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What is a Linear Pair of Angles? Discover the concept of a linear pair of angles, where two adjacent angles combine to form a straight line. Discover the properties, examples and meaning of this fundamental geometric relationship.
Contents
What is a Linear Pair of Angles?
A line pair of angles is a pair of adjacent angles formed when two lines intersect. These two angles are also known as supplementary angles because their measures add up to 180 degrees. In a linear pair, the angles are positioned so that they form a straight line, with one angle on each side of the intersection point. Since the sum of the measures of the angles in a straight line is always 180 degrees, the two angles in a linear pair are always complementary to each other.
Common practical examples of a linear pair of angles include
- Angles formed by the hands of a clock
- Angles formed by the slices of pizza
- The angle formed by the scissors
What is a Linear Pair with an Example?
A linear pair is a pair of adjacent angles that are formed when two lines intersect. To illustrate this concept, let’s consider an example:
Imagine two lines, line AB and line CD, intersecting at point P. This intersection creates four angles: angle APD, angle DPC, angle CPB, and angle APB.
In this scenario, a line pair is formed by angles APD and DPC. These two angles are adjacent, meaning that they share a common vertex (point P) and a common side (line PD). Moreover, they are positioned so that they form a straight line (lines AB and CD are collinear), with an angle APD on one side of the intersection and an angle DPC on the other side.
Since angles APD and DPC form a straight line, they are supplementary angles, which means that the sum of their measures is equal to 180 degrees. If, for example, angle APD measures 80 degrees, angle DPC would measure 100 degrees because 80 + 100 equals 180.
Thus, in this example, angles APD and DPC form a linear pair.
How to Find the Linear Pair of Angles?
To find a linear pair of angles, you must follow these steps:
- Identify the intersecting lines: Look for the lines that intersect or cross each other.
- Find the adjacent angles: Focus on the angles that are formed on either side of the intersection point. These angles should share a common vertex and a common side.
- Check if they form a straight line: Determine if the lines on which the angles lie are collinear, that means they form a straight line. This can be confirmed by examining the shape and alignment of the lines.
- Check that the angles are supplementary: Make sure that the sum of the measures of the two adjacent angles is equal to 180 degrees. If they satisfy this condition, they form a linear pair.
By following these steps, you can identify and confirm the presence of a linear pair of angles.
Is a Linear Pair always Supplementary?
Yes, a linear pair of angles is always supplementary. Supplementary angles are defined as two angles whose measures add up to 180 degrees. In the case of a linear pair, the two adjacent corners are positioned so that they form a straight line. Since the sum of the measures of the angles in a straight line is always 180 degrees, the angles in a linear pair are always complementary to each other. Thus, if two angles form a linear pair, they will always be supplementary.
Properties of a Linear Pair of Angles
The properties of a linear pair of angles are as follows:
- Supplementary angles: The angles in a linear pair are always supplementary, which means that their measures add up to 180 degrees.
- Adjacent angles: A line pair consists of two adjacent angles, meaning they share a common vertex and a common side.
- Common arm: The two angles in a line pair share a common arm, which is the line segment between their vertices.
- Formation of a straight line: The angles in a linear pair are positioned so that they form a straight line when the lines on which they lie intersect.
- Non-overlapping: The angles in a line pair do not overlap or share any interior points. They are separate and distinct angles.
- Unique solution: Given a line and a point on it, there is exactly one angle that forms a linear pair with a given angle.
These properties define and describe the nature of a linear pair of angles.
Solved Examples on a Linear Pair of Angles
Here are some solved examples of linear pairs of angles:
Example 1:
In a linear pair of angles, if one angle measures 120 degrees, what is the measure of the other angle?
Solution:
In a linear pair of angles, the sum of their measures is always 180 degrees.
Let’s denote the measure of the other angle as x degrees.
Therefore, we have the equation:
120 + x = 180
Subtracting 120 from both sides, we get:
x = 180 – 120
x = 60
So, the measure of the other angle is 60 degrees.
Example 2:
In a linear pair of angles, if one angle measures 100 degrees, what is the measure of the other angle?
Solution:
In a linear pair of angles, the sum of their measures is always 180 degrees.
Let’s denote the measure of the other angle as x degrees.
Therefore, we have the equation:
100 + x = 180
Subtracting 100 from both sides, we get:
x = 180 – 100
x = 80
So, the measure of the other angle is 80 degrees.
Example 3:
In a linear pair of angles, if one angle measures 70 degrees, what is the measure of the other angle?
Solution:
In a linear pair of angles, the sum of their measures is always 180 degrees.
Let’s denote the measure of the other angle as x degrees.
Therefore, we have the equation:
70 + x = 180
Subtracting 70 from both sides, we get:
x = 180 – 70
x = 110
So, the measure of the other angle is 110 degrees.
Types of Linear pair of Angles
A line pair of angles refers to a pair of adjacent angles formed when two lines intersect. The sum of the measures of these angles is always 180 degrees. There are different types of linear pairs of angles based on their properties and relationships. Here are some common types:
Adjacent Supplementary Angles: In this type, the two angles are adjacent (share a common side and vertex) and their measures add up to 180 degrees. For example, if one angle will measure 120 degrees, the other angle will measure 60 degrees.
Vertical Angles: When two lines intersect, they form two pairs of vertical angles. Vertical angles are opposite angles formed by the intersecting lines. Every pair of vertical angles is a linear pair because their measures add up to 180 degrees. For example, if one angle measures 80 degrees, the other angle formed by the intersecting lines will measure 100 degrees.
Complementary Angles: Although not strictly a linear pair, complementary angles can be considered a special case. Complementary angles are two angles whose measures add up to 90 degrees. If two angles are complementary, they can form a linear pair with two more angles, each at 90 degrees. For example, if one angle measures 30 degrees, the other complementary angle will measure 60 degrees.
It is important to note that linear pairs of angles are always adjacent, meaning that they share a common side and vertex. Their sum is always 180 degrees, but the specific type of line pair depends on their relationship, such as being adjacent and complementary or forming vertical angles.
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