Quadrilateral.

Any closed polygon with four sides, four angles and four vertices are known as Quadrilateral. It could be a regular or irregular polygon.

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  1. Trapezium:

If a quadrilateral has one pair of parallel sides then it is a Trapezium.

Here AD∥BC in quadrilateral ABCD, hence it is a trapezium.

Remark:  If the non   parallel sides of a trapezium are equal then it is called Isosceles Trapezium.

  1. Kite:

If the two pairs of adjacent sides are equal in a quadrilateral then it is called a Kite.

Here AB   =   BC and AD   =   CD

Properties of a kite:

The two diagonals are perpendicular to each other.

One of the diagonal bisects the other one.

∟ A   =   ∟ C but ∟ B ≠∟ D

  1. Parallelogram:

If the two pairs of opposite sides are parallel in a quadrilateral then it is called a Parallelogram.

Here, AB ∥ DC and BC ∥ AD, hence ABCD is a parallelogram.

Elements of a Parallelogram:  

Some terms related to a parallelogram ABCD

  1. Opposite Sides:  –   Pair of opposite sides are

AB and DC,       AD and BC

  1. Opposite Angles:  –   Pair of opposite angles are

∟  A and ∟ C       ∟ B and ∟ D

  1. Adjacent Sides : –  Pair of adjacent sides are

AB and BC        BC and DC        DC and AD        AB and AD

  1. Adjacent Angles  –   Pair of adjacent angles are

∠A and  ∠B          ∠B and  ∠C      ∠C and  ∠D          ∠A and  ∠D

Properties of a Parallelogram:

  1. The opposite sides of a parallelogram will always be equal.
    Here, AB   =   DC and AD   =   BC.

    1. The opposite angles of a parallelogram will always be of equal measure.

    As in the above figure,  ∠A   =  ∠ C    and    ∠D   =   ∠B.

    1. The two diagonals of a parallelogram bisect each other.

    Here in ABCD, AC and BD bisect each other at point E.

    So that AE  =  EC and DE  =   EB.

    1. The pair of adjacent angles in a parallelogram will always be a supplementary angle.

    Example:        Three angles of a quadrilateral are in the ratio 3: 4: 5. The difference of the least and the greatest of these angles is 50. Find all the four angles of the quadrilateral.

    Solution:  The ratio of the three angles of quadrilateral   =   3:   4:   5

    Let the angles be 3 x,   4 x  and  5 x.

    The greatest angle among these is 5 x and the least is  3 x.

    According to the question, 5 x   –   3 x   =   50   ⇒   2 x   =   50

    ⇒   x   =   50 ÷ 2   ⇒   x   =   25

    Hence, the three angles of quadrilateral are 3 × 25  =   75°,                        4  ×  25°   =  100°   and    5  ×  25°  =  125°.

    Fourth angle of quadrilateral   =   360°   –   ( 75°   +   100°   +   125° )  =   360°   –   300°   =   60°