Class 11 Notes - Lines and Angles

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Lines and Angles

In this chapter, you will learn about the fundamental concepts of lines and angles, their types, properties, and relationships. Understanding lines and angles is essential for solving geometric problems and forms the basis for advanced geometry.

1. Basic Terms and Definitions

  • Line: A straight path 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.
  • Collinear Points: Points that lie on the same straight line.
  • Non-collinear Points: Points that do not lie on the same straight line.

2. Types of Angles

  • Acute Angle: Less than 90°
  • Right Angle: Exactly 90°
  • Obtuse Angle: Greater than 90° but less than 180°
  • Straight Angle: Exactly 180°
  • Reflex Angle: Greater than 180° but less than 360°
  • Complete Angle: Exactly 360°

3. Pairs of Angles

  • Complementary Angles: Two angles whose sum is 90°.
  • Supplementary Angles: Two angles whose sum is 180°.
  • Adjacent Angles: Two angles that have a common vertex and a common arm but do not overlap.
  • Linear Pair: A pair of adjacent angles whose non-common arms form a straight line (sum is 180°).
  • Vertically Opposite Angles: Angles opposite each other when two lines intersect; they are always equal.

4. Properties and Theorems

  • When two lines intersect, the vertically opposite angles are equal.
  • The sum of angles on a straight line is 180°.
  • If two lines are parallel and are cut by a transversal, then:
    • Corresponding angles are equal.
    • Alternate interior angles are equal.
    • Co-interior (consecutive) angles are supplementary.

5. Parallel and Transversal Lines

  • Parallel Lines: Lines in a plane that never meet, no matter how far they are extended.
  • Transversal: A line that intersects two or more lines at distinct points.

When a transversal cuts two parallel lines, several pairs of angles are formed, such as corresponding, alternate interior, and co-interior angles.

6. Examples

Example 1: Find the value of x if two supplementary angles are (2x + 10)° and (3x – 10)°.

Solution: (2x + 10) + (3x – 10) = 180 ⇒ 5x = 180 ⇒ x = 36°.

Example 2: If two lines intersect and form one angle of 70°, what are the measures of the other three angles?

Solution: Vertically opposite angles are equal, so two angles are 70°. The other two are 110° each (since 180° – 70° = 110°).

7. Practice Questions

  1. Define complementary and supplementary angles with examples.
  2. If two angles form a linear pair and one is 45°, what is the other?
  3. Draw two parallel lines and a transversal. Mark all pairs of corresponding and alternate interior angles.
  4. Prove that vertically opposite angles are always equal.
  5. If two lines are cut by a transversal and a pair of alternate interior angles are equal, what can you say about the lines?

8. Summary

  • Lines and angles are fundamental concepts in geometry.
  • There are different types of angles and relationships between them.
  • Understanding properties of parallel lines and transversals is crucial for solving geometric problems.