Class 9 Mathematics: Triangles

• Introduction to Triangles

  • Definition: A closed figure formed by three intersecting lines.
  • Parts: 3 sides, 3 angles, and 3 vertices.
  • Classification:
    • By Sides: Scalene, Isosceles, Equilateral.
    • By Angles: Acute, Right, Obtuse.

• Congruence of Triangles

  • Definition: Figures that have the same shape and same size.
  • Congruent Triangles ($\cong$): Corresponding sides and corresponding angles are equal (CPCT: Corresponding Parts of Congruent Triangles).

• Criteria for Congruence

  • SAS Criterion (Side-Angle-Side): Two sides and the included angle are equal.
  • ASA Criterion (Angle-Side-Angle): Two angles and the included side are equal.
  • AAS Criterion (Angle-Angle-Side): Two angles and any non-included side are equal.
  • SSS Criterion (Side-Side-Side): All three sides of one triangle are equal to the corresponding sides of another.
  • RHS Criterion (Right angle-Hypotenuse-Side): For right-angled triangles, the hypotenuse and one side are equal.

• Inequalities in a Triangle

  • Theorem 1: Angle opposite to the longer side is larger.
  • Theorem 2: Side opposite to the larger angle is longer.
  • Theorem 3: Sum of any two sides is greater than the third side ($a + b > c$).