Quick Revision Notes: Class 9 Mathematics

Chapter: Lines and Angles (रेखाएँ और कोण)

1. Basic Terms and Definitions (मूल पद एवं परिभाषाएँ)

• Line (रेखा): A line extends indefinitely in both directions. It has no endpoints.
• Line Segment (रेखाखंड): A part of a line with two endpoints.
• Ray (किरण): A part of a line with one endpoint, extending indefinitely in one direction.
• Collinear Points (संरेख बिंदु): Three or more points that lie on the same straight line.
• Non-collinear Points (असंरेख बिंदु): Points that do not lie on the same straight line.

2. Types of Angles (कोणों के प्रकार)

• Acute Angle (न्यून कोण): An angle measuring between 0° and 90° (0° < x < 90°).
• Right Angle (समकोण): An angle measuring exactly 90°.
• Obtuse Angle (अधिक कोण): An angle measuring between 90° and 180° (90° < x < 180°).
• Straight Angle (ऋजु कोण): An angle measuring exactly 180°.
• Reflex Angle (प्रतिवर्ती कोण): An angle measuring greater than 180° and less than 360° (180° < x < 360°).

3. Pairs of Angles (कोणों के युग्म)

• Complementary Angles (कोटिपूरक कोण / पूरक कोण): Two angles whose sum is 90°.
  • Formula: If angle A and angle B are complementary, Angle A + Angle B = 90°.
• Supplementary Angles (संपूरक कोण): Two angles whose sum is 180°.
  • Formula: If angle A and angle B are supplementary, Angle A + Angle B = 180°.
• Adjacent Angles (आसन्न कोण): Two angles that have a common vertex, a common arm, and their non-common arms are on opposite sides of the common arm.
• Linear Pair of Angles (रेखीय युग्म): Two adjacent angles whose non-common arms form a line. The sum of angles in a linear pair is always 180°.
  • Formula: Angle 1 + Angle 2 = 180°
• Vertically Opposite Angles (शीर्षाभिमुख कोण): When two lines intersect each other, the vertically opposite angles formed are equal.
  • Rule: If line AB and CD intersect at O, then Angle AOD = Angle BOC and Angle AOC = Angle BOD.

4. Intersecting and Non-Intersecting Lines (प्रतिच्छेदी और अप्रतिच्छेदी रेखाएँ)

• Intersecting Lines (प्रतिच्छेदी रेखाएँ): Lines that meet at a common point.
• Parallel Lines (समांतर रेखाएँ): Non-intersecting lines that are always at the same perpendicular distance from each other everywhere.

5. Axioms and Theorems (अभिकृहीत और प्रमेय)

• Linear Pair Axiom (रैखिक युग्म अभिकृहीत):
  • If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
  • Converse: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
• Vertically Opposite Angles Theorem (शीर्षाभिमुख कोण प्रमेय):
  • If two lines intersect each other, then the vertically opposite angles are equal.

6. Lines Parallel to the Same Line (एक ही रेखा के समांतर रेखाएँ)

• Theorem: Lines which are parallel to the same line are parallel to each other.
  • If Line l || Line m and Line m || Line n, then Line l || Line n.

7. Angle Sum Property of a Triangle (त्रिभुज का कोण योग गुण)

• Theorem: The sum of the angles of a triangle is 180°.
  • Formula: In ΔABC, Angle A + Angle B + Angle C = 180°.
• Exterior Angle Property (बहिष्कोण गुण):
  • If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
  • Formula: Exterior Angle = Sum of two interior opposite angles.