Class 9 Mathematics Notes Chapter 6 (Chapter 6) – Examplar Problem (Englisha) Book

Alright class, let's focus on Chapter 6: Lines and Angles from your NCERT Exemplar book. This is a foundational chapter for geometry, and its concepts are frequently tested in various government exams. Pay close attention to the definitions, axioms, and theorems.

Chapter 6: Lines and Angles - Detailed Notes for Exam Preparation

1. Basic Terms and Definitions

• Line: A collection of points extending infinitely in both directions. Represented as $\overleftrightarrow{AB}$. It has no endpoints and no definite length.
• Line Segment: A part of a line with two distinct endpoints. Represented as $\overline{AB}$. It has a definite length.
• Ray: A part of a line with one endpoint (starting point) and extending infinitely in one direction. Represented as $\overrightarrow{AB}$.
• 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 (called the vertex). The rays are called the arms of the angle. Measured in degrees (°).
  • Acute Angle: An angle greater than 0° but less than 90° (0° < θ < 90°).
  • Right Angle: An angle exactly equal to 90° (θ = 90°).
  • Obtuse Angle: An angle greater than 90° but less than 180° (90° < θ < 180°).
  • Straight Angle: An angle exactly equal to 180° (θ = 180°). Forms a straight line.
  • Reflex Angle: An angle greater than 180° but less than 360° (180° < θ < 360°).
• Complementary Angles: Two angles whose sum is 90°. Each angle is the complement of the other.
• Supplementary Angles: Two angles whose sum is 180°. Each angle is the supplement of the other.
• Adjacent Angles: Two angles that have a common vertex, a common arm, and their non-common arms are on different sides of the common arm.
• Linear Pair of Angles: A pair of adjacent angles whose non-common arms form a straight line. The sum of angles in a linear pair is always 180°. (They are supplementary).
• Vertically Opposite Angles: When two lines intersect, the angles opposite to each other at the point of intersection are called vertically opposite angles. They are always equal.

2. Intersecting Lines and Non-intersecting Lines

• Intersecting Lines: Two lines that cross each other at a single common 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 remains constant.

3. Pairs of Angles Axioms and Theorems

• Axiom 6.1 (Linear Pair Axiom): If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.
• Axiom 6.2 (Converse of Linear Pair Axiom): If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
• Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equal. (Crucial for problem-solving).

4. Parallel Lines and a Transversal

• Transversal: A line that intersects two or more distinct lines at distinct points.
• When a transversal intersects two lines (parallel or not), eight angles are formed. These angles have special names based on their positions:
  • Corresponding Angles: Pairs of angles in the same relative position at each intersection. (e.g., ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 in standard diagrams).
  • Alternate Interior Angles: Pairs of angles on opposite sides of the transversal and between the two lines. (e.g., ∠4 and ∠6, ∠3 and ∠5).
  • Alternate Exterior Angles: Pairs of angles on opposite sides of the transversal and outside the two lines. (e.g., ∠1 and ∠7, ∠2 and ∠8).
  • Consecutive Interior Angles (or Co-interior Angles / Allied Angles): Pairs of angles on the same side of the transversal and between the two lines. (e.g., ∠4 and ∠5, ∠3 and ∠6).

5. Axioms and Theorems Related to Parallel Lines

• Axiom 6.3 (Corresponding Angles Axiom): If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
• Axiom 6.4 (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 6.2: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
  • Proof Idea: Use Corresponding Angles Axiom and Vertically Opposite Angles Theorem.
• Theorem 6.3 (Converse of Theorem 6.2): If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
• Theorem 6.4: If a transversal intersects two parallel lines, then each pair of consecutive interior angles is supplementary (sum is 180°).
  • Proof Idea: Use Corresponding Angles Axiom and Linear Pair Axiom.
• Theorem 6.5 (Converse of Theorem 6.4): If a transversal intersects two lines such that a pair of consecutive interior angles is supplementary, then the two lines are parallel.
• Theorem 6.6: 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).

6. Angle Sum Property of a Triangle

• Theorem 6.7: The sum of the angles of a triangle is 180°.
  • Proof Idea: Draw a line parallel to one side of the triangle through the opposite vertex. Use alternate interior angles and angles on a straight line.

7. Exterior Angle Theorem of a Triangle

• Theorem 6.8: 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.
  • Proof Idea: Use the Angle Sum Property and Linear Pair Axiom.

Key Takeaways for Exams:

• Master the definitions and properties of different angle pairs (Linear Pair, Vertically Opposite, Corresponding, Alternate Interior, Consecutive Interior).
• Understand the conditions required to prove lines are parallel (equality of corresponding angles, equality of alternate interior angles, supplementary consecutive interior angles).
• Be proficient in applying the Angle Sum Property and Exterior Angle Theorem for triangles.
• Many problems involve combining these concepts, especially using parallel lines to find angles within triangles or other figures.
• Practice problems from the NCERT Exemplar thoroughly, as they often involve multi-step reasoning.

Multiple Choice Questions (MCQs)

1. If two lines intersect such that one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are:
   (a) Acute angles
   (b) Right angles
   (c) Obtuse angles
   (d) Straight angles

2. Two complementary angles are such that one angle is 10° more than the other. The angles are:
   (a) 50°, 40°
   (b) 40°, 50°
   (c) 85°, 95°
   (d) 35°, 45°

3. In the figure, if AB || CD, then the value of x is:
   (Assume a figure where a transversal intersects parallel lines AB and CD, forming an angle of 75° and an alternate interior angle marked as x)
   (a) 75°
   (b) 105°
   (c) 90°
   (d) 65°

4. An exterior angle of a triangle is 110° and one of the interior opposite angles is 40°. The other interior opposite angle is:
   (a) 70°
   (b) 110°
   (c) 140°
   (d) 50°

5. If a ray stands on a line, the sum of the adjacent angles formed is:
   (a) 90°
   (b) 180°
   (c) 270°
   (d) 360°

6. In a triangle ABC, if ∠A = 55° and ∠B = 75°, then ∠C is:
   (a) 50°
   (b) 55°
   (c) 75°
   (d) 130°

7. Lines parallel to the same line are:
   (a) Perpendicular to each other
   (b) Intersecting
   (c) Parallel to each other
   (d) None of the above

8. If a transversal intersects two lines such that the consecutive interior angles are supplementary, then the lines are:
   (a) Parallel
   (b) Intersecting
   (c) Perpendicular
   (d) Cannot be determined

9. The angle which is equal to its supplement is:
   (a) 45°
   (b) 90°
   (c) 180°
   (d) 0°

10. In the figure, if line l || line m, what is the value of y?
    (Assume a figure where parallel lines l and m are intersected by a transversal. An angle corresponding to y is shown as 60°)
    (a) 60°
    (b) 120°
    (c) 90°
    (d) 30°

Answers to MCQs:

1. (c) Obtuse angles (If one pair is acute, their sum is < 180°. The other pair must form linear pairs with them, making them obtuse).
2. (a) 50°, 40° (Let angles be x and x+10. x + x + 10 = 90 => 2x = 80 => x = 40. Angles are 40° and 50°).
3. (a) 75° (Alternate interior angles are equal when lines are parallel).
4. (a) 70° (Exterior angle = sum of interior opposite angles. 110° = 40° + y => y = 70°).
5. (b) 180° (Linear Pair Axiom).
6. (a) 50° (Angle sum property: 55° + 75° + ∠C = 180° => 130° + ∠C = 180° => ∠C = 50°).
7. (c) Parallel to each other (Theorem 6.6).
8. (a) Parallel (Converse of Theorem 6.4).
9. (b) 90° (Let angle be x. x = 180° - x => 2x = 180° => x = 90°).
10. (a) 60° (Corresponding angles are equal when lines are parallel).

Make sure you understand the reasoning behind each answer. Revise these concepts regularly, and practice solving problems from the Exemplar book and previous exam papers. Good luck!