Triangles and Its Properties
1. What Is a Triangle?
A TRIANGLE is a closed figure formed by THREE line segments.
Parts of a Triangle:
- Vertices: The three points where the sides meet (A, B, C).
- Sides: The three line segments (AB, BC, CA).
- Angles: The three interior angles (∠A, ∠B, ∠C).
Notation: ΔABC.
2. Types of Triangles
Based on Sides
| Type | Property | Example |
|---|---|---|
| Equilateral | All 3 sides EQUAL. All angles = 60°. | Sides: 5 cm, 5 cm, 5 cm |
| Isosceles | 2 sides EQUAL. Base angles EQUAL. | Sides: 6 cm, 6 cm, 4 cm |
| Scalene | ALL sides DIFFERENT. All angles different. | Sides: 3 cm, 4 cm, 5 cm |
Based on Angles
| Type | Property | Example |
|---|---|---|
| Acute-angled | ALL angles < 90° | 50°, 60°, 70° |
| Right-angled | ONE angle = 90° | 90°, 45°, 45° |
| Obtuse-angled | ONE angle > 90° | 110°, 35°, 35° |
3. Angle Sum Property
The sum of the three interior angles of a triangle is ALWAYS 180°.
∠A + ∠B + ∠C = 180°
Proof (Simple)
Draw a triangle. Draw a line through one vertex parallel to the opposite side. Alternate interior angles show that the three angles form a straight line = 180°.
Worked Example (ICSE 2024, 2 marks)
'In a triangle, angles are in the ratio 2 : 3 : 4. Find each angle.'
Solution: Let angles be 2x, 3x, 4x. 2x + 3x + 4x = 180° 9x = 180° x = 20°. Angles: 2x = 40°, 3x = 60°, 4x = 80°.
Worked Example 2 (ICSE 2023, 2 marks)
'In ΔABC, ∠A = 65° and ∠B = 40°. Find ∠C.' ∠C = 180° - 65° - 40° = 75°.
4. Exterior Angle Property
An EXTERIOR ANGLE is formed when one side of a triangle is extended.
Exterior angle = Sum of two interior OPPOSITE angles.
∠ACD (exterior) = ∠A + ∠B
Worked Example (ICSE 2024, 3 marks)
'In ΔABC, side BC is extended to D. If ∠A = 50° and ∠B = 70°, find ∠ACD.'
Solution: ∠ACD = ∠A + ∠B = 50° + 70° = 120°.
Also
∠ACD + ∠ACB = 180° (linear pair). Check: 120° + 60° = 180°. ✓
5. Pythagoras Theorem
In a RIGHT-ANGLED triangle: (Hypotenuse)² = (Base)² + (Height)²
Hypotenuse: The side OPPOSITE the right angle. It is the LONGEST side.
Formula
c² = a² + b², where c = hypotenuse, a and b = other two sides.
Worked Example (ICSE 2024, 3 marks)
'A ladder 13 m long reaches a window 12 m above the ground. Find the distance of the foot of the ladder from the wall.'
Solution: Hypotenuse = 13 m, Height = 12 m. Base² = 13² - 12² = 169 - 144 = 25. Base = √25 = 5 m.
Common Pythagorean Triplets (Memorise)
(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (6, 8, 10).
6. Triangle Inequality Property
The sum of any TWO sides of a triangle is GREATER than the THIRD side.
Converse: If this is true for all three combinations, a triangle CAN be formed.
Checking if Triangle is Possible
'Can sides of 3 cm, 4 cm, 7 cm form a triangle?' 3 + 4 = 7. NOT greater than 7. So NO triangle is possible. 'Can sides of 4 cm, 5 cm, 6 cm form a triangle?' 4 + 5 > 6 ✓, 5 + 6 > 4 ✓, 4 + 6 > 5 ✓. YES.
7. Altitude and Median
| Term | Definition |
|---|---|
| Altitude | Perpendicular from a vertex to the OPPOSITE side (or its extension). |
| Median | Line segment from a vertex to the MIDPOINT of the opposite side. |
- Every triangle has 3 altitudes. They are CONCURRENT (meet at one point — the ORTHOCENTRE).
- Every triangle has 3 medians. They are CONCURRENT (meet at the CENTROID).
- In an EQUILATERAL triangle, altitude = median.
8. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Angle sum property | 2-3 marks | Very High |
| Exterior angle property | 2-3 marks | High |
| Pythagoras theorem | 3 marks | Very High |
| Triangle inequality | 2 marks | Medium |
| Types of triangles | 1-2 marks | Low |
Common Mistakes
- Pythagoras theorem applied to ANY triangle — NO, it applies ONLY to RIGHT triangles.
- Confusing hypotenuse: it is opposite the right angle, NOT the longest side (which happens to be the same thing).
- Forgetting to take SQUARE ROOT after applying Pythagoras.
- Triangle inequality: check ALL THREE combinations, not just one.
Self-Test (5 Questions)
Q1. 'Find the third angle if two angles of a triangle are 48° and 72°.' (1 mark)
- A) 50°
- B) 60°
- C) 70°
- D) 80°
Q2. 'Is a triangle with sides 5, 7, 12 possible?' (1 mark)
Q3. 'In a right triangle with legs 6 cm and 8 cm, find the hypotenuse.' (2 marks)
Q4. 'The exterior angle of a triangle is 120° and its interior opposite angles are equal. Find each angle.' (3 marks)
Q5. 'A 25 m ladder reaches a window 20 m high. Find distance of foot from the wall.' (2 marks)
Answers
A1. B) 60°. (180° - 48° - 72° = 60°.) A2. NO. (5 + 7 = 12, NOT greater than 12.) A3. 10 cm. (c² = 6² + 8² = 36 + 64 = 100. c = 10 cm.) A4. Each interior opposite angle = 60°. Third angle = 60°. So all angles are 60°. (Exterior = sum of two opposite. 2x = 120°, x = 60°. Third = 180 - 120 = 60°.) A5. 15 m. (b² = 25² - 20² = 625 - 400 = 225. b = 15 m.)
