Class 9 Mathematics: Lines and Angles

• Basic Terms and Definitions

  • Line segment: A part of a line with two end points
  • Ray: A part of a line with one end point
  • Collinear points: Three or more points lying on the same line
  • Non-collinear points: Points that do not lie on the same line

• Types of Angles

  • Acute angle: Measures between $0^\circ$ and $90^\circ$
  • Right angle: Exactly $90^\circ$
  • Obtuse angle: Measures between $90^\circ$ and $180^\circ$
  • Straight angle: Exactly $180^\circ$
  • Reflex angle: Measures between $180^\circ$ and $360^\circ$

• Pairs of Angles

  • Complementary angles: Sum of two angles is $90^\circ$
  • Supplementary angles: Sum of two angles is $180^\circ$
  • Adjacent angles: Have a common vertex, a common arm, and non-common arms on opposite sides
  • Linear pair: Adjacent angles whose sum is $180^\circ$ (form a straight line)
  • Vertically opposite angles: Formed by two intersecting lines; are equal

• Intersecting and Non-Intersecting Lines

  • Intersecting lines: Meet at a common point
  • Parallel lines: Never meet; perpendicular distance between them is always equal

• Axioms and Theorems

  • Linear Pair Axiom: If a ray stands on a line, the sum of two adjacent angles is $180^\circ$ (and converse)
  • Vertically Opposite Angles Theorem: If two lines intersect, the vertically opposite angles are equal

• Angles Made by a Transversal with Parallel Lines

  • Transversal line: A line that intersects two or more lines at distinct points
  • Angle types:
    • Corresponding angles (Equal if lines are parallel)
    • Alternate interior angles (Equal if lines are parallel)
    • Alternate exterior angles (Equal if lines are parallel)
    • Interior angles on the same side of the transversal (Sum is $180^\circ$ / Supplementary if lines are parallel)
  • Lines parallel to the same line: Lines parallel to the same line are parallel to each other

• Angle Sum Property of a Triangle

  • Theorem: The sum of the angles of a triangle is $180^\circ$
  • Exterior Angle Theorem: If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles