Triangles

🎯 Learning Objective

Apply the angle-sum, exterior-angle, Pythagoras and triangle-inequality rules to exam questions.

💡 Concept

• Angles of a triangle always add to 180°
• Exterior angle = sum of the two opposite interior angles
• Right triangle (Pythagoras): H² = P² + B²
• Triplets to memorise: 3-4-5, 5-12-13, 7-24-25, 8-15-17 — and all their multiples
• Triangle inequality: sum of any two sides must exceed the third side

🧮 Key Formulas

A + B + C = 180°

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Exterior angle = sum of opposite interior angles

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H² = P² + B²

✏️ Easy Example

Q. Two angles of a triangle are 65° and 45°. Find the third angle.

1. 65 + 45 = 110
2. Third angle = 180 − 110

Answer: 70°

🇮🇳 Real-Life Example

Masons (राजमिस्त्री) check a perfect corner with the 3-4-5 rope trick: mark 3 m and 4 m along two walls — the diagonal must be exactly 5 m.

📝 Exam-Level Example

Q. A 13 m ladder rests against a wall with its foot 5 m from the wall. How high up the wall does it reach?

1. H² = P² + B² → 13² = h² + 5²
2. h² = 169 − 25 = 144
3. h = √144

Answer: 12 m

📝 Exam-Level Example

Q. An exterior angle of a triangle is 110° and one opposite interior angle is 40°. Find the other opposite interior angle.

1. Exterior = sum of opposite interiors
2. 110 = 40 + x
3. x = 70

Answer: 70°

🪄 Memory Trick

Two sides of a right triangle given? Check triplet multiples before squaring — sides 39 and 15 are 3 × (13, 5), so the third side is 3 × 12 = 36.

⚠️ Common Mistakes

• ❌ Applying Pythagoras to a triangle that is not right-angled
• ❌ Accepting impossible triangles — sides 2, 3, 6 cannot form one (2 + 3 < 6)

🏆 Exam Tips

• ✅ Exterior-angle rule skips finding the third angle — saves a full step
• ✅ In exams, triplet multiples (6-8-10, 9-12-15, 10-24-26) appear more than raw triplets

📌 Summary

• Angle sum = 180°
• Exterior angle = sum of opposite interiors
• H² = P² + B²; memorise the triplets
• Any two sides together must beat the third