 |
VOOZH | about |
|
VOOZH | about |
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.
Table of Content
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°.
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 Diagram
These are some of the most important properties of a linear pair of angles:
Read More On
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°.
There are two axioms which are related to Linear Pair of angles.
Let's learn about each of them in detail.
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 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.
Note: An axiom is a statement which is universally true and doesn't need any proof.
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 |
|---|---|---|
| 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. |
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 |
|---|---|---|
| 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. |
Read More on,
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°.
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.
Related Topics
