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    Let ABCD be a quadrilateral which satisfies any of the following conditions.( i ) opposite sides are equal( ii ) each diagonal of ABCD divides it into
    Question

    Let ABCD be a quadrilateral which satisfies any of the following conditions.
    ( i ) opposite sides are equal
    ( ii ) each diagonal of ABCD divides it into congruent triangles
    ( iii ) each pair of opposite angles are equal
    ( iv ) diagonals of ABCD bisect each other
    Then ABCD has to be a _____________

    A.

    Kite

    B.

    Trapezium

    C.

    Parallelogram

    D.

    Rectangle

    Correct option is C

    Solution:
    (i) Opposite sides are equal:
    If AB = CD and BC = AD, then by joining diagonal AC, triangles ABC and CDA are congruent (SSS). Hence, corresponding angles are equal, which implies opposite sides are parallel. Therefore, ABCD is a parallelogram.

    (ii) Each diagonal divides it into congruent triangles:
    If a diagonal divides the quadrilateral into two congruent triangles, then corresponding sides are equal. This reduces to condition (i), so ABCD is a parallelogram.

    (iii) Each pair of opposite angles are equal:
    If ∠A = ∠C and ∠B = ∠D, then using the angle sum property (360°), we get consecutive angles supplementary. Thus, opposite sides are parallel, and ABCD is a parallelogram.

    (iv) Diagonals bisect each other:
    If diagonals intersect at a point and bisect each other, then triangles formed are congruent. This gives parallel opposite sides. Hence, ABCD is a parallelogram.

    Conclusion: Any one of the given conditions ensures that ABCD is a parallelogram.

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