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the angle of quadrilaterals are in the ratio 3 ratio 5 ratio 9 ratio thirteen ratio. find all the the angles of quadrilaterals​

Answer»

Question-:

  • The angles of quadrilateral are in ratio 3 : 5 : 9 : 13 .Find all angles of quadrilateral.

AnswEr-:

  • \underline{\boxed{\mathrm {\dag{\red{ The\:measure \:\:of\;all\:angles\:of\:Quadrilateral \:are\:36^{0} ,60^{⁰} , 108^{⁰}\:and\:156^{⁰}}}}}}

Explanation-:

  • \sf{<klux>GIVEN</klux> \:That-:}

  • The angles of quadrilateral are in ratio 3 : 5 : 9 : 13 .

  • \sf{To \:Find-:}

  • The all angles of Quadrilateral.

\dag{\mathrm{Solution \:For \:Question \:-: }}

  • \sf{Let's \:Assume-:}

  • The all angles of quadrilateral be 3X , 5X , 9x and 13 x .

Then ,

  • \frak{All\:Angle \: -:} \begin{cases} \sf{Angle\:1\:or\:\angle A= \frak{3x^{0}}} & \\\\ \sf{ Angle\:2\:or\:\angle B\:=\:\frak{5x^{0}}}& \\\\ \sf{ Angle\:3:or\:\angle C\:=\:\frak{9x^{0}}}& \\\\ \sf{ Angle\:4\:or\:\angle D\:=\:\frak{13x^{0}}}\end{cases} \\\\

  • As we know that ,

  • \underline{\boxed{\mathrm {\dag{\red{ The\:toatl\:sum\:of\;angles\:of\:Quadrilateral \:is\:360^{0}}}}}}

  • Or ,

  • \underline{\boxed{\mathrm {\dag{\red{\angle A + \angle B + \angle C + \angle D \:=\:360^{0}}}}}}

Here ,

  • \frak{Here\: -:} \begin{cases} \sf{Angle\:1\:or\:\angle A= \frak{3x^{0}}} & \\\\ \sf{ Angle\:2\:or\:\angle B\:=\:\frak{5x^{0}}}& \\\\ \sf{ Angle\:3:or\:\angle C\:=\:\frak{9x^{0}}}& \\\\ \sf{ Angle\:4\:or\:\angle D\:=\:\frak{13x^{0}}}\end{cases} \\\\

  • Now by putting known or Given Values-:

  • \longrightarrow {\mathrm {3x + 5x + 9x + 13 x = 360^{⁰}}}

  • \longrightarrow {\mathrm {8x + 9x + 13 x = 360^{⁰}}}

  • \longrightarrow {\mathrm {17x + 13 x = 360^{⁰}}}

  • \longrightarrow {\mathrm {30 x = 360^{⁰}}}

  • \longrightarrow {\mathrm {x = \dfrac{\cancel {360}}{\cancel {30}}}}

  • \longrightarrow {\mathrm { x = 12 }}

Therefore-:

  • \longrightarrow{\mathrm{ x = 12 }}

Now ,

  • \frak{By\:Putting \:x =12-:} \begin{cases} \sf{Angle\:1\:or\:\angle A= \frak{3x=3\times 12=36^{⁰}}} & \\\\ \sf{ Angle\:2\:or\:\angle B\:=\:\frak{5x=5\times 12 = 60^{⁰}}}& \\\\ \sf{ Angle\:3:or\:\angle C\:=\:\frak{9x= 9 \times 12 = 108^{⁰}}}& \\\\ \sf{ Angle\:4\:or\:\angle D\:=\:\frak{13x= 13 \times 12 = 156^{⁰}}}\end{cases} \\\\

Hence ,

  • \underline{\boxed{\mathrm {\dag{\red{ The\:measure \:\:of\;all\:angles\:of\:Quadrilateral \:are\:36^{0} ,60^{⁰} , 108^{⁰}\:and\:156^{⁰}}}}}}

________________________________________

\huge {\mathrm { ♡Verification-: }}

  • \underline{\boxed{\mathrm {\dag{\red{ The\:toatl\:sum\:of\;angles\:of\:Quadrilateral \:is\:360^{0}}}}}}
  • Or ,
  • \underline{\boxed{\mathrm {\dag{\red{\angle A + \angle B + \angle C + \angle D \:=\:360^{0}}}}}}

\frak{Here-:} \begin{cases} \sf{Angle\:1\:or\:\angle A= \frak{36^{⁰}}} & \\\\ \sf{ Angle\:2\:or\:\angle B\:=\:\frak{ 60^{⁰}}}& \\\\ \sf{ Angle\:3:or\:\angle C\:=\:\frak{ 108^{⁰}}}& \\\\ \sf{ Angle\:4\:or\:\angle D\:=\:\frak{156^{⁰}}}\end{cases} \\\\

Now , By Putting known and Given Values-:

  • \longrightarrow {\mathrm {36 + 60 + 108 + 156 = 360^{⁰}}}

  • \longrightarrow {\mathrm {96+ 264 = 360^{⁰}}}

  • \longrightarrow {\mathrm {360^{⁰} = 360^{⁰}}}

Therefore ,

  • \longrightarrow {\mathrm {LHS = RHS}}

  • \longrightarrow {\mathrm {Hence\:Verified!}}

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