
and form a straight line.
Note: An axiom is a statement which is universally true and doesn’t need any proof.
Linear Pair vs. Supplementary Angles
Linear Pair of Angles and Supplementary Angles both sum to give 180°. However, there is significant difference between them. Let’s learn more about them in the table below:
| Characteristic | Linear Pair of Angles | Supplementary Angles |
|---|---|---|
|
Difference between Linear Pair and Supplementary Angles | ||
| Definition | Two adjacent angles with non-common sides forming a straight line. | Two angles whose sum equals 180°. |
| Sum of Measures | Always adds up to 180°. | Always adds up to 180°. |
| Formation | Always formed on a straight line. | Not necessarily formed on a straight line. |
| Adjacent Angles | All linear pairs are adjacent, but not all adjacent angles form a linear pair. |
All supplementary angles are adjacent. |
| Non-Common Arms | The non-common arms do not necessarily form a line. | The non-common arms form a straight line. |
| Examples | If ∠A and ∠B are a linear pair, then ∠A + ∠B = 180°. | If ∠C and ∠D are supplementary, then ∠C + ∠D = 180°. |
| Common Vertex and Arm | Linear pairs share a common vertex and a common arm. | Supplementary angles may or may not share a common vertex. |
Linear pair and Adjacent Angles
Let’s also discuss some key differences between a linear pair of angles and Adjacent Angles, which are:
| Aspect | Linear Pair of Angles | Adjacent Angles |
|---|---|---|
|
Differences between Linear Pair and Adjacent Angles | ||
| Definition | A linear pair of angles consists of two adjacent angles whose non-common sides form a straight line. | Adjacent angles are two angles that have a common vertex and a common side but do not overlap. |
| Angle Sum | The sum of the angles in a linear pair is always 180 degrees. | The sum of adjacent angles can be any value. There is no specific sum requirement. |
| Configuration | The non-common sides of the angles form a straight line (180 degrees). | The non-common sides do not necessarily form a straight line and can be oriented in any direction. |
| Relationship | Linear pairs are a specific type of adjacent angles with an additional condition about their orientation and sum. | All linear pairs are adjacent angles, but not all adjacent angles form a linear pair. |
| Example | If two lines intersect and form a right angle, the other two angles forming the straight line are a linear pair. | Two angles sharing a common side in a triangle are adjacent but not necessarily a linear pair. |
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Linear Pair of Angles Examples
We have solved some questions on Linear Pair of Angles to enhance your understanding of the concepts.
Example 1: If ∠PQR and ∠RQS form a linear pair, and the measure of ∠PQR is 75∘, what is the measure of ∠RQS?
Solution:
Since ∠PQR and ∠RQS form a linear pair, their measures add up to 180°.
Let’s denote the measure of ∠RQS as x
The equation representing the linear pair is: 75°+x=180°
Subtract 75° from both sides to find x=180°−75°
Therefore, x=105°
Example 2: In a linear pair of angles, if ∠A measures 70°, what is the measure of the adjacent angle, ∠B?
Solution:
According to the given ratio, the sum of the measures of the two angles is 2x + 3x = 5x.
Since these angles form a linear pair, the sum of their measures is 180°.
So, 5x = 180
Solving for (x):
x = 180/5
x = 36
Now, the measures of the angles:
∠1 = 2x = 2 × 36 = 72°
∠2 = 3x = 3 × 36 = 108°
Therefore, the measures of the two angles are 72° and 108°.
Linear Pair of Angles Worksheet
Here are some questions on Linear Pair of Angles for your practice:
Q1: If the measure of one angle in a linear pair is 120°, find the measure of its adjacent angle.
Q2: In a linear pair of angles, if one angle measures 2y and the other measures 3y−10, determine the value of y and the measures of both angles.
Q3: If the measures of two angles forming a linear pair are in the ratio of 5:8, and the larger angle is 144°, find the measure of the smaller angle.
Q4: The measures of two angles forming a linear pair are consecutive even integers. If the smaller angle is 60°, find the measures of both angles.
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Linear Pair Meaning- FAQs
1. What is a Linear Pair of Angles?
A linear pair of angles refers to two adjacent angles sharing a common arm and forming a straight line. In this arrangement, the sum of the measures of the two angles always equals 180°.
2. How can you identify a Linear Pair of an Angle?
To identify the linear pair of an angle, look for an adjacent angle that shares a common arm, and together, they create a straight line when combined.
3. Is a Linear Pair of Angles Always Supplementary?
Yes, a linear pair of angles is always supplementary, meaning the sum of their measures is consistently 180°.
4. How many Angles make up a Linear Pair?
A linear pair consists of two angles, specifically two adjacent angles forming a straight line.
5. Are Angles in a Linear Pair always Congruent?
No, angles in a linear pair are not always congruent. While they share a common arm, their measures may differ, contributing to the supplementary nature of the pair.
6. Is it Possible for Three Angles to form a Linear Pair?
No, a linear pair is exclusively formed by two angles. Three angles cannot create a linear pair, as a linear pair involves two adjacent angles along a straight line.
7. Do Supplementary Angles form a Linear Pair?
No, Supplemenatry Angles do not always form a linear pair
8. What is difference between Linear Pair and Supplementary Angles?
The difference between Linear Pair and Supplementary Angle is that Linear Pair always joins to form a straight line while Supplementary Angles may or may not form a straight line