Linear Pair of Angles
Before learning about the Linear Pair of Angles, let us first understand a little bit about Adjacent Angles and Opposite Rays. You will understand Linear Pair better once you understand Adjacent Angles and Opposite Rays. Adjacent angles are formed when two angles have a common vertex and a common side but do not overlap. Both the angles are represented by a common arm. Opposite rays are formed when two rays start from a common point and go off in exactly opposite directions.
|1.||Definition of Linear Pair of Angles|
|2.||Relationship Between Pair of Angles|
|3.||Linear Pair Axiom|
Definition of Linear Pair of Angles
When two lines intersect each other at a single point, linear pair of angles are formed. If the angles so formed are adjacent to each other after the intersection of the two lines, the angles are said to be linear. If two angles form a linear pair, the angles are supplementary, whose measures add up to 180°. Hence, a linear pair of angles always add up to 180°.
Relationship Between Pair of Angles
Two angles are considered to be adjacent angles when they share a common vertex, a common side, but no common interior points. (They do not overlap and they only share a vertex and side)
∠1 and ∠2 are adjacent angles. ∠PQR and ∠1 are NOT adjacent angles. (∠PQR overlaps ∠1.)
A linear pair is formed when the two adjacent angles and their non-common sides form opposite rays. A classic example of a linear pair of angles is the pair of scissors. In a scissor, the flanks of scissors, which are adjacent to each other and have a common vertex R, and form an angle of 180°
In the below-given figure, ∠1 and ∠2 forms a linear pair. P, Q, and R are collinear points and line PQ is a straight line. ∠2 and ∠1 are supplementary.
Two angles are said to be vertical angles when their sides form two pairs of opposite rays (straight lines). Vertical angles are always equal, they are not adjacent. Vertical angles are also called vertically opposite angles.
In the image provided below, the vertical angles are located across from one another in the corners of the "X" formed by the two straight lines.
- ∠1 and ∠2 are vertical angles.
- ∠3 and ∠4 are vertical angles.
- ∠1 and ∠3 form a linear pair, and they are not vertical angles.
Two angles are said to be supplementary when the sum of their angles is 180º. Supplementary angles can be placed in a way that they form a linear pair (straight line), or they can be two separate angles too.
In the figure given below:
- ∠1 and ∠2 are supplementary.
- ∠X and ∠Y are supplementary.
The line through points H, I and J is a straight line.
Linear Pair Axiom
Axiom: Linear pair axiom states that if a ray stands on a line, then the sum of two adjacent angles is 180º. Will the converse of this statement be true? That is if the sum of a pair of adjacent angles is 180º, will the non-common arms of the two angles form a line? Yes, this is can be formalized as an axiom as well. These two axioms are grouped together as the Linear Pair Axiom.
Example: 1) In the given figure, ∠POR and ∠ QOR form a linear pair if ∠1 - ∠2 = 60°, find the value of ∠1 and ∠2.
Given ∠1 - ∠2 = 60° ………… (i)
We know that, ∠1 + ∠2 = 180° ………… (ii)
Adding (i) and (ii)
2 ∠1 = 240° ⇒ ∠1 = 240°/2 = 120°. Therefore, ∠1 = 120°
Since, ∠1 - ∠2 = 60° or, 120° - ∠2 = 60° or, 120° - 120° - ∠2 = 60° - 120° ⇒ -∠2 = -60°. Therefore, ∠2 = 60°
Example: 2) Do the angles form a linear pair?
Here, the two angles do not have a common vertex. Hence, they do not form a linear pair.
- In a linear pair, if the two angles have a common vertex, common side then the non-common side makes a straight line and the sum of the measure of angles is 180°
- Linear pairs are always supplementary
- Linear pairs of angles are not always congruent
Examples of Linear Pair of Angles
Example 1: If one of the angles forming a linear pair is a right angle, then what can you say about its other angle?
Solution: Let one of the angles forming a linear pair be 'a' and the other be 'b'
Given that ∠a = 90° and we already know that linear pairs of angles are supplementary ⇒ ∠a + ∠b = 180°
90° + ∠b = 180° ⇒ ∠b = 180° - 90° ⇒ ∠b = 90°
In a linear pair of angles, if one of the angles is a right angle then another angle is also a right angle.
Example 2: ∠POC = ∠COQ , then show that ∠POC = 90°
Since ray OC stands on line PQ. ∴∠POC + ∠COQ = 180° (Linear-Pair )
But ∠POC = ∠COQ (given) ⇒ ∠POC + ∠POC = 180° ⇒ 2∠POC = 180°
∠POC = 180°/2 = 90° ⇒ ∠POC = 90° (Hence Proved)
Example 3: If two angles are in the ratio of 4:5 and these two angles form a linear pair angles, then find the measure of each.
Solution: Let the two angles be 4y and 5y.
We know that linear pair angles are supplementary ⇒ 4y + 5y = 180°
9y = 180°
y = 180°/9
y = 20
Therefore, the two angles are: 4y = 4 × 20 = 80° and 5y = 5 × 20 = 100°
Practice Questions on Linear Pair of Angles
FAQs on Linear Pair of Angles
How Do you Find the Linear Pair of an Angle?
If there is a pair of adjacent angles, then this pair is a linear pair, that is if the sum of the (measures of the) two angles will be 180°. So, linear pair of angles always add up to 180°
Is a Linear Pair always Supplementary?
Supplementary is one of the conditions for being a linear pair. Hence, linear pairs will always be supplementary. As per their definition, a linear pair forms a straight angle which measures 180º
How Many Angles are there in a Linear Pair?
In a linear pair, two adjacent angles are formed by two intersecting lines. A straight angle has an angle of 180°, so a linear pair of angles must add up to 180°.
Are Linear Pair Angles always Congruent?
Linear pairs of angles are not always congruent. Only when the measure of each of the angles is 90°, a linear pair of angles is said to be congruent.
Can 3 Angles Form a Linear Pair?
A linear pair can be defined as two adjacent angles that add up to 180° or two angles which when combined together form a line or a straight angle. Three angles can be supplementary, but not necessarily adjacent. For instance, angles in any triangle add up to 180° but they don't necessarily form a linear pair.