Question

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

Answer 1

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-:

• $\sf{Given \:That-:}$

• 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 ,

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

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

___________________________♡___________________________

Answer 2

Given:

• The four angles of a quadrilateral are in the ratio- 3 : 5 : 9 : 13.

To do:

• Specify the measure of angles of the quadrilateral.

Concept:

‎ ‎ ‎ ‎ ‎ ‎Here, we will be using the angle sum property of a quadrilateral to find the answer. This property can be defined as, “The sum of all the angles i.e, the sum of four angles of a quadrilateral equals 360°”.

Solution:

Let the four angles be-

• $\sf{\angle A=3z}$
• $\sf{\angle B=5z}$
• $\sf{\angle C=9z}$
• $\sf{\angle D=13z}$

As we know-

${\boxed{\bf{\pink{\angle A+\angle B+\angle C+\angle D=360\degree}}}}$

$\:$

On substituting the values-

$\:\:\:\:\:\:\:\longmapsto{\bf{3z+5z+9z+13z=360}}$

$\\\:\:\:\:\:\:\:\longmapsto{\bf{8z+22z=360}}$

$\\\:\:\:\:\:\:\:\longmapsto{\bf{30z=360}}$

$\\\:\:\:\:\:\:\:\longmapsto{\bf{z=\dfrac{36\cancel{0}}{3\cancel{0}}}}$

$\\\:\:\:\:\:\:\:\longmapsto{\bf{z=\dfrac{\cancel{36}}{\cancel{3}}}}$

$\\\:\:\:\:\:\:\:\longmapsto\underline{\bf{\orange{z=12}}}$

$\:$

Since the value of z is 12, the value of-

• $\bf { \red{\angle A }\rightarrow3z=3(12)= \red{36 \degree}}$
• $\bf { \purple{\angle B }\rightarrow5z=5(12)= \purple{60 \degree}}$
• $\bf { \red{\angle C}\rightarrow9z=9(12)= \red{108 \degree}}$
• $\bf { \purple{\angle D }\rightarrow13z=13(12)= \purple{156 \degree}}$

Verification:

$\rightarrowtail{\sf{\angle A+\angle B+\angle C+\angle D=360}}$

Substituting the obtained measures-

$\rightarrowtail{\sf{36+60+108+156=360}}$

$\rightarrowtail{\sf{96+108+156=360}}$

$\rightarrowtail{\sf{204+156=360}}$

$\rightarrowtail{\sf{360=360}}$

$\rightarrowtail{\sf{L.H.S=R.H.S}}$

$\\\therefore\underline{\sf{The\:measures\:of\:angle\:A, \:B,\:C,\:D\:are\:36\degree,\:60\degree,\:108\degree\:and\:156\degree,\: respectively.}}$

____________________________________

Quadrilaterals:-

‎ ‎ ‎ ‎ ‎ ‎They are simple plane figures bounded by four line segments. Squares, rectangles, trapeziums, rhombi, parallelograms are some quadrilaterals. Now, let's learn about it!

Let's consider EFGH as a quadrilateral.

• It has four edges - (EF, FG, GH, EH)
• It has four vertices - (E, F, G, H)
• ∠E + ∠F + ∠G + ∠H = 360°

The above listed properties are the three common properties of any quadrilateral.

____________________________________