the angle of a quadrilateral are in the ratio of 3:4:5:6:. find the measure of each angle About the author Natalia
Given : The angles of Quadrilateral is in ratio 3:4:5:6 . Exigency To Find : Measures of all angles of Quadrilateral. ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀ ❍ Let’s Consider measure of all four angles of quadrilateral be 3x , 4x, 5x & 6x respectively. . [tex]\frak{\underline { \dag As \: We \:Know \:that \:,}}\\[/tex] [tex]\underline {\boxed {\sf{ \star The \:sum\:of \:all\:angles \:of\:Quadrilateral \:is \:360\degree}}}\\\\\\[/tex] Or , [tex]\underline {\boxed {\sf{ \star \angle A + \angle B + \angle C + \angle D =\:360\degree}}}\\\\\\[/tex] Where , [tex]\angle A , \angle B , \angle C \:and\: \angle D [/tex] are the all four angles of Quadrilateral. ⠀⠀⠀⠀⠀⠀[tex]\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 3x + 4x + 5x + 6x =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 7x + 5x + 6x =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 12x + 6x =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 18x =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ x =\:\dfrac{\cancel {360}}{\cancel {18}}}\\\\\\[/tex] ⠀⠀⠀⠀⠀[tex]\underline {\boxed{\pink{ \mathrm { x = 20\:\degree}}}}\:\bf{\bigstar}\\\\\\[/tex] 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, ⠀⠀⠀⠀⠀[tex]\therefore {\underline{ \mathrm { Hence,\: Measure \:of\:all\:four\:angles \:of\:Quadrilateral \:are\:60\degree, \:80\degree ,\:100\degree \:\&\:120\degree \: }}}\\\\\\[/tex] ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀ V E R I F I C A T I O N : [tex]\frak{\underline { \dag As \: We \:Know \:that \:,}}\\[/tex] [tex]\underline {\boxed {\sf{ \star The \:sum\:of \:all\:angles \:of\:Quadrilateral \:is \:360\degree}}}\\\\\\[/tex] Or , [tex]\underline {\boxed {\sf{ \star \angle A + \angle B + \angle C + \angle D =\:360\degree}}}\\\\\\[/tex] Where , [tex]\angle A , \angle B , \angle C \:and\: \angle D [/tex] are the all four angles of Quadrilateral. ⠀⠀⠀⠀⠀⠀[tex]\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 100\degree + 80\degree + 60\degree+ 120\degree =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 1 80\degree + 180\degree =\:360\degree}\\\\\\[/tex] ⠀⠀⠀⠀⠀[tex]\underline {\boxed{\pink{ \mathrm { 360\degree = 360\:\degree}}}}\:\bf{\bigstar}\\\\\\[/tex] ⠀⠀⠀⠀⠀[tex]\therefore {\underline {\bf{ Hence, \:Verified \:}}}\\\\\\[/tex] ⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀ Reply
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. ________________________________ 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° [tex]\therefore[/tex] Yes the sum of all the angles of this quadrilateral is 360°. Hence all the angles we got are correct. ________________________________ Reply
Given : The angles of Quadrilateral is in ratio 3:4:5:6 .
Exigency To Find : Measures of all angles of Quadrilateral.
⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀
❍ Let’s Consider measure of all four angles of quadrilateral be 3x , 4x, 5x & 6x respectively. .
[tex]\frak{\underline { \dag As \: We \:Know \:that \:,}}\\[/tex]
Or ,
Where ,
⠀⠀⠀⠀⠀⠀[tex]\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 3x + 4x + 5x + 6x =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 7x + 5x + 6x =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 12x + 6x =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 18x =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ x =\:\dfrac{\cancel {360}}{\cancel {18}}}\\\\\\[/tex]
⠀⠀⠀⠀⠀[tex]\underline {\boxed{\pink{ \mathrm { x = 20\:\degree}}}}\:\bf{\bigstar}\\\\\\[/tex]
Therefore,
Therefore,
⠀⠀⠀⠀⠀[tex]\therefore {\underline{ \mathrm { Hence,\: Measure \:of\:all\:four\:angles \:of\:Quadrilateral \:are\:60\degree, \:80\degree ,\:100\degree \:\&\:120\degree \: }}}\\\\\\[/tex]
⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀
V E R I F I C A T I O N :
[tex]\frak{\underline { \dag As \: We \:Know \:that \:,}}\\[/tex]
Or ,
Where ,
⠀⠀⠀⠀⠀⠀[tex]\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 100\degree + 80\degree + 60\degree+ 120\degree =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀⠀[tex]:\implies \tt{ 1 80\degree + 180\degree =\:360\degree}\\\\\\[/tex]
⠀⠀⠀⠀⠀[tex]\underline {\boxed{\pink{ \mathrm { 360\degree = 360\:\degree}}}}\:\bf{\bigstar}\\\\\\[/tex]
⠀⠀⠀⠀⠀[tex]\therefore {\underline {\bf{ Hence, \:Verified \:}}}\\\\\\[/tex]
⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀
Given:–
To Find:–
Solution:-
Let the ratio common be x.
Hence,
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°
Putting the value of x in all the angles:-
∴ 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°
[tex]\therefore[/tex] Yes the sum of all the angles of this quadrilateral is 360°. Hence all the angles we got are correct.
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