In geometry, a linear pair of angles is a pair of surrounding angles formed when two lines intersect each other. Adjacent angles are developed when 2 angles have actually a usual vertex and also a typical arm but do no overlap. The linear pair of angles are always supplementary as they kind on a directly line. In various other words, the amount of two angles in a straight pair is always 180 degrees.

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1.Definition of direct Pair the Angles
2.Properties of direct Pair of Angles
3.Linear Pair of angle Vs Supplementary Angles
4.Linear Pair Postulate

When two lines intersect each other at a single point, linear pairs of angles space formed. If the angle so created are adjacent to each other after the intersection that the 2 lines, the angle are claimed to it is in linear. If two angles type a straight pair, the angles are supplementary, whose measures include up come 180°. Hence, a direct pair of angle always add up to 180°.


There space some properties of linear pair of angle that make them unique and also different indigenous other species of angles. Look in ~ the straight pair of angle properties noted below:

The sum of two angles in a linear pair is constantly 180°.

In geometry, there room two types of angle whose amount is 180 degrees. They are straight pairs that angles and supplementary angles. We often say that the linear pair of angles room supplementary, but do you recognize that these two types of angles room not the same? allow us understand the difference between supplementary angles and also linear pair the angles with the table offered below:

Linear Pair that AnglesSupplementary Angles
These angles are always nearby to every other. It means, a pair of angles whose amount is 180 degrees and also they lie next to each other sharing a common vertex and a common arm are known as linear pair of angles.These angles need not it is in adjacent. Their amount is additionally 180°.
All linear pairs are supplementary angle too.All supplementary angles room not straight pairs.
Example: ∠1 and ∠2 in the image given below.Example: ∠A and also ∠B, ∠1 and ∠2 (in the picture below).

In the photo below, it have the right to be clearly seen the both the pairs of angles space supplementary, however ∠A and ∠B are not straight pairs due to the fact that they room not adjacent angles.


The direct pair postulate says that if a beam stands ~ above a line, then the amount of two nearby angles is 180º. Will certainly the converse the this statement it is in true? that is if the sum of a pair of adjacent angles is 180º, will the non-common arms of the 2 angles type a line? Yes, the converse is additionally true. These two axioms space grouped with each other as the direct pair axiom. In the figure below, beam QS stand on a heat PR forming a direct pair of angle ∠1 and also ∠2.


Important Notes

In a linear pair, if the two angles have actually a usual vertex and a typical arm, climate the non-common side makes a directly line and also the sum of the measure of angles is 180°.Linear pairs are always supplementary.

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Check these interesting short articles related to the linear pair of angle in geometry.

Example 1: If among the angles developing a direct pair is a best angle, then what have the right to you say around its other angle?Solution: Let one of the angles forming a direct pair it is in 'a' and the various other be 'b'.Given that ∠a = 90° and also we already know that linear pairs the angles space supplementary ⇒ ∠a + ∠b = 180°.⇒ 90° + ∠b = 180°⇒ ∠b = 180° - 90°⇒ ∠b = 90°Therefore, in a straight pair of angles, if among the angles is a ideal angle then one more angle is additionally a ideal angle.

Example 2: In the provided figure, if POQ is a straight line and ∠POC = ∠COQ, then show that ∠POC = 90°.


Since ray OC stands on heat PQ. So, by linear pair axiom, ∠POC + ∠COQ = 180°. However ∠POC = ∠COQ (given).⇒ ∠POC + ∠POC = 180°⇒ 2∠POC = 180°

⇒ ∠POC = 180°/2 = 90°⇒ ∠POC = 90°Hence Proved.

Example 3: If 2 angles developing a straight pair are in the ratio of 4:5, then discover the measure of each of the angles.Solution: let the 2 angles it is in 4y and 5y.

We recognize that direct pair of angles room supplementary ⇒ 4y + 5y = 180°.

9y = 180°

y = 180/9

y = 20

Therefore, the 2 angles are: 4y = 4 × 20 = 80° and also 5y = 5 × 20 = 100°.

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