Question

The measures of the angles of a quadrilateral are in the ratio 2:3:7:6 .find their measures in degrees and radian. state ,with reasons ,whther quadrilateral is cycilc ans in step by step

Answer 1

Answer:

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Step-by-step explanation:

$to \: find = \\ measures \: of \: angles \: of \: quadrilateral \: \\ \\ so \: for \: a \: certain \: quadrilateral \\ its \: measures \: of \: angles \: are \\ in \: ratio \: \: 2 : 3 : 7 : 6 \: \\ thus \: then \\ let \: common \: multiple \: be \: x \\ \\ 2x + 3x + 7x + 6x = 360 \\ 18x = 360 \\ \\ x = 20 \\ so \: hence \: angles \: are \\ \\ 2x = (2 \times 20) = 40° \\ \\ 3x = (3 \times 20) = 60° \\ \\ 7x = (7 \times 20) = 140° \\ \\ 6x = (6 \times 20) = 120°$

$similarly \: \\ their \: measures \: in \: radian \: is : - \\ \\ 120° = \frac{2 \pi {}^{c} }{3} \\ \\ 140° = \frac{7\pi {}^{c} }{9} \\ \\ 40° = \frac{2\pi {}^{c} }{9} \\ \\ 60° = \frac{\pi {}^{c} }{3}$

$so \: now \ \\ adding \: opposite \: angles \: according \\ to \: given \: ratio\\ we \: get \\ \\ 120° + 60° = 180° \\ \\ 140° + 40° = 180° \\ \\ so \: as \: then \\ \: opposite \: angles \: are \: supplementary \\ by \: converse \: of \: \\ cyclic \: quadrilateral \: theorem \\ \\ we \: can \: conclude \: that \\ given \: quadrilateral \: is \: cyclic$