Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines Infographic | NCF 2023 Ganita Prakash

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✦ NCF 2023 GANITA PRAKASH • GRADE 7 SERIES ✦

Chapter 7: A Tale of Three Intersecting Lines

The World of Triangles: Triangle Inequality, Angle Sum Property, Altitudes & Constructions

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📖 Triangle Families

By Sides

• Equilateral: All 3 sides equal (all angles 60°)
• Isosceles: Exactly 2 sides equal
• Scalene: All 3 sides different lengths

By Angles

• Acute-angled: All 3 angles < 90°
• Right-angled: Exactly one angle = 90°
• Obtuse-angled: Exactly one angle > 90°

📐 Fundamental Laws

1. Triangle Inequality Theorem:

The sum of lengths of any two sides must be strictly greater than the third side (i.e. a + b > c, b + c > a, c + a > b). Quick test: Sum of 2 smaller sides > Longest side.

2. Angle Sum Property:

The sum of all interior angles is always 180° (proven via auxiliary parallel lines by Euclid in The Elements).

⛰️ Altitudes & Angles

Exterior Angle Property:

An exterior angle equals the sum of its two interior opposite angles: ∠ACD = ∠A + ∠B.

Altitude (Height):

A perpendicular line segment drawn from a vertex to the opposite side (or extended base). A triangle has 3 altitudes.

Right Triangle Altitude:

In a right-angled triangle, the two perpendicular legs serve as two of the altitudes directly!

🛠️ Construction Conditions & Logic

• Three Sides Given (SSS): Use compass arcs of given radii. Two circles intersect internally if and only if the lengths satisfy the triangle inequality.

• Two Sides & Included Angle (SAS): Triangle is uniquely formed as long as the included angle is strictly less than 180°.

• Two Angles & Included Side (ASA): A triangle exists only if the sum of the two given angles is strictly less than 180° (i.e. ∠A + ∠B < 180°).

• Range of Third Side: For any two known sides a and b, the third side c must always satisfy: |a − b| < c < a + b.

⚠️ Common Triangle Traps

Trap 1: The "Sum Equals Third Side" Fallacy

Lengths like (3, 6, 9) or (2, 3, 5) CANNOT form a triangle because 3 + 6 = 9. The arcs merely touch on a straight line segment without forming a third vertex.

Trap 2: Obtuse Altitude Location

In an obtuse-angled triangle, altitudes from acute vertices lie outside the triangle on the extended base lines, not inside!

Designed by Akash Srivastva | Mathematics Hub • NCF 2023 Visual Learning Series

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