Angles on a Straight Line
When two or more rays originate from a point on a straight line, the angles formed on one side of the line always add up to 180°. This property is known as the straight angle property or the angles on a straight line property.
A straight angle measures exactly 180°. When a ray stands on a straight line, it divides the straight angle into two parts. The sum of these two parts is always 180°.
This concept is fundamental to Class 9 geometry. It is used to derive the linear pair axiom, prove the vertically opposite angles theorem, and establish results about parallel lines and transversals.
In the NCERT Class 9 textbook, the angles on a straight line property is stated as the Linear Pair Axiom: "If a ray stands on a line, then the sum of two adjacent angles so formed is 180°." This axiom, together with its converse, is the starting point for many geometric proofs in the chapter on Lines and Angles.
Two angles whose sum is 180° are called supplementary angles. When these supplementary angles are also adjacent (share a common arm), they form a linear pair. Every linear pair is supplementary, but not every pair of supplementary angles forms a linear pair — they must be adjacent.
What is Angles on a Straight Line?
Definition: If a ray stands on a straight line, then the sum of the two adjacent angles formed is 180°.
∠AOB + ∠BOC = 180°
Where:
- AOC is a straight line
- OB is a ray standing on the line AOC
- ∠AOB and ∠BOC are adjacent angles on the same side of the line
Converse:
- If two adjacent angles add up to 180°, then the non-common arms of the angles form a straight line.
- This converse is equally important and is used in proofs to establish collinearity.
Important:
- A straight angle = 180°. It is the angle formed by two opposite rays.
- This property extends to multiple rays: if several rays stand on a line from the same point, the sum of ALL angles on one side is still 180°.
- This is also called the supplementary property of adjacent angles on a line.
Angles on a Straight Line Formula
Key Formulas:
1. Two angles on a straight line:
∠AOB + ∠BOC = 180°
(when AOC is a straight line and OB is a ray)
2. Three or more angles on a straight line:
∠₁ + ∠₂ + ∠₃ + ... = 180°
(when all angles are on the same side of the straight line)
3. Finding an unknown angle:
- If one angle is known: Unknown angle = 180° − known angle
- If multiple angles are known: Unknown angle = 180° − (sum of known angles)
4. Related property — Angles at a point:
- Angles around a complete point = 360° (full rotation)
- Angles on a straight line = 180° (half rotation)
Derivation and Proof
Proof: The Sum of Angles on a Straight Line is 180°
Given: AOC is a straight line. Ray OB stands on AOC.
To prove: ∠AOB + ∠BOC = 180°
Proof:
- A straight line AOC forms a straight angle at O.
- A straight angle measures 180°. Therefore, ∠AOC = 180°.
- Ray OB divides ∠AOC into two adjacent angles: ∠AOB and ∠BOC.
- Since OB lies between OA and OC (on one side): ∠AOB + ∠BOC = ∠AOC
- Substituting: ∠AOB + ∠BOC = 180°
Hence proved.
Proof of the Converse:
Given: ∠AOB + ∠BOC = 180° and ∠AOB and ∠BOC are adjacent.
To prove: A, O, C are collinear (AOC is a straight line).
- Assume AOC is NOT a straight line. Then OC′ is the ray such that AOC′ is a straight line.
- Then ∠AOB + ∠BOC′ = 180° (angles on straight line AOC′)
- But ∠AOB + ∠BOC = 180° (given)
- Subtracting: ∠BOC′ = ∠BOC
- This is possible only if OC′ and OC are the same ray.
- Therefore, AOC is indeed a straight line.
Types and Properties
Different Configurations of Angles on a Straight Line:
1. Two Angles (Linear Pair)
- One ray stands on a straight line, creating exactly two adjacent angles.
- The two angles are supplementary (sum = 180°).
- Example: If one angle is 70°, the other is 110°.
2. Three Angles on a Straight Line
- Two rays stand on the same straight line from the same point.
- Three adjacent angles are formed.
- Sum of all three = 180°.
- Example: ∠₁ = 50°, ∠₂ = 60°, ∠₃ = 70° (total = 180°).
3. Four or More Angles
- Multiple rays from the same point on a line.
- All angles on one side sum to 180°.
- This is used in problems involving protractors and compass constructions.
4. Equal Angles on a Straight Line
- If two equal angles are on a straight line: each = 90°.
- The ray is then perpendicular to the line.
- If three equal angles: each = 60°.
5. Angles Involving Variables
- Common exam pattern: angles given as algebraic expressions (e.g., 3x°, (2x + 10)°).
- Set their sum equal to 180° and solve for x.
- Then substitute back to find each angle.
Solved Examples
Example 1: Example 1: Find the unknown angle
Problem: A ray OB stands on line AC. If ∠AOB = 125°, find ∠BOC.
Solution:
Given:
- AOC is a straight line
- ∠AOB = 125°
Using angles on a straight line:
- ∠AOB + ∠BOC = 180°
- 125° + ∠BOC = 180°
- ∠BOC = 180° − 125° = 55°
Answer: ∠BOC = 55°
Example 2: Example 2: Three angles on a straight line
Problem: Three angles on a straight line are 40°, x°, and 60°. Find x.
Solution:
Using angles on a straight line:
- 40° + x° + 60° = 180°
- 100° + x° = 180°
- x° = 180° − 100° = 80°
Answer: x = 80°
Example 3: Example 3: Algebraic expressions
Problem: Two angles on a straight line are (3x + 15)° and (2x + 25)°. Find x and both angles.
Solution:
Sum = 180°:
- (3x + 15) + (2x + 25) = 180
- 5x + 40 = 180
- 5x = 140
- x = 28
The angles:
- First angle = 3(28) + 15 = 84 + 15 = 99°
- Second angle = 2(28) + 25 = 56 + 25 = 81°
Verification: 99° + 81° = 180° ✓
Answer: x = 28; angles are 99° and 81°.
Example 4: Example 4: Perpendicular ray
Problem: A ray OB is perpendicular to line AC. Find ∠AOB and ∠BOC.
Solution:
- Perpendicular means the ray makes a 90° angle with the line.
- ∠AOB = 90°
- Using angles on a straight line: ∠BOC = 180° − 90° = 90°
Answer: ∠AOB = 90° and ∠BOC = 90°.
Example 5: Example 5: Angles in ratio
Problem: Two angles on a straight line are in the ratio 2 : 3. Find both angles.
Solution:
Let the angles be 2k and 3k.
- 2k + 3k = 180°
- 5k = 180°
- k = 36°
The angles:
- First angle = 2 × 36 = 72°
- Second angle = 3 × 36 = 108°
Answer: The angles are 72° and 108°.
Example 6: Example 6: Four angles on a straight line
Problem: Four angles on a straight line are 35°, 45°, 55°, and y°. Find y.
Solution:
- 35 + 45 + 55 + y = 180
- 135 + y = 180
- y = 45
Answer: y = 45°
Example 7: Example 7: Proving collinearity using the converse
Problem: Rays OA and OC are such that ∠AOB = 110° and ∠BOC = 70°, where OB is a ray between OA and OC. Are A, O, and C collinear?
Solution:
- ∠AOB + ∠BOC = 110° + 70° = 180°
Using the converse: If the sum of two adjacent angles is 180°, the non-common arms form a straight line.
- Since ∠AOB + ∠BOC = 180°, rays OA and OC are opposite rays.
- Therefore, A, O, and C are collinear.
Answer: Yes, A, O, C are collinear (they lie on the same straight line).
Example 8: Example 8: Supplement of an angle
Problem: Find the supplement of 63°.
Solution:
- Two angles are supplementary if their sum is 180°.
- Supplement = 180° − 63° = 117°
Answer: The supplement of 63° is 117°.
Example 9: Example 9: Angle bisector on a straight line
Problem: Ray OB stands on line AC making ∠AOB = 120°. Ray OD bisects ∠BOC. Find ∠AOD.
Solution:
- ∠AOB + ∠BOC = 180° → ∠BOC = 180° − 120° = 60°
- OD bisects ∠BOC → ∠BOD = ∠DOC = 60°/2 = 30°
- ∠AOD = ∠AOB + ∠BOD = 120° + 30° = 150°
Answer: ∠AOD = 150°
Example 10: Example 10: Difference of angles on a straight line
Problem: Two supplementary angles on a straight line differ by 36°. Find both angles.
Solution:
Let the angles be x° and y° where x > y.
- x + y = 180 (angles on a straight line)
- x − y = 36 (given)
- Adding: 2x = 216 → x = 108
- Substituting: y = 180 − 108 = 72
Answer: The angles are 108° and 72°.
Real-World Applications
Applications of Angles on a Straight Line:
- Proving Vertically Opposite Angles: The vertically opposite angles theorem is proved using the straight line angle property. If two lines intersect, each pair of adjacent angles on a straight line sums to 180°, leading to the conclusion that opposite angles are equal. This is one of the first formal proofs students encounter in Class 9.
- Parallel Lines and Transversals: The co-interior angles property (sum = 180° for parallel lines) directly uses the straight line angle sum. Many problems on parallel lines rely on this result. The alternate interior angles theorem is also derived from this property.
- Triangle Angle Sum: The proof that the sum of angles in a triangle is 180° uses the angles-on-a-straight-line property. A line parallel to the base is drawn through the vertex, and alternate interior angles are used along with the straight line property.
- Construction and Design: Architects and engineers verify that structural angles on a beam or straight support sum to 180° to ensure correct alignment. Misalignment of even 1° can cause structural stress in buildings.
- Clock Angles: The angle between clock hands at various times can be computed by treating the 12-hour dial as 360° and using supplementary angle relationships. For example, at 4 o’clock, the angle is 120° and its supplement is 240° (reflex).
- Navigation and Bearing: Bearings in navigation use the concept of supplementary angles. A bearing and its back-bearing differ by 180°. If a ship sails on bearing N30°E, the return bearing is S30°W (180° apart).
- Optics — Reflection: The angle of incidence and angle of reflection on a mirror surface are measured from the normal. The incident ray, normal, and reflected ray all relate through the straight-line property at the point of incidence.
- Folding and Origami: When a sheet of paper is folded along a crease, the angle between the two parts of the fold on one side of the crease sums to 180°, applying the straight line property.
Key Points to Remember
- A straight angle measures exactly 180°.
- If a ray stands on a straight line, the two adjacent angles formed are supplementary (sum = 180°).
- This property extends to multiple angles: if several rays stand on a line from the same point, all angles on one side sum to 180°.
- The converse is also true: if adjacent angles sum to 180°, the non-common arms form a straight line.
- Two equal angles on a straight line are each 90° (perpendicular).
- This property is the basis for the linear pair axiom.
- It is used to prove vertically opposite angles are equal.
- In problems with variables, set the sum of angles equal to 180° and solve the equation.
- Angles in a ratio on a straight line: let them be k-multiples and use sum = 180°.
- Do NOT confuse with angles at a point (which sum to 360°).
Practice Problems
- A ray OB stands on line AC. If ∠AOB = 148°, find ∠BOC.
- Three angles on a straight line are x°, 2x°, and 3x°. Find all three angles.
- Two supplementary angles are in the ratio 5 : 7. Find both angles.
- Is it possible for two angles on a straight line to both be obtuse? Explain.
- Angles on a straight line are (4x − 10)° and (6x + 10)°. Find x and both angles.
- Ray OD bisects ∠AOB. If ∠AOB and ∠BOC are on a straight line with ∠BOC = 70°, find ∠AOD.
- Five angles on a straight line are 20°, 30°, 40°, 50°, and x°. Find x.
- The supplement of an angle is four times the angle. Find the angle.
Frequently Asked Questions
Q1. What is the sum of angles on a straight line?
The sum of all angles on one side of a straight line, at a common point, is always 180°.
Q2. What is a straight angle?
A straight angle is an angle that measures exactly 180°. It is formed by two opposite rays and looks like a straight line.
Q3. What is a linear pair of angles?
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°.
Q4. Can three angles be on a straight line?
Yes. If multiple rays originate from the same point on a line, three or more angles can be formed on one side. Their sum is still 180°.
Q5. What is the converse of the straight line angle property?
If two adjacent angles sum to 180°, then the non-common arms of the angles form a straight line. This is used to prove that three points are collinear.
Q6. What is the difference between supplementary angles and angles on a straight line?
Supplementary angles are any two angles that sum to 180° (they do not need to be adjacent). Angles on a straight line are specifically adjacent supplementary angles formed when a ray stands on a line.
Q7. If one angle on a straight line is 90°, what is the other?
The other angle is also 90°. The ray is perpendicular to the line, and both angles are right angles.
Q8. Can angles on a straight line be used to find unknown angles in parallel line problems?
Yes. When a transversal cuts parallel lines, co-interior angles on the same side sum to 180°. This directly uses the straight line angle property.
Q9. Is this property the same as angles at a point?
No. Angles on a straight line sum to 180° (half turn). Angles at a point sum to 360° (full turn). They are different properties.
Q10. Is angles on a straight line in the CBSE Class 9 syllabus?
Yes. This is a core concept in the CBSE Class 9 chapter on Lines and Angles. It is used throughout the chapter and in proofs of other theorems.
