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Quadrilaterals

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Some special type of Quadrilaterals

Quadrilaterals are the polygons having four sides. Parallelogram, rectangle, square, rhombus, trapezium, and kite have some special properties. So, they are called special types of a quadrilateral. Some types of quadrilateral are as follows

  1. Parallelogram


    Its opposite sides are equal and parallel.
    \(\therefore\) AB= DC and AB//DC, AD = BC and AD//BC.
    Its opposite angles are equal.
    \(\therefore\) \(\angle\)A = \(\angle\)C and \(\angle\)B = \(\angle\)D
    Its diagonal bisect each other.
    \(\therefore\) Diagonals AC and BD bisect each other ar O.
    i.e AO = OC and BO = OD.
  2. Rectangle


    Its opposite sides are equal and parallel.
    \(\therefore\) AB = DC and AB//DC, AD = BC and AD//BC.
    Its all angles are equal andeach of them is 90°.
    \(\angle\)A = \(\angle\)B = \(\angle\)C = \(\angle\)D = 90°.
    Its diagonalsare equal and they bisect each other.
    \(\therefore\) Diagonals AC = BD, AO = OC and BO = OD.
  3. Square


    ts all sides are equal.
    \(\therefore\) AB = BC = CD = DA
    Its all angles are equal and they are 90°.
    \(\therefore\) \(\angle\)A = \(\angle\)B = \(\angle\)C = \(\angle\)D = 90.
    Its diagonal are equal and they bisect each other at right angle.
    \(\therefore\) AC = BD and they bisect each other at O at a right angle.
    i.e AO = OC, BO = OD and BD⊥AC at O.
  4. Rhombus
    Its all sides are equal and opposite sides are parallel.

    \(\therefore\) AB = BC = CD = DA and AB//DC, AD//BC
    Its opposite angles are equal.
    \(\therefore\) \(\angle\)A = \(\angle\)C and \(\angle\)B = \(\angle\)D
    Its diagonals are not equal but they bisect each other at right angle.
    \(\therefore\) AC and BD bisect each other at O at right angle.
    i.e AO = OC, BO = OD and BD⊥AC at O.
  5. Trapezium


    Its any one pair of opposite sides are parallel.
    \(\therefore\) AB//DC.
  6. Kite

    Its particular pairs of adjacent sides are equal.
    AB = AD and BC = DC
    The opposite angles formed by each pair of unequal adjacent sides are equal.
    \(\angle\)ABC = \(\angle\)ADC
    Diagonals intersect each other at O at a right angle.
    i.e BD⊥AC at O.



  • Opposite sides of a parallelogram are equal and parallel. 
  • The sides of rhombus are equal and opposite sides are parallel.
  • The particular pairs of adjacent sides are equal.
.

Very Short Questions

Solution:

Let the angles of the quadrilateral be x°, 2x°, 3x° and 4x° respectively.

Here, x° + 2x° + 3x° + 4x° = 360°

or, 10x° = 360°

or, x° = \(\frac{360°}{10}\) = 36°

\(\therefore\) x° = 36°, 2 × 36° =72°, 3x° = 3 × 36° = 108° and 4x° = 4 × 36° = 144°.

Solution:

Fig: Quadrilateral 

 y° + 105° = 180° [Being the sum of a straight angle]

or, y° = 180° - 105° = 75°

Now, x° + (2x° + 10°) + 2x° + y° = 360° [The sum of the angles of a quadrilateral]

or, 5x° + 10° + 75° = 360°

or, 5x° = 360° - 85°

or x° = \(\frac{275°}{5}\) = 55°

\(\therefore\) x° = 55°, 2x° + 10° = 2×55° + 10° = 120°, 2x° = 2×55° = 110° and y° = 75°.

 

Solution:

Fig: Quadrilateral 

Here, 2x° + 3x° + (2x+10)° + (x+30)° = 360° [Sum of the angles of quadrilateral]

or, (8x + 40)° = 360°

or, 8x° = 360° -40°

or, x = \(\frac{320°}{8}\) = 40°

\(\therefore\) 2x° = 2×40° = 80°, 3x° = 3×40° = 120°, (2x + 10)° = 2×40° + 10° = 90° and (x + 30)° = 40° + 30° = 70°

Solution:

Here,

w° = 25° [Being alternate angles]

x° = 20° [Being alternate angles]

Now, w° + 20° = 25° + 20° = 45°

x° + 25° = 20° + 25° = 45°

Again, y° + 45° = 180° [Being the sum of co-interior angles]

or, y° = 180° - 45°

or, y° = 135°

Also, z° = y° = 135° [Being the opposite angles of a parallelogram]

\(\therefore\) w° = 25°, x° = 20°, y° = z° = 135°.

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  • The sides of the square are ______ .

    equal
    joint
    diffierent
    adjacent
  • In parallelogram the opposite sides are ______ .

    joint
    equal
    different
    adjacent
  • Rhombus all sides are equal and opposite sides are ______ .

    disjoint
    adjacent
    perpendicular
    parallel
  • In the kite, opposite angles formed by each pair of unequal adjacent sides are _______ .

    equal
    parallel
    perpendicular
    adjacent
  • The diagonals of a rectangle are equal and they ______  each other.

    perpendicular
    circular
    parallel
    bisect
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