Chapter 7: A Tale of Three Intersecting Lines
The World of Triangles: Triangle Inequality, Angle Sum Property, Altitudes & Constructions
📖 Triangle Families
• Equilateral: All 3 sides equal (all angles 60°)
• Isosceles: Exactly 2 sides equal
• Scalene: All 3 sides different lengths
• 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
An exterior angle equals the sum of its two interior opposite angles: ∠ACD = ∠A + ∠B.
A perpendicular line segment drawn from a vertex to the opposite side (or extended base). A triangle has 3 altitudes.
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!