Linear Pair of Angles
When two angles are formed on a straight line and share a common arm and a common vertex, they form a linear pair. The two angles together make a straight angle.
A straight line measures 180°. So if two adjacent angles sit on a straight line, their sum is always 180°. This is the Linear Pair Axiom.
For example, if one angle is 60°, the other must be 180° − 60° = 120°. This relationship is used throughout geometry to find unknown angles.
What is Linear Pair of Angles - Grade 7 Maths (Lines and Angles)?
Definition: A linear pair of angles is a pair of adjacent angles whose non-common arms form a straight line (opposite rays).
Properties:
- The two angles are adjacent — they share a common vertex and a common arm.
- Their non-common arms are opposite rays — they form a straight line.
- The sum of a linear pair is always 180°.
- Each angle in a linear pair is the supplement of the other.
Linear Pair Axiom: If a ray stands on a line, then the sum of the two adjacent angles formed is 180°.
Linear Pair of Angles Formula
Formula:
Angle 1 + Angle 2 = 180°
Where Angle 1 and Angle 2 form a linear pair.
To find one angle when the other is known:
- Angle 2 = 180° − Angle 1
Types and Properties
Special Cases of Linear Pairs:
- Equal linear pair: Both angles are 90° each. This happens when the ray is perpendicular to the line.
- Unequal linear pair: One angle is acute (less than 90°) and the other is obtuse (greater than 90°). Example: 50° and 130°.
- One angle is 0°: The other is 180° — both rays overlap the line (trivial case).
Difference from Supplementary Angles:
- All linear pairs are supplementary (sum = 180°).
- But NOT all supplementary angles form a linear pair.
- Supplementary angles need not be adjacent. Linear pairs MUST be adjacent with non-common arms forming a line.
Solved Examples
Example 1: Finding the Missing Angle
Problem: Two angles form a linear pair. One angle is 65°. Find the other.
Solution:
- Linear pair sum = 180°
- Other angle = 180° − 65° = 115°
Answer: The other angle is 115°.
Example 2: Finding Angles Using a Variable
Problem: Two angles forming a linear pair are (2x + 10)° and (3x − 5)°. Find both angles.
Solution:
- (2x + 10) + (3x − 5) = 180
- 5x + 5 = 180
- 5x = 175
- x = 35
- First angle = 2(35) + 10 = 80°
- Second angle = 3(35) − 5 = 100°
- Check: 80 + 100 = 180 ✓
Answer: The angles are 80° and 100°.
Example 3: Equal Linear Pair
Problem: Two angles forming a linear pair are equal. Find each angle.
Solution:
- Let each angle = x
- x + x = 180°
- 2x = 180°
- x = 90°
Answer: Each angle is 90°. The ray is perpendicular to the line.
Example 4: Ratio Problem
Problem: Two angles forming a linear pair are in the ratio 2:3. Find both angles.
Solution:
- Let the angles be 2x and 3x.
- 2x + 3x = 180°
- 5x = 180°
- x = 36°
- Angles: 2(36) = 72° and 3(36) = 108°
Answer: The angles are 72° and 108°.
Example 5: Identifying a Linear Pair
Problem: Angle A = 120° and Angle B = 60°. They share a common vertex and a common arm, and their other arms point in opposite directions. Do they form a linear pair?
Solution:
- They are adjacent (common vertex, common arm).
- Non-common arms form a straight line (opposite directions).
- Sum = 120° + 60° = 180°.
Answer: Yes, they form a linear pair.
Example 6: Word Problem — Clock Angles
Problem: At 6 o'clock, the hour and minute hands form a straight line. The upper angle is 180°. A fly sits on the minute hand, creating a ray that makes a 70° angle with the 12. What angle does it make with the 6?
Solution:
- The ray from the centre to the fly and the line from 12 to 6 form angles on both sides.
- The angle on one side = 70°.
- The angle on the other side = 180° − 70° = 110°.
Answer: The angle with the 6 is 110°.
Example 7: Multiple Linear Pairs
Problem: Three rays start from a point on a line, forming angles of x°, 2x°, and 3x° on one side. Find x.
Solution:
- All angles on one side of a line sum to 180°.
- x + 2x + 3x = 180°
- 6x = 180°
- x = 30°
- The angles are 30°, 60°, and 90°.
Answer: x = 30°.
Example 8: Difference Between Supplementary and Linear Pair
Problem: Angles P = 110° and Q = 70° are supplementary. Do they form a linear pair?
Solution:
- P + Q = 110 + 70 = 180° → They are supplementary.
- But to be a linear pair, they must also be adjacent (share a vertex and an arm) with non-common arms forming a line.
- Without this information, we cannot confirm they form a linear pair.
Answer: They are supplementary but not necessarily a linear pair unless they are also adjacent with opposite non-common arms.
Real-World Applications
Real-world uses of linear pairs:
- Road intersections: When two roads meet, the angles formed on a straight road are linear pairs.
- Door opening: The angle a door makes with the wall and the remaining angle to the opposite wall form a linear pair.
- Clock hands: Angles formed by clock hands on either side of a straight line are linear pairs.
- Carpentry: When cutting wood along a straight edge, the angles on both sides of the cut form a linear pair.
- Geometry proofs: Linear pair axiom is fundamental for proving many angle relationships.
Key Points to Remember
- A linear pair consists of two adjacent angles whose non-common arms form a straight line.
- The sum of a linear pair is always 180°.
- Each angle in a linear pair is the supplement of the other.
- All linear pairs are supplementary, but all supplementary angles are NOT linear pairs.
- If both angles in a linear pair are equal, each is 90°.
- The Linear Pair Axiom: If a ray stands on a line, the two adjacent angles formed add up to 180°.
- The converse is also true: if two adjacent angles sum to 180°, their non-common arms form a straight line.
Practice Problems
- Two angles form a linear pair. One is 72°. Find the other.
- Angles (x + 20)° and (2x + 40)° form a linear pair. Find x.
- Two angles forming a linear pair are in the ratio 5:4. Find both.
- Can two acute angles form a linear pair? Explain.
- Can two obtuse angles form a linear pair? Explain.
- If one angle of a linear pair is 90°, what is the other? What name do we give this?
- Three angles on a straight line are 40°, x°, and 60°. Find x.
Frequently Asked Questions
Q1. What is a linear pair of angles?
A linear pair is formed when two adjacent angles have their non-common arms pointing in opposite directions (forming a straight line). Their sum is always 180°.
Q2. Are all supplementary angles linear pairs?
No. Supplementary angles add up to 180°, but they need not be adjacent. A linear pair must be adjacent with non-common arms forming a line. For example, 110° and 70° are supplementary, but they only form a linear pair if they share a vertex and a common arm.
Q3. Can two acute angles form a linear pair?
No. Two acute angles (each less than 90°) cannot sum to 180°. At most their sum is less than 180°. So they cannot form a linear pair.
Q4. Can two right angles form a linear pair?
Yes. If both angles are 90°, their sum is 180°, and they can form a linear pair. This happens when a ray is perpendicular to a line.
Q5. How is a linear pair related to vertically opposite angles?
When two lines intersect, each angle forms a linear pair with its adjacent angle. The angles that are NOT adjacent (across the intersection) are vertically opposite and equal.
Related Topics
- Supplementary Angles
- Adjacent Angles
- Vertically Opposite Angles
- Angles on a Straight Line
- Complementary Angles
- Transversal and Parallel Lines
- Corresponding Angles
- Alternate Interior Angles
- Co-Interior Angles
- Angles at a Point
- Alternate Exterior Angles
- Proving Lines are Parallel
- Word Problems on Lines and Angles
