Question

One angle of a quadrilateral is
$\frac{2 \: \pi}{9}$
radian and the measures of the other three angles are in the ratio 3:5:8, find
their measures in degree.
​

Answer 1

Question:-

One angle of a quadrilateral is

$\sf \frac{2 \: \pi}{9}$ radian and the measures of the other three angles are in the ratio 3:5:8, find their measures in degree.

Given:-

One angle of a quadrilateral is

$\sf \frac{2 \: \pi}{9}$ radian and the measures of the other three angles are in the ratio 3:5:8.

To Find:-

• measures in degree.

Solution:-

The sum of angles of a quadrilateral is 360°.

$\sf one \: of \: the \: angles \: is \: given \: to \: be( \frac{2\pi}{9} ) {}^{c}</p><p></p><p> <strong>=</strong><strong> </strong><strong>(</strong><strong>2</strong><strong>π</strong><strong>/</strong><strong>9</strong><strong> </strong><strong>×</strong><strong> </strong><strong>180</strong><strong>/</strong><strong>π</strong><strong>)</strong><strong>°</strong><strong> </strong><strong>=</strong><strong> </strong><strong>40</strong><strong>°</strong><strong>.</strong></p><p></p><p>∴ Sum of the remaining three angles is </p><p> 360° – 40° = 320°. </p><p></p><p>Since, these three angles are in the ratio of 3:5:8.</p><p></p><p>∴ Degree measures of these angles are 3k,5k,8k where k is constant. </p><p></p><p>∴ 3k + 5k + 8k = 320°.</p><p></p><p>∴ 16k = 320°</p><p></p><p>[tex] \sf∴k = \frac{{ \cancel{32 0}}}{{ \cancel{16} }}$

∴ k = 20°.

∴ The measures of three angles are

• (3k)° = (3 × 20)° = 60°.

• (5k)° = (5 × 20)° = 100°. and,

• (8k)° = (8 × 20)° = 160°.

Answer:-

• 1st angle = 60°.

• 2nd angle = 100°. and,

• 3rd angle = 160°.

[Refer to the above attachment for your answer]

Hope you have satisfied. ⚘

Answer 2

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:One \: angle \: of \: quadrilateral = \dfrac{2\pi}{9}$

and three angles are in the ratio 3 : 5 : 8

Let we assume that

The quadrilateral be ABCD such that

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{\angle A = \dfrac{2\pi}{9} = \dfrac{2 \times 180 \degree \: }{9} = 40\degree} \\ \\ &\sf{\angle B = 3x\degree}\\ \\ &\sf{\angle C = 5x\degree} \\ \\ &\sf{\angle D = 8x\degree} \end{cases}\end{gathered}\end{gathered}$

We know,

Sum of all interior angles of a quadrilateral is 360°.

$\rm \implies\:\angle A + \angle B + \angle C + \angle D = 360\degree$

$\rm \implies\:40\degree + 3x\degree + 5x\degree + 8x\degree = 360\degree$

$\rm \implies\:40\degree + 16x\degree = 360\degree$

$\rm \implies\: 16x\degree = 320\degree$

$\bf\implies \:x = 20\degree$

Hence,

The angles of a quadrilateral ABCD are

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{\angle A = 40\degree} \\ \\ &\sf{\angle B = 60\degree}\\ \\ &\sf{\angle C = 100\degree} \\ \\ &\sf{\angle D = 160\degree} \end{cases}\end{gathered}\end{gathered}$

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More to Know

$\boxed{\tt{ {\pi}^{c} \: = \: 180\degree \: }}$

$\boxed{\tt{ \: 1 \: right \: angle \: = \: 90\degree \: = \: {\bigg[\dfrac{\pi}{2} \bigg]}^{c} \: }}$

$\boxed{\tt{ \: {1}^{c} = \bigg[\dfrac{180}{\pi} \bigg]^{\degree} \: }}$

$\boxed{\tt{ \: 1\degree = \: {\bigg[\dfrac{\pi}{180} \bigg]}^{c} \: }}$

$\boxed{\tt{ 1\degree = 60' \: }}$

$\boxed{\tt{ 1' = 60'' \: }}$