Lines and Angles Class 7 Notes: Chapter 5

Introduction to Geometry

Line, line segment and ray

  • If we take a point and draw a straight path that extends endlessly on both the sides, then the straight path is called as a line.
  • A ray is a part of a line with one endpoint.
  • A line segment is a part of a line with two endpoints.

Line, line segment and ray

Angles

  • An angle is formed when two rays originate from the same end point.
  • The rays making an angle are called the arms of the angle.
  • The end point is called the vertex of the angle.
  • Angles

Complementary Angles

  • Two angles whose sum is 90∘ are called complementary angles.
    Example: 50∘+40∘=90∘
    ∴ 50∘ and 40∘ angles are complementary angles.
    Complementary Angles

Parallel Lines and a Transversal

Transversal intersecting two lines

  • Transversal is a line that intersects two or more lines at different points.

Transversal intersecting two lines

  • Corresponding Angles:
    (i) ∠1 and ∠5 (ii) ∠2 and ∠6
    (iii) ∠3 and ∠7 (iv) ∠4 and ∠8
  • Alternate Interior Angles:
    (i) ∠3 and ∠6 (ii) ∠4 and ∠5
  • Alternate Exterior Angles:
    (i) ∠1 and ∠8 (ii) ∠2 and ∠7
  • Interior angles on the same side of the transversal:
    (i) ∠3 and ∠5 (ii) ∠4 and ∠6

Transversal of Parallel Lines

Transversal of Parallel Lines

  • If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
    (i) ∠1=∠5 (ii) ∠2=∠6
    (iii) ∠3=∠7 (iv) ∠4=∠8
  • If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
    (i) ∠3=∠6 (ii) ∠4=∠5
  • If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
    (i) ∠3+∠5=180∘ (ii) ∠4+∠6=180∘

Checking if two or more lines are parallel

  • There are three conditions to check whether the two lines are parallel. They are:
    (i) If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.
    (ii) If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
    (iii) If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Intersecting Lines and Pairs of Angles

Supplementary angles

  • Two angles whose sum is 180∘ are called supplementary angles.
    Example: 110∘+70∘=90∘
    ∴ 110∘ and 70∘ angles are supplementary angles.
    Supplementary angles

Adjacent Angles

  • Two angles are adjacent, if they have
    (i) A common vertex
    (ii) A common arm
    (iii) Their non-common arms on different sides of the common arm.
    Adjacent Angles
    Here ∠ABD and ∠DBC are adjacent angles.

Linear Pair

  • Linear pair of angles are adjacent angles whose sum is equal to 180∘.
    Linear Pair
    Here, 1 and 2 are linear pair of angles.

Vertically Opposite Angles

  • Vertically opposite angles are formed when two straight lines intersect each other at a common point.
  • Vertically opposite angles are equal.
    Vertically Opposite Angles
    Here, the following pairs of angles are vertically opposite angles.
    (i) a and c
    (ii) b and d

Intersecting and Non-Intersecting lines

  • Intersecting lines are lines which intersect at a common point called the point of intersection.
    Intersecting lines
  • Parallel lines are lines which do not intersect at any point. Parallel lines are also known as non- intersecting lines.
     Non-Intersecting lines

Basic Properties of a Triangle

Sum of Interior Angles in a Triangle

  • Angle sum property of a triangle: Sum of all interior angles of a triangle is 180∘.
    Sum of Interior Angles in a Triangle
    In △ABC, ∠1+∠2+∠3=180∘

The exterior angle of a triangle = Sum of opposite internal angles

  • 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.
    The exterior angle of a triangle = Sum of opposite internal angles
    In △ABC, ∠CAB+∠ABC=∠ACD.

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