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Angles

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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

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.



  • Definition of angles
  • Different pairs of angle
  • Pairs of an angle made by transversal lines.
.

Very Short Questions

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°

0%
  • If x° and 112 are adjacent angles in linear pair, what is the value of x° ?

    66°
    68°
    69°
    67°
  • If 2a° and 3a° formed a linear pair, what will be the value of them?

    74° and 109°
    73° and 107°
    72° and 108°
    75° and 106°
  • Find the size of an angle which is four times its complement.

    80° and 90°
    70° and 80°
    80° and 100°
    70° and 90°
  • Find the size of an angle which is five times its supplement.

    30° and 180°
    20° and 100°
    30° and 150°
    20° and 120°
  • The amount of turn between two straight lines that have a common endpoint that is a ______ .

    base
    vertex
    edges
    angles
  • The sum of two angles in supplementary angles are ______ .

    240°
    360°
    180° 
    90°
  • If the one angle of supplementary angle is 85°, then what will be other angles?

    100°
    105° 
    90°
    95°
  • The sum of angles in complementary angles are ______ .

    90°
    360°
    180°
    240°
  • If the one angle of a complementary angle is 68°, then what will be other angle?

    26°
    28°
    22°
    24°
  • The sum of straight line is _______ .

    180°
    240°
    360°
    90°
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