Question

Three angles of a quadrilateral are in the ratio 4:5:3 .The difference of the least and the greatest of these angles is 42degree .Find all the four angles of the quadrilateral?​

Answer 1

Given : Three angles of a quadrilateral are in the ratio 4:5:3 & the Difference between the greatest and least of these angle of Quadrilateral is 42⁰ .

Need To Find : All four angles of Quadrilateral ?

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❍ Let's say , that the three angles of Quadrilateral be 4x , 5x & 3x , respectively .

$\bigstar\:\underline{\pmb{\boldsymbol {\: According \:To\:the\:Question \:\::\:}}}$

⠀▪︎⠀The Difference b/w the greatest and least of these angle of Quadrilateral is 42⁰ .

$\\:\implies \sf \: \Big\{ \: Greatest\:_{\:(Angle)\:}\:\Big\}\:-\: \Big\{ \: Least\:_{\:(Angle)\:}\:\Big\}\:\:=\: 42^\circ \:$

$\\:\implies \sf \: \Big\{ \: 5x\:\Big\}\:-\: \Big\{ \: 3x\:\Big\}\:\:=\: 42^\circ \:$

$\\:\implies \sf \: \: 5x\:\:- \: 3x\:\:=\: 42^\circ \:$

$\\:\implies \sf \: \: 2x\:\:=\: 42^\circ \:$

$\\:\implies \sf \: \: x\:\:=\: \cancel {\dfrac{42}{2}} \:$

$\\:\implies \pmb{\sf \: \: x\:\:=\: 21^\circ\:} \:$

Therefore,

• ∠A of Quadrilateral , 4x = 4 × 21 = 84⁰ ,
• ∠B of Quadrilateral , 5x = 5 × 21 = 105⁰ &
• ∠C of Quadrilateral , 3x = 3 × 21 = 63⁰ .

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$\qquad \bigstar \:\;\underline{\pmb{ \cal{ ANGLE \:\:SUM \:\:PROPERTY \:\:\::\:}}}$

⠀⠀▪︎⠀⠀Angle Sum Property of Quadrilateral states , that the all four angles of Quadrilateral will sum up to 360⁰ .

$\\ \dashrightarrow \pmb{\sf \angle A \:+\:\angle B \:+\: \angle C \:+\:\angle D\:=\:360^\circ\:}\\\\\\ \dashrightarrow \sf 84 \:+\:105 \:+\: 63 \:+\:\angle D\:=\:360^\circ\:\\\\\\ \dashrightarrow \sf 84 \:+\:168\:+\:\angle D\:=\:360^\circ\:\\\\\\ \dashrightarrow \sf 252\:+\:\angle D\:=\:360^\circ\:\\\\\\ \dashrightarrow \sf \angle D\:=\:360 - 252 \:\\\\\\ \dashrightarrow \pmb {\underline {\boxed {\purple {\:\frak{ \:\angle D\:\:=\:108^\circ\:}}}}}\:\bigstar \:$

$\therefore\: \underline {\sf Hence, \:All\:four\:angles\:of\:Quadrilateral \:are\:\pmb{\sf 84^\circ\:,\:105^\circ\:,\:63^\circ\:\&\:108^\circ \:}\:,respectively \:.}$