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Linear Pair of Angles

Linear Pair of Angles

A pair of adjacent angles formed by intersecting lines. Angles 1 and 2 below are a linear pair. So are angles 2 and 4, angles 3 and 4, and angles 1 and 3. Linear pairs of angles are supplementary.

 

Two intersecting lines forming 4 angles labeled 1, 2, 3, 4, where adjacent angles (e.g., 1&2, 3&4) are linear pairs.

Key Formula

∠A+∠B=180°\angle A + \angle B = 180°
Where:
  • ∠A\angle A = One angle of the linear pair
  • ∠B\angle B = The other angle of the linear pair, adjacent to ∠A with their non-common sides forming a straight line

Worked Example

Problem: Two angles form a linear pair. One angle measures 65°. Find the measure of the other angle.
Step 1: Write the linear pair relationship. Since the two angles form a linear pair, they are supplementary.
∠A+∠B=180°\angle A + \angle B = 180°
Step 2: Substitute the known angle measure.
65°+∠B=180°65° + \angle B = 180°
Step 3: Solve for the unknown angle by subtracting 65° from both sides.
∠B=180°−65°=115°\angle B = 180° - 65° = 115°
Answer: The other angle measures 115°.

Another Example

This example uses algebraic expressions instead of a single known angle, which is how linear pair problems typically appear on tests and homework.

Problem: Two angles form a linear pair. One angle is represented by the expression (3x + 10)° and the other by (2x + 20)°. Find the value of x and the measure of each angle.
Step 1: Set up the equation using the linear pair property.
(3x+10)+(2x+20)=180(3x + 10) + (2x + 20) = 180
Step 2: Combine like terms on the left side.
5x+30=1805x + 30 = 180
Step 3: Subtract 30 from both sides.
5x=1505x = 150
Step 4: Divide both sides by 5 to find x.
x=30x = 30
Step 5: Substitute x = 30 back into each expression to find the angle measures.
3(30)+10=100°and2(30)+20=80°3(30) + 10 = 100° \quad \text{and} \quad 2(30) + 20 = 80°
Answer: x = 30, and the two angles measure 100° and 80°. Check: 100° + 80° = 180°. ✓

Frequently Asked Questions

What is the difference between a linear pair and supplementary angles?
All linear pairs are supplementary, but not all supplementary angles form a linear pair. Supplementary angles are any two angles that add up to 180°, even if they are in completely different locations. A linear pair has a stricter requirement: the two angles must be adjacent (share a common vertex and a common side) and their non-common sides must form a straight line.
Can a linear pair have two right angles?
Yes. If both angles in a linear pair each measure 90°, they satisfy the requirement that their sum is 180°. This occurs when the two lines intersect at right angles, meaning they are perpendicular.
How many linear pairs are formed when two lines intersect?
When two straight lines intersect, they form four angles at the intersection point. These four angles create exactly four linear pairs. Each angle forms a linear pair with each of its two adjacent angles.

Linear Pair vs. Vertical Angles

Linear PairVertical Angles
DefinitionTwo adjacent angles whose non-common sides form a straight lineTwo non-adjacent angles formed by two intersecting lines, sitting across from each other
Angle relationshipSupplementary (sum = 180°)Congruent (equal in measure)
PositionAdjacent — share a common sideOpposite — do not share a common side
Number per intersection4 linear pairs2 pairs of vertical angles

Why It Matters

Linear pairs appear frequently in geometry proofs and standardized tests whenever you work with intersecting lines, triangles, or parallel lines cut by a transversal. Recognizing a linear pair lets you immediately set up an equation to find an unknown angle. This concept also serves as a foundation for proving that vertical angles are congruent — one of the earliest and most important theorems in geometry.

Common Mistakes

Mistake: Assuming any two angles that add to 180° form a linear pair.
Correction: Two angles must also be adjacent and have their non-common sides form a straight line. Supplementary angles that are not next to each other are not a linear pair.
Mistake: Confusing linear pairs with vertical angles at an intersection.
Correction: At an intersection, adjacent angle pairs are linear pairs (supplementary), while the angles across from each other are vertical angles (congruent). Pick the correct relationship based on the position of the angles.

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