Angles

Subject: Compulsory Maths

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Overview

The amount of turn between two straight lines that have a common endpoint that is a vertex is called angle. This note includes the pairs of angles made by a transversal with lines and different Pairs of angles.
Angles

Angles

The amount of turn between two straight lines that have a common endpoint that is a vertex.

Different Pairs of angles

  1. Adjacent Angles

    \(\angle\)AOB and\(\angle\)BOC have common vertex O and a common arm OB. They are called adjacent angles.


  2. Linear Pair

    \(\angle\)AOB and\(\angle\)BOC are a pair of adjacent angles. Their sum is a straight angle (180°).
    i.e\(\angle\)AOB +\(\angle\)BOC = 180°
    \(\angle\)AOB and\(\angle\)BOC are called a linear pair.




  3. Vertically Opposite Angles

    \(\angle\)AOC and\(\angle\)BOD are formed by the intersected line segments and they lie to the opposite side of the common vertex. They are called vertically opposite angles.\(\angle\)AOD and\(\angle\)BOC are another pairs of vertically opposite angles. Vertically opposite angles are always equal.
    \(\therefore\)\(\angle\)AOC =\(\angle\)BOD and\(\angle\)AOD =\(\angle\)BOC.



  4. Complementary Angles

    The sum of\(\angle\)AOB and\(\angle\)BOC is a right angle (90°). i.e\(\angle\)AOB +\(\angle\)BOC = 90°
    \(\angle\)AOB and\(\angle\)BOC are called complementary angles.
    Here, complement of\(\angle\)AOB = 90° -\(\angle\)BOC
    Complement of\(\angle\)BOC = 90° - AOB




  5. Supplementary Angles

    The sum of \(\angle\)AOB and\(\angle\)BOC is two right angles (180°).
    i.e\(\angle\)AOB +\(\angle\)BOC = 180°
    \(\angle\)AOB and\(\angle\)BOC are called supplementary angles.
    Here, supplement of\(\angle\)AOB = 180° -\(\angle\)BOC
    Supplement of\(\angle\)BOC = 180° -\(\angle\)AOB

Pairs of angles made by a transversal with lines

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In the given figure, AB and CD and two parallel lines (AB//CD). PQ is the transversal that intersects AB at R and CD at S.






      1. Exterior and alternate exterior angles

        \(\angle\)a,\(\angle\)b,\(\angle\)c and\(\angle\)d are lying outside the parallel lines. They are called exterior angles.\(\angle\)a and\(\angle\)d are lying to the opposite side of the transversal. They are called alternate exterior angles.
        The alternate exterior angles made by a transversal with parallel lines are always equal.
        \(\therefore\)\(\angle\)a =\(\angle\)d and\(\angle\)b =\(\angle\)c
      2. Interior and Co-interior angles

        \(\angle\)a,\(\angle\)b,\(\angle\)c and\(\angle\)d are lying inside the parallel lines. They are called interior angles. \(\angle\)a and\(\angle\)c are the pair of interior angles lying to the same side of the transversal. They are called Co-interior angles. The sum of a pair of co-interior angles made by a transversal with parallel lines is always 180°.
        \(\therefore\)\(\angle\)a + \(\angle\)c = 180° and\(\angle\)b +\(\angle\)d = 180°
      3. Alternate angles

        \(\angle\)a and\(\angle\)d are a pair of interior angles lying to the opposite side of a transversal and they are not adjacent to each other. They are called alternate angles.\(\angle\)b and\(\angle\)c are another pairs of alternate angles.
        A pair of alternate angles made by a transversal with parallel lines is always equal.
        \(\therefore\)\(\angle\)a =\(\angle\)d and\(\angle\)b =\(\angle\)c



      4. Corresponding angles

        \(\angle\)a is an exterior and\(\angle\)d is an interior angle lying to the same side of the transversal and they are not adjacent to each other.; They are called corresponding angles. b and d are another pairs of corresponding angles.
        A pair of corresponding angles made by a transversal with parallel lines is always equal.
Things to remember
  • Definition of angles
  • Different pairs of angle
  • Pairs of an angle made by transversal lines.
  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.
Videos for Angles
Finding Right, Acute, and Obtuse Angles: Grade 4 Module 4 Lesson
Types of angles
Questions and Answers

Solution:

Here, x° + (x+10)° = 90° [The sum of a pair of complementary angles]

or, 2x° = 90° - 10°

or, x° = \(\frac{80°}{2}\) = 40°

\(\therefore\) x° = 40° and (x+10)° = 40° + 10° = 50°

 

Solution:

Let the required supplementary angles be 3x° and 2x°.

\(\therefore\) 3x° + 2x° = 180° [The sum of a pair of supplementary angles]

or, 5x° = 180°

x° = \(\frac{180°}{5}\) = 36°

\(\therefore\) 3x° = 3 × 36° = 108°

2x° = 2 × 36° = 72°

Solution:

Here, x° + 2x° + 3x° = 180° [Being the sum a straight angle]

or, 6x° = 180°

or, x° = \(\frac{180°}{6}\) = 30°

\(\therefore\) x° = 30°, 2x° = 2×30° = 60° and 3x° = 3×30° = 90°

Again, a° = x° = 30°, b = 2x° = 60° and c° = 3x° = 90° [Each pair is vertically opposite angles]

Solution:

w = 110° [Being vertically opposite angles]

x = w = 110° [Being alternate angles]

y = x = 110° [Being vertically opposite angles]

y + z = 180° [Being the sum of a pair of co-interior angles]

or, 110 + z = 180°

or, z = 180° - 110° = 70°

So, w = x= y = 110° and z = 70°

The different pairs of angles are:

  1. Adjacent angles
  2. Linear Pair
  3. Vertically opposite angles
  4. Complementary angles
  5. Supplementary angles
Quiz

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