What is Linear Pair of Angles? Definition, Properties and FAQs
What is Linear Pair of Angles?
A linear pair of angles refers to a pair of adjacent angles formed when two straight lines intersect. These angles are supplementary, meaning that the sum of their measures is always 180 degrees.
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In other words, if you have two angles, A and B, that form a linear pair, then:
∠A+∠B=180∘
The angles in a linear pair share a common arm (side) and vertex, but their other arms form a straight line. Visualizing it, you can imagine a straight line with a point of intersection, and the two adjacent angles on either side of the intersection point form a linear pair.
Understanding linear pairs is essential in geometry and helps in solving problems related to angles formed by intersecting lines or angles within polygons.
Properties of Linear Pair of Angles
In geometry, a linear pair of angles refers to a pair of adjacent angles whose non-common sides are opposite rays, and the sum of their measures is always 180 degrees. Here are some properties of linear pairs of angles:
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Adjacent Angles: Linear pairs consist of two adjacent angles, which means they share a common side and a common vertex.
Opposite Rays: The non-common sides of the two angles form a straight line, and they are considered opposite rays. This implies that the rays extend indefinitely in opposite directions.
Angle Sum Property: The sum of the measures of the two angles forming a linear pair is always 180 degrees. This property is known as the angle sum property of a straight line.
∠A+∠B=180∘
where ∠A and ∠B are the two angles forming the linear pair.
Supplementary Angles: Because the sum of the angles in a linear pair is 180 degrees, the angles are supplementary. Supplementary angles are two angles whose measures add up to 180 degrees.
Linear Pair Theorem: If two angles form a linear pair, then they are supplementary. In other words, if ∠A and ∠B form a linear pair, then ∠A+∠B=180∘.
Understanding the properties of linear pairs is fundamental in geometry, especially when dealing with angles formed by intersecting lines and angles on a straight line.
Linear Pair Postulate
The Linear Pair Postulate, also known as the Linear Pair Axiom, is a fundamental concept in geometry related to angles formed by two intersecting lines. The postulate states that when two lines intersect, they form a pair of adjacent angles (angles that share a common vertex and a common side) that add up to 180 degrees. In other words, the sum of the measures of the two adjacent angles, known as a linear pair, is always 180 degrees.
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Mathematically, if two angles, say ∠A and ∠B, are formed by the intersection of two lines, and they are adjacent, then the linear pair postulate can be expressed as:
m∠A+m∠B=180∘
Here:
- m∠A is the measure of angle ∠A,
- m∠B is the measure of angle ∠B.
This postulate is based on the fact that a straight line forms an angle of 180 degrees, and when two lines intersect, the adjacent angles on one side of the intersection together create a straight line.
Understanding the Linear Pair Postulate is crucial in solving problems involving angles formed by intersecting lines, and it is a key concept in elementary geometry.
Linear Pair Axioms
In geometry, a linear pair refers to a pair of adjacent angles whose non-common sides are opposite rays. Linear pairs are formed when two lines intersect. There are two main axioms or properties related to linear pairs:
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Linear Pair Axiom (Postulate): If two angles form a linear pair, then they are supplementary. In other words, the sum of the measures of the two angles is always 180∘180∘. Mathematically, if ∠�∠A and ∠�∠B form a linear pair, then:
m∠A+m∠B=180∘
Here, m∠A represents the measure of angle A, and m∠B represents the measure of angle B.
Converse of the Linear Pair Axiom: If the sum of the measures of two angles is 180∘180∘, then the angles form a linear pair. Mathematically, if m∠A+m∠B=180∘, then ∠A and ∠B form a linear pair.
These axioms are fundamental in understanding the relationships between angles formed when two lines intersect. The concept of linear pairs is often used in geometry proofs and problem-solving involving angles.
Linear Pair of Angles Vs Supplementary Angles
Here's a tabular comparison between linear pair of angles and supplementary angles:
| Feature | Linear Pair of Angles | Supplementary Angles |
|---|---|---|
| Definition | Two angles that are adjacent and their sum is 180 degrees. | Two angles whose sum is 180 degrees. |
| Relationship | Adjacent angles sharing a common arm and a common vertex. | Angles that do not necessarily share a common arm or vertex. |
| Sum of Measures | The sum of the measures of the two angles is always 180 degrees. | The sum of the measures of the two angles is always 180 degrees. |
| Example | ∠ABC and ∠CBD are a linear pair if the common side is BC. | ∠PQR and ∠XYZ are supplementary angles. |
| Notation | m∠ABC + m∠CBD = 180 degrees. | m∠PQR + m∠XYZ = 180 degrees. |
| Intersection of Lines | Formed when two lines intersect. | Angles on either side of the transversal. |
| Common Arm | Have a common arm and a common vertex. | Do not necessarily have a common arm. |
| Types of Angles | Can include acute, obtuse, or right angles. | Can also include acute, obtuse, or right angles. |
Solved Examples on Linear Pair of Angles
Here are some examples of linear pairs of angles.
Example 1: Consider two angles, ∠ABC and ∠CBD, such that ∠ABC+∠CBD=180∘. These angles form a linear pair.
If ∠ABC=120∘, find∠CBD.
Solution:
∠CBD=180∘−∠ABC
∠CBD=180∘−120∘
∠CBD=60∘
So, ∠CBD is 60 degrees.
Example 2: Let ∠PQR and ∠RQS be two angles forming a linear pair. If ∠PQR=90∘, find ∠RQS.
Solution:
Since ∠PQR+∠RQS=180∘ for a linear pair,
∠RQS=180∘−∠PQR
∠RQS=180∘−90∘
∠RQS=90∘
So, RQS is 90 degrees.
These examples illustrate how to use the property of linear pairs, where the sum of two angles forming a linear pair is always 180 degrees.
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