Class 9 Notes

Class 9 Mathematics Notes Chapter 6 (Lines and Angles) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Alright class, let's focus on Chapter 6: Lines and Angles. This is a fundamental chapter in geometry, and its concepts are crucial not just for your Class 9 understanding but also form the basis for many questions in government competitive exams. Pay close attention to the definitions and theorems.

Chapter 6: Lines and Angles - Detailed Notes

1. Basic Terms and Definitions

  • Line: A straight path extending indefinitely in both directions. It has no endpoints. Represented as AB↔.
  • Line Segment: A part of a line with two distinct endpoints. Represented as AB¯¯¯¯¯¯¯¯.
  • Ray: A part of a line with one endpoint, extending indefinitely in one direction. Represented as AB→.
  • Collinear Points: Three or more points that lie on the same straight line.
  • Non-collinear Points: Three or more points that do not lie on the same straight line.
  • Angle: Formed when two rays originate from the same endpoint (vertex). The rays are called the arms of the angle. Measured in degrees (°).

2. Types of Angles

  • Acute Angle: An angle measuring greater than 0° and less than 90°. (0° < θ < 90°)
  • Right Angle: An angle measuring exactly 90°. (θ = 90°)
  • Obtuse Angle: An angle measuring greater than 90° and less than 180°. (90° < θ < 180°)
  • Straight Angle: An angle measuring exactly 180°. Forms a straight line. (θ = 180°)
  • Reflex Angle: An angle measuring greater than 180° and less than 360°. (180° < θ < 360°)

3. Related Angles

  • Complementary Angles: Two angles whose sum is 90°. Each angle is the complement of the other.
    • Example: 30° and 60° are complementary.
  • Supplementary Angles: Two angles whose sum is 180°. Each angle is the supplement of the other.
    • Example: 110° and 70° are supplementary.
  • Adjacent Angles: Two angles that have:
    • A common vertex.
    • A common arm.
    • Their non-common arms are on different sides of the common arm.
    • Important Note: Adjacent angles don't necessarily sum to 180° or 90° unless specified.
  • Linear Pair of Angles: A pair of adjacent angles whose non-common arms form a straight line.
    • Axiom 1 (Linear Pair Axiom): If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.
    • Axiom 2 (Converse): If the sum of two adjacent angles is 180°, then their non-common arms form a line.
    • Angles in a linear pair are always supplementary.
  • Vertically Opposite Angles: Angles formed when two lines intersect. They are opposite to each other at the vertex.
    • Theorem 1: If two lines intersect each other, then the vertically opposite angles are equal.

4. Intersecting Lines and Non-Intersecting Lines

  • Intersecting Lines: Two lines that cross each other at a single point.
  • Non-Intersecting Lines (Parallel Lines): Two lines in the same plane that never intersect, no matter how far they are extended. The perpendicular distance between them is always constant.

5. Pairs of Angles formed by a Transversal

  • Transversal: A line that intersects two or more distinct lines at distinct points.
  • When a transversal intersects two lines (say 'l' and 'm'), eight angles are formed. These angles have special names based on their positions:
    • Interior Angles: Angles between the lines l and m (e.g., ∠3, ∠4, ∠5, ∠6 in standard diagrams).
    • Exterior Angles: Angles outside the lines l and m (e.g., ∠1, ∠2, ∠7, ∠8).
    • Corresponding Angles: Angles in the same relative position at each intersection. Pairs: (∠1, ∠5), (∠2, ∠6), (∠4, ∠8), (∠3, ∠7).
    • Alternate Interior Angles: Angles on opposite sides of the transversal and between the two lines. Pairs: (∠4, ∠6), (∠3, ∠5).
    • Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the two lines. Pairs: (∠1, ∠7), (∠2, ∠8).
    • Consecutive Interior Angles (or Co-interior Angles / Allied Angles): Angles on the same side of the transversal and between the two lines. Pairs: (∠4, ∠5), (∠3, ∠6).

6. Parallel Lines and a Transversal

This is where the properties become very important for problem-solving. If a transversal intersects two parallel lines:

  • Axiom 3 (Corresponding Angles Axiom): If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
  • Axiom 4 (Converse of Corresponding Angles Axiom): If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.
  • Theorem 2: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
  • Theorem 3 (Converse): If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
  • Theorem 4: If a transversal intersects two parallel lines, then each pair of consecutive interior angles is supplementary (sum is 180°).
  • Theorem 5 (Converse): If a transversal intersects two lines such that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
  • Note: If lines are parallel, alternate exterior angles are also equal.

7. Lines Parallel to the Same Line

  • Theorem 6: Lines which are parallel to the same line are parallel to each other. (If line l || line m and line n || line m, then line l || line n).

8. Angle Sum Property of a Triangle

  • Theorem 7: The sum of the three interior angles of a triangle is always 180°.
    • Proof idea: Draw a line parallel to one side of the triangle through the opposite vertex and use alternate interior angle properties.

9. Exterior Angle Property of a Triangle

  • Theorem 8: 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.
    • Example: In ΔABC, if side BC is extended to D, then ∠ACD (exterior angle) = ∠BAC + ∠ABC (interior opposite angles).

Key Takeaways for Exams:

  • Memorize definitions precisely.
  • Understand the conditions for angles (complementary, supplementary, linear pair, adjacent).
  • Master the angle relationships when a transversal cuts parallel lines (Corresponding, Alternate Interior, Consecutive Interior). Know both the property and its converse.
  • Remember the Angle Sum Property and Exterior Angle Property of triangles.
  • Practice identifying angle pairs in diagrams quickly.
  • Be prepared to use multiple properties together to solve a problem.

Multiple Choice Questions (MCQs)

  1. If two complementary angles are in the ratio 4:5, the angles are:
    (A) 40°, 50°
    (B) 50°, 40°
    (C) 36°, 54°
    (D) 45°, 45°

  2. In the figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE. (Assume A-O-B and C-O-D are straight lines, E is another point).
    (A) 30°
    (B) 40°
    (C) 35°
    (D) 25°

  3. An angle is 20° more than its complement. What is the measure of the angle?
    (A) 55°
    (B) 65°
    (C) 35°
    (D) 70°

  4. Two parallel lines are intersected by a transversal. If one of the interior angles on the same side of the transversal is 110°, what is the measure of the other angle?
    (A) 110°
    (B) 70°
    (C) 90°
    (D) 20°

  5. In ΔABC, ∠A = 50° and ∠B = 60°. If the side BC is extended to D, what is the measure of the exterior angle ∠ACD?
    (A) 110°
    (B) 120°
    (C) 70°
    (D) 180°

  6. If two lines intersect, how many pairs of vertically opposite angles are formed?
    (A) 1
    (B) 2
    (C) 3
    (D) 4

  7. Which of the following statements ensures that lines 'l' and 'm' are parallel when intersected by a transversal 't'?
    (A) A pair of adjacent angles are equal.
    (B) A pair of vertically opposite angles are equal.
    (C) A pair of alternate interior angles are equal.
    (D) The sum of angles in a linear pair is 180°.

  8. The measure of a reflex angle must be:
    (A) Between 0° and 90°
    (B) Between 90° and 180°
    (C) Exactly 180°
    (D) Between 180° and 360°

  9. In a triangle, one angle is 90°. What is the sum of the other two angles?
    (A) 180°
    (B) 90°
    (C) 270°
    (D) Cannot be determined

  10. If a ray stands on a line, the sum of the adjacent angles formed is:
    (A) 90°
    (B) 180°
    (C) 360°
    (D) Depends on the angles

Answer Key for MCQs:

  1. (A) 40°, 50° (Let angles be 4x and 5x. 4x + 5x = 90 => 9x = 90 => x = 10. Angles are 40° and 50°)
  2. (A) 30° (∠AOC = ∠BOD = 40° vertically opposite. 40° + ∠BOE = 70° => ∠BOE = 30°)
  3. (A) 55° (Let angle be x. Complement is 90-x. x = (90-x) + 20 => 2x = 110 => x = 55°)
  4. (B) 70° (Consecutive interior angles are supplementary. 180° - 110° = 70°)
  5. (A) 110° (Exterior angle = sum of interior opposite angles = ∠A + ∠B = 50° + 60° = 110°)
  6. (B) 2
  7. (C) A pair of alternate interior angles are equal. (This is a condition for parallel lines).
  8. (D) Between 180° and 360°
  9. (B) 90° (Sum of all three is 180°. 180° - 90° = 90°)
  10. (B) 180° (Linear Pair Axiom)

Make sure you thoroughly understand these concepts and practice solving problems based on them. Good luck with your preparation!

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