Linear Pair of Angles: Definition & Axiom

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Linear Pair of Angles: Definition & Axiom

TL;DR

A linear pair of angles is two adjacent angles whose non-common sides form a straight line, so they always add to 180°. This article covers the definition, the linear pair axiom and its converse, how a linear pair differs from supplementary and vertical angles, and six worked examples.

What Is a Linear Pair of Angles?

A linear pair of angles is a pair of adjacent angles formed when one ray stands on a straight line. Adjacent angles are two angles that share a common vertex and a common arm but do not overlap, and a linear pair adds one more condition: their two non-common arms point in exactly opposite directions, forming a single straight line.

Because the two non-common arms make a straight line, and a straight line is a straight angle of 180°, the two angles of a linear pair always sum to 180°. That is the defining feature: a linear pair is adjacent and sits on a straight line. Both conditions must hold.

The concept appears in NCERT Class 7, Chapter 5 (Lines and Angles) and across CCSS-M 7.G.B.5, where students first use angle relationships to solve for unknowns.

The Linear Pair Axiom

The relationship is formalised in the linear pair axiom, sometimes called the linear pair postulate:

If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.

An axiom is a statement accepted as true without proof, because it is taken as one of geometry's starting rules. The converse is also true and is just as useful:

If two adjacent angles add up to 180°, then their non-common arms form a straight line.

The converse is what lets you work backwards: if you can show two adjacent angles are supplementary, you have proved that their outer arms lie on one straight line, which is a standard step in geometry proofs about points being collinear.

Linear Pair vs Supplementary Angles

This is the distinction that trips up the most students, so it is worth pinning down precisely. Are all linear pairs supplementary? Yes. Are all supplementary angles a linear pair? No.

Supplementary angles are any two angles whose measures add to 180°, with no requirement that they touch. Two angles drawn on opposite sides of a page, one 110° and one 70°, are supplementary, but they are not a linear pair because they are not adjacent.

A linear pair is the special case of supplementary angles that are also adjacent and sit on one straight line. Every linear pair is supplementary; only the adjacent, straight-line supplementary pairs are linear pairs.

A second comparison is worth a line: a linear pair is not the same as a pair of vertical angles. Vertical angles are the opposite angles formed when two lines cross; they are equal, not supplementary. When two lines intersect, each angle forms a linear pair with each of its neighbours (summing to 180°) and a vertical pair with the angle across from it (equal).

Examples of the Linear Pair of Angles

With the definition, the axiom, and the supplementary distinction in hand, here is the linear pair doing real work. The problems build from a direct subtraction up to an algebraic ratio.

Example 1 - Two angles form a linear pair, and one of them is 110°. Find the other.

A linear pair sums to 180°, so subtract:

180°−110°=70°.

The other angle is 70°.

Example 2 - Ray O stands on line AB. One angle (∠AOC) is given as 70°. A student finds the vertical angle to ∠AOC across the intersection and writes the linear-pair partner ∠COB as 70° too, reasoning "they're both at O." Find ∠COB correctly.

The correct relationship is the linear pair sum:

∠AOC + ∠COB = 180°; ⇒; 70° + ∠COB = 180°; ⇒; ∠COB = 110°.

So ∠COB = 110°.

Example 3 - Two angles of a linear pair are equal. Find each angle.

Equal angles summing to 180° split it evenly:

180° / 2 = 90°.

Each angle is 90°. (This is the only case where a linear pair is also a pair of right angles, the special moment where the standing ray is perpendicular to the line.)

Example 4 - The angles of a linear pair are in the ratio 4 : 5. Find both angles.

Let the angles be 4x and 5x. Their sum is 180°:

4x + 5x = 180°; ⇒; 9x = 180°; ⇒; x = 20°.

So the angles are 4(20°) = 80° and 5(20°) = 100°. Check: 80° + 100° = 180°.

Example 5 - Two adjacent angles measure (2x + 10)° and (3x − 5)° and form a linear pair. Find x and both angles.

A linear pair sums to 180°:

(2x + 10) + (3x − 5) = 180; ⇒; 5x + 5 = 180; ⇒; 5x = 175; ⇒; x = 35.

The angles are 2(35) + 10 = 80° and 3(35) − 5 = 100°. Check: 80° + 100° = 180°.

Example 6 - Three rays OA, OC, and OB are drawn so that A, O, B lie on a straight line. If ∠AOC = 3y and ∠COB = 2y, find y, then explain why this proves A, O, B are collinear only if the sum is 180°.

Since the rays around the straight line give a linear pair:

3y + 2y = 180°; ⇒; 5y = 180°; ⇒; y = 36°.

So ∠AOC = 108° and ∠COB = 72°. By the converse of the linear pair axiom, because these adjacent angles add to 180°, the arms OA and OB must form one straight line, confirming A, O, B are collinear.

Key Takeaways

Practice These Problems to Solidify Your Understanding

  1. Two angles form a linear pair. One is 47°. Find the other.

    • Answer: 180°−47°=133°.

  2. The angles of a linear pair are in the ratio 7 : 11. Find both angles.

    • Answer: 7x + 11x = 180° gives x = 10°, so the angles are 70° and 110°.

  3. Two adjacent angles (5x)° and (4x)° form a linear pair. Find x and both angles.

    • Answer: 9x = 180° gives x = 20°, so the angles are 100° and 80°.

Frequently Asked Questions

What is a linear pair of angles?
It is two adjacent angles whose non-common arms form a straight line, so the two angles always add to 180°.

Is a linear pair always supplementary?
Yes. Because the two angles sit on a straight line, they always sum to 180°, which is the definition of supplementary.

Is a linear pair always equal?
No. They are equal only in the special case where each is 90°. In every other case the two angles differ while still summing to 180°.

What is the difference between a linear pair and supplementary angles?
Every linear pair is supplementary, but not every supplementary pair is a linear pair. A linear pair must also be adjacent and lie on one straight line; supplementary angles only need to add to 180°.

Can a linear pair add up to something other than 180 degrees?
No. By the linear pair axiom, two angles on a straight line always total exactly 180°. If a pair does not total 180°, it is not a linear pair.