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Question

# Which of the following statements are true and which are false? (i) The diagonals of a parallelogram are equal. (ii) The diagonals of a rectangle are perpendicular to each other. (iii) The diagonals of a rhombus are equal. (iv) Every rhombus is a kite. (v) Every rectangle is a square. (vi) Every square is a a parallelogram. (vii) Every square is a rhombus. (viii) Every rectangle is a parallelogram. (ix) Every parallelogram is a rectangle. (x) Every rhombus is a parallelogram.

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Solution

## $\left(\mathrm{i}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{false}.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{diagonals}\mathrm{of}\mathrm{a}\mathrm{parallelogram}\mathrm{bisect}\mathrm{each}\mathrm{other},\mathrm{but}\mathrm{they}\mathrm{are}\mathrm{not}\mathrm{equal}\mathrm{in}\mathrm{length}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{false}.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{diagonals}\mathrm{of}\mathrm{a}\mathrm{rectangle}\mathrm{are}\mathrm{equal}\mathrm{and}\mathrm{bisect}\mathrm{each}\mathrm{other},\mathrm{but}\mathrm{they}\mathrm{are}\mathrm{not}\mathrm{perpendicular}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{iii}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{false}.\phantom{\rule{0ex}{0ex}}\mathrm{All}\mathrm{the}\mathrm{sides}\mathrm{of}\mathrm{a}\mathrm{rhombus}\mathrm{are}\mathrm{equal},\mathrm{but}\mathrm{the}\mathrm{diagonals}\mathrm{are}\mathrm{not}\mathrm{equal}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{iv}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{true}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{v}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{false}.\phantom{\rule{0ex}{0ex}}\mathrm{Every}\mathrm{square}\mathrm{is}\mathrm{a}\mathrm{rectangle},\mathrm{but}\mathrm{every}\mathrm{rectangle}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{square}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{vi}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{true}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{vii}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{true}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{viii}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{true}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{ix}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{false}.\phantom{\rule{0ex}{0ex}}\mathrm{A}\mathrm{rectangle}\mathrm{is}\mathrm{a}\mathrm{special}\mathrm{type}\mathrm{of}\mathrm{parallelogram},\mathrm{but}\mathrm{every}\mathrm{parallelogram}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{rectangle}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{x}\right)\mathrm{The}\mathrm{given}\mathrm{statement}\mathrm{is}\mathrm{true}.\phantom{\rule{0ex}{0ex}}$

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