Lines and Angles
1. Basic Concepts
Point
A POINT is an exact location. It has NO size. Denoted by a dot.
Line Segment
A LINE SEGMENT has TWO end points. Denoted as AB.
Line
A LINE extends infinitely in BOTH directions. Denoted as AB ↔.
Ray
A RAY has ONE end point and extends infinitely in ONE direction. Denoted as AB →.
2. Types of Angles
An angle is formed when TWO RAYS meet at a common point (the VERTEX).
| Type | Measure | Diagram |
|---|---|---|
| Acute | Between 0° and 90° | |
| Right | Exactly 90° | Marked with a small square |
| Obtuse | Between 90° and 180° | |
| Straight | Exactly 180° | A straight line |
| Reflex | Between 180° and 360° | |
| Complete | Exactly 360° | Full rotation |
Naming Angles
∠ABC means angle with vertex at B, with BA and BC as arms.
3. Special Pairs of Angles
Complementary Angles
Two angles whose SUM is 90°.
Example: 30° and 60° are complementary. If ∠A = 42°, complement of ∠A = 90° - 42° = 48°.
Supplementary Angles
Two angles whose SUM is 180°.
Example: 110° and 70° are supplementary. If ∠B = 65°, supplement of ∠B = 180° - 65° = 115°.
Adjacent Angles
Two angles that share:
- A COMMON vertex
- A COMMON arm
- They do NOT overlap.
Vertically Opposite Angles
When two lines INTERSECT, the angles OPPOSITE each other are called vertically opposite angles. They are ALWAYS EQUAL.
Worked Example (ICSE 2024, 2 marks)
Two lines AB and CD intersect at O. If ∠AOC = 55°, find ∠BOD, ∠AOD, and ∠BOC.
Solution: ∠BOD = ∠AOC = 55° (vertically opposite). ∠AOD = 180° - 55° = 125° (linear pair with ∠AOC). ∠BOC = ∠AOD = 125° (vertically opposite).
4. Angles Formed by a Transversal
A TRANSVERSAL is a line that cuts TWO or more lines at distinct points.
When Transversal Cuts PARALLEL Lines
| Angle Pair | Relationship | Example |
|---|---|---|
| Corresponding angles (same position) | EQUAL | ∠1 and ∠5 |
| Alternate interior angles | EQUAL | ∠3 and ∠6 |
| Alternate exterior angles | EQUAL | ∠1 and ∠8 |
| Co-interior angles (interior on same side) | SUM = 180° | ∠3 and ∠5 |
Diagram Reference
If two parallel lines are cut by a transversal:
l
1 2 |
4 3 |
---------|------- m (parallel)
5 6 |
8 7 |
|------- n (parallel)
Worked Example (ICSE 2023, 3 marks)
In the figure, l ∥ m and t is a transversal. If ∠4 = 70°, find all other angles.
Solution:
- ∠3 = ∠4 = 70° (vertically opposite)
- ∠1 = 180° - 70° = 110° (linear pair with ∠4)
- ∠2 = ∠1 = 110° (vertically opposite)
- ∠5 = ∠4 = 70° (alternate interior)
- ∠8 = ∠5 = 70° (vertically opposite)
- ∠6 = 180° - 70° = 110° (co-interior with ∠4, or linear pair with ∠5)
- ∠7 = ∠6 = 110° (vertically opposite)
5. Checking Parallelism
If a transversal cuts two lines and:
- A pair of corresponding angles are EQUAL, OR
- A pair of alternate interior angles are EQUAL, OR
- A pair of co-interior angles are SUPPLEMENTARY, Then the two lines are PARALLEL.
Worked Example
'Check if lines are parallel: a transversal gives ∠1 = 65° and ∠5 = 65° (corresponding).' YES — corresponding angles are equal, so lines are parallel.
6. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Complementary and supplementary angles | 2 marks | High |
| Vertically opposite angles | 2 marks | Very High |
| Parallel lines and transversal (finding angles) | 3-4 marks | Very High |
| Proving lines parallel | 2-3 marks | Medium |
Common Mistakes
- Confusing alternate interior with co-interior (alternate = EQUAL, co-interior = SUM 180°).
- Thinking all pairs of equal angles formed by a transversal mean lines are parallel — ONLY true if the lines are known to be parallel OR specific angle pairs are equal.
- Forgetting linear pair property when solving for missing angles.
- Not writing '°' (degree symbol) with angle measures.
Angle Fact Summary
- Vertically opposite angles are EQUAL.
- Linear pair is SUPPLEMENTARY (sum = 180°).
- Angles around a point sum to 360°.
- Corresponding angles (∥ lines): EQUAL.
- Alternate angles (∥ lines): EQUAL.
- Co-interior angles (∥ lines): Supplementary.
Self-Test (5 Questions)
Q1. If an angle is 36°, find its complement. (1 mark)
- A) 54°
- B) 144°
- C) 36°
- D) 64°
Q2. Two angles are supplementary. If one is 72°, find the other. (1 mark)
Q3. State TRUE/FALSE: 'Vertically opposite angles are always supplementary.' (1 mark)
Q4. In the figure with parallel lines and a transversal, if one interior angle is 110°, find the co-interior angle. (2 marks)
Q5. Two parallel lines are cut by a transversal. If one corresponding angle is 58°, find ALL angles. (4 marks)
Answers
A1. A) 54°. (90° - 36° = 54°.) A2. 108°. (180° - 72° = 108°.) A3. FALSE. Vertically opposite angles are EQUAL. They are supplementary only when each is 90°. A4. 70°. (Co-interior angles are supplementary: 180° - 110° = 70°.) A5. The corresponding angle is 58°. All acute angles = 58°, all obtuse angles = 122°. (4 acute, 4 obtuse.)
