Chapter: The Triangle and its Properties

• Introduction to Triangle

  • Definition: Closed figure with 3 sides, 3 angles, and 3 vertices
  • Basic Parts: Sides ($AB, BC, CA$), Angles ($\angle A, \angle B, \angle C$)
  • Types based on Sides: Scalene, Isosceles, Equilateral
  • Types based on Angles: Acute-angled, Right-angled, Obtuse-angled

• Medians of a Triangle

  • Definition: Line segment joining a vertex to the mid-point of the opposite side
  • Key Feature: A triangle has exactly 3 medians
  • Centroid: Point of concurrency of all three medians

• Altitudes of a Triangle

  • Definition: Perpendicular line segment from a vertex to the opposite side
  • Key Feature: Also called the height; a triangle has 3 altitudes
  • Location: Can lie inside, outside, or on the boundary of the triangle

• Exterior Angle of a Triangle

  • Definition: Angle formed when a side of a triangle is produced
  • Exterior Angle Property: Equal to the sum of its two interior opposite angles
  • Inequality Property: Greater than either of its interior opposite angles

• Angle Sum Property of a Triangle

  • Theorem: The sum of the three interior angles of a triangle is always $180^\circ$
  • Verification: Can be proven by drawing a line parallel to one side through the opposite vertex

• Special Triangles

  • Equilateral Triangle: All sides are equal; all interior angles are $60^\circ$
  • Isosceles Triangle: Two sides are equal; angles opposite to equal sides are equal

• Sum of the Lengths of Two Sides

  • Triangle Inequality Theorem: The sum of the lengths of any two sides is greater than the length of the third side
  • Application: Used to check if a triangle is possible with given side lengths

• Right-Angled Triangle and Pythagoras Property

  • Hypotenuse: The side opposite to the right angle (longest side)
  • Pythagoras Theorem: $(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Perpendicular})^2$ ($c^2 = a^2 + b^2$)
  • Converse: If the property holds, the triangle is guaranteed to be a right-angled triangle