Question

the angle of a quadrilateral are in the ratio of 3:4:5:6:. find the measure of each angle​

Answer 1

Given : The angles of Quadrilateral is in ratio 3:4:5:6 .

Exigency To Find : Measures of all angles of Quadrilateral.

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❍ Let's Consider measure of all four angles of quadrilateral be 3x , 4x, 5x & 6x respectively. .

$\frak{\underline { \dag As \: We \:Know \:that \:,}}$

• $\underline {\boxed {\sf{ \star The \:sum\:of \:all\:angles \:of\:Quadrilateral \:is \:360\degree}}}$

Or ,

• $\underline {\boxed {\sf{ \star \angle A + \angle B + \angle C + \angle D =\:360\degree}}}$

Where ,

• $\angle A  , \angle B , \angle C \:and\: \angle D$ are the all four angles of Quadrilateral.

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  3x + 4x + 5x + 6x =\:360\degree}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  7x + 5x + 6x =\:360\degree}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  12x + 6x =\:360\degree}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  18x =\:360\degree}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{ x =\:\dfrac{\cancel {360}}{\cancel {18}}}$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  x = 20\:\degree}}}}\:\bf{\bigstar}$

Therefore,

• First Angle of Quadrilateral is 3x = 3 × 20 = 60⁰

• Second angle of Quadrilateral is 4x = 4 × 20 = 80⁰

• Third angle of Quadrilateral is 5x = 5 × 20 = 100⁰

• Fourth Angle of Quadrilateral is 6x = 6 × 20 = 120⁰

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm { Hence,\: Measure \:of\:all\:four\:angles \:of\:Quadrilateral \:are\:60\degree, \:80\degree ,\:100\degree \:\&\:120\degree \: }}}$

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V E R I F I C A T I O N :

$\frak{\underline { \dag As \: We \:Know \:that \:,}}$

• $\underline {\boxed {\sf{ \star The \:sum\:of \:all\:angles \:of\:Quadrilateral \:is \:360\degree}}}$

Or ,

• $\underline {\boxed {\sf{ \star \angle A + \angle B + \angle C + \angle D =\:360\degree}}}$

Where ,

• $\angle A  , \angle B , \angle C \:and\: \angle D$ are the all four angles of Quadrilateral.

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  100\degree + 80\degree + 60\degree+ 120\degree =\:360\degree}$

⠀⠀⠀⠀⠀⠀$:\implies \tt{  1 80\degree + 180\degree =\:360\degree}$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{ \mathrm {  360\degree = 360\:\degree}}}}\:\bf{\bigstar}$

⠀⠀⠀⠀⠀$\therefore {\underline {\bf{ Hence, \:Verified \:}}}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

Answer 2

Given:-

• Ratio of the angle of a quadrilateral = 3:4:5:6

To Find:-

• Measure of each angle.

Solution:-

Let the ratio common be x.

Hence,

• 1st angle = 3x
• 2nd angle = 4x
• 3rd angle = 5x
• 4th angle = 6x

According to the angle sum property of a quadrilateral,

∠1 + ∠2 + ∠3 + ∠4 = 360°

= 3x + 4x + 5x + 6x = 360°

= 18x = 360°

=> x = 360/18

=> x = 20°

• ∴ The value of x is 20°

Putting the value of x in all the angles:-

• ∠1 = 3x = 3 × 20 = 60°
• ∠2 = 4x = 4 × 20 = 80°
• ∠3 = 5x = 5 × 20 = 100°
• ∠4 = 6x = 6 × 20 = 120°

∴ The four angles of the quadrilateral are 60°, 80°, 100°, 120° respectively.

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Check Point!!!

Let us check whether the sum of all the angles of this quadrilateral is 360° or not.

= ∠1 + ∠2 + ∠3 + ∠4 = 360°

= 60° + 80° + 100° + 120° = 360°

= 140° + 220° = 360°

= 360° = 360°

$\therefore$ Yes the sum of all the angles of this quadrilateral is 360°. Hence all the angles we got are correct.

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