Lines and Angles Class 9 ||Maths|| Chapter 6 Notes

Lines and Angles ||Maths|| Chapter 6 NCERT Notes

1. Basic Definitions

  • Line: A straight one-dimensional figure with no thickness that extends infinitely in both directions.
  • Line Segment: A part of a line with two endpoints.
  • Ray: A part of a line that starts at one point and extends infinitely in one direction.
  • Angle: Formed by two rays with a common endpoint called the vertex.

2. Types of Angles

  • Acute Angle: An angle measuring less than 90°.
  • Right Angle: An angle measuring exactly 90°.
  • Obtuse Angle: An angle measuring more than 90° but less than 180°.
  • Straight Angle: An angle measuring exactly 180°.
  • Reflex Angle: An angle measuring more than 180° but less than 360°.
  • Complementary Angles: Two angles whose sum is 90°.
  • Supplementary Angles: Two angles whose sum is 180°.
  • Adjacent Angles: Two angles that have a common arm and vertex but no common interior points.
  • Linear Pair: A pair of adjacent angles whose non-common arms form a straight line, summing up to 180°.
  • Vertically Opposite Angles: The angles opposite each other when two lines intersect. These angles are equal.

3. Pair of Lines

  • Intersecting Lines: Two lines that meet or cross each other at a point.
  • Parallel Lines: Lines in a plane that never meet or intersect, no matter how far they are extended. The distance between them remains constant.

4. Angles Formed by a Transversal

A transversal is a line that intersects two or more lines at different points. When a transversal crosses two lines, several angles are formed:

  • Corresponding Angles: Angles that are in the same position relative to the two lines and the transversal. If two parallel lines are cut by a transversal, corresponding angles are equal.

  • Alternate Interior Angles: Angles on opposite sides of the transversal and between the two lines. If the two lines are parallel, alternate interior angles are equal.

  • Alternate Exterior Angles: Angles on opposite sides of the transversal but outside the two lines. If the lines are parallel, alternate exterior angles are equal.

  • Interior Angles on the Same Side of the Transversal (Co-Interior/Consecutive Interior Angles): Angles on the same side of the transversal and between the two lines. If the lines are parallel, these angles are supplementary (sum equals 180°).

5. Properties of Parallel Lines with a Transversal

When two parallel lines are intersected by a transversal, the following properties hold:

  1. Corresponding Angles are equal.
  2. Alternate Interior Angles are equal.
  3. Alternate Exterior Angles are equal.
  4. Co-Interior Angles are supplementary.

6. Important Theorems

  • Theorem 1 (Corresponding Angles Postulate): If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

  • Theorem 2 (Alternate Interior Angles Theorem): If a transversal intersects two parallel lines, each pair of alternate interior angles is equal.

  • Theorem 3 (Converse of Corresponding Angles Postulate): If a transversal intersects two lines such that a pair of corresponding angles is equal, then the lines are parallel.

  • Theorem 4 (Converse of Alternate Interior Angles Theorem): If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the lines are parallel.

  • Theorem 5 (Converse of Co-Interior Angles Theorem): If a transversal intersects two lines such that a pair of co-interior angles is supplementary, then the lines are parallel.

7. Linear Pair Axiom

If two angles form a linear pair, then their sum is always 180°. This property is used to prove several theorems in geometry.

8. Angle Sum Property of a Triangle

The sum of the interior angles of any triangle is always 180°. This is a fundamental property used in various geometric proofs.

Summary:

  • A line extends infinitely in both directions, while an angle is formed when two rays meet at a common point.
  • Pairs of angles, such as complementary, supplementary, and adjacent angles, have specific properties.
  • When a transversal cuts two lines, it forms different types of angles (corresponding, alternate interior, alternate exterior, and co-interior), with special properties if the lines are parallel.
  • Understanding the properties of angles and lines is essential for solving geometrical problems involving triangles, polygons, and other figures.