Class 7 Maths Lines and Angles Notes

Class 7 Maths Lines and Angles Notes

Introduction:

The chapter “Lines and Angles” in Class 7 Maths explores fundamental concepts related to lines, angles, and the relationship between them. Understanding lines and angles is crucial for geometry, as it lays the foundation for studying more complex shapes and figures. In this chapter, students learn about various types of angles, properties of lines, and how they interact in different geometric contexts.

Key Concepts Covered:

1. What is a Line?

  • A line is a straight path that extends infinitely in both directions. It has no endpoints.
  • A line segment is a part of a line with two endpoints.
  • A ray is a part of a line that has one fixed point (the endpoint) and extends infinitely in one direction.

Example:

  • A line can be represented as AB↔\overleftrightarrow{AB}AB, a line segment as AB‾\overline{AB}AB, and a ray as AB→\overrightarrow{AB}AB.

2. What is an Angle?

  • An angle is formed when two lines or rays meet at a common point, called the vertex.
  • The amount of rotation between the two rays is measured in degrees.
  • Angles are denoted by the symbol ∠\angle∠.

Example:
If two lines ABABAB and BCBCBC meet at point BBB, the angle formed is written as ∠ABC\angle ABC∠ABC.

3. Types of Angles:

There are four main types of angles:

  • Acute Angle: An angle that is less than 90°.
    • Example: ∠ABC=30∘\angle ABC = 30^\circ∠ABC=30∘
  • Right Angle: An angle that is exactly 90°.
    • Example: ∠ABC=90∘\angle ABC = 90^\circ∠ABC=90∘
  • Obtuse Angle: An angle that is greater than 90° but less than 180°.
    • Example: ∠ABC=120∘\angle ABC = 120^\circ∠ABC=120∘
  • Straight Angle: An angle that is exactly 180°.
    • Example: ∠ABC=180∘\angle ABC = 180^\circ∠ABC=180∘

4. Complementary and Supplementary Angles:

  • Complementary Angles: Two angles that add up to 90°.
    • Example: If ∠ABC=30∘\angle ABC = 30^\circ∠ABC=30∘, then its complement is 60∘60^\circ60∘ because 30∘+60∘=90∘30^\circ + 60^\circ = 90^\circ30∘+60∘=90∘.
  • Supplementary Angles: Two angles that add up to 180°.
    • Example: If ∠ABC=120∘\angle ABC = 120^\circ∠ABC=120∘, then its supplement is 60∘60^\circ60∘ because 120∘+60∘=180∘120^\circ + 60^\circ = 180^\circ120∘+60∘=180∘.

5. Linear Pair of Angles:

  • When two angles are adjacent (share a common vertex and a common arm) and their sum is 180°, they form a linear pair.
  • Example: If ∠ABC=120∘\angle ABC = 120^\circ∠ABC=120∘ and ∠CBD=60∘\angle CBD = 60^\circ∠CBD=60∘, then the two angles together form a linear pair because 120∘+60∘=180∘120^\circ + 60^\circ = 180^\circ120∘+60∘=180∘.

6. Vertically Opposite Angles:

  • When two lines intersect, the opposite angles formed are called vertically opposite angles. These angles are always equal.

Example: If two lines intersect at point OOO, then ∠AOC=∠BOD\angle AOC = \angle BOD∠AOC=∠BOD and ∠AOD=∠BOC\angle AOD = \angle BOC∠AOD=∠BOC, meaning the opposite angles are equal.

7. Parallel Lines and Transversals:

  • Parallel Lines: Two lines that never meet and are always equidistant from each other. They are denoted by the symbol ∥\parallel∥.
  • A transversal is a line that intersects two or more lines at distinct points.

Example: If two parallel lines lll and mmm are cut by a transversal ttt, different pairs of angles are formed, such as alternate angles, corresponding angles, and co-interior angles.

8. Properties of Angles Formed by a Transversal:

  • Alternate Angles: Angles that lie on opposite sides of the transversal but inside the parallel lines. These angles are equal.
  • Corresponding Angles: Angles that are on the same side of the transversal and in corresponding positions with respect to the parallel lines. These angles are equal.
  • Co-interior Angles: Angles that lie on the same side of the transversal and inside the parallel lines. These angles are supplementary (add up to 180°).

Example: If two parallel lines are cut by a transversal:

  • Alternate angles are equal.
  • Corresponding angles are equal.
  • Co-interior angles add up to 180°.

Important Questions with Answers:

  1. What is an angle?
    • Answer: An angle is formed when two lines or rays meet at a common point, called the vertex.
  2. What is the difference between a line and a line segment?
    • Answer: A line extends infinitely in both directions, while a line segment has two endpoints.
  3. What are complementary angles?
    • Answer: Complementary angles are two angles that add up to 90°.
  4. What are vertically opposite angles?
    • Answer: Vertically opposite angles are the angles that are opposite each other when two lines intersect. They are always equal.
  5. What are parallel lines?
    • Answer: Parallel lines are lines that never meet and are equidistant from each other at all points.
  6. What is the sum of angles in a linear pair?
    • Answer: The sum of angles in a linear pair is 180°.
  7. What are alternate angles formed by a transversal?
    • Answer: Alternate angles are the angles formed on opposite sides of the transversal but inside the parallel lines. They are equal.
  8. How are co-interior angles related?
    • Answer: Co-interior angles lie on the same side of the transversal and inside the parallel lines. They are supplementary (add up to 180°).