Question

in a quadrilateral ABCD the angles at the verticals A B and C are in the 1:3:7. find the angle at the vertex C and the ratio of angle C to angle D given that the sum of angles at the verticals A B and C is 242°​

Answer 1

$\sf\large\underline\blue{Given:}$

$\rm{\implies Ratio\:_{(Vertex\:A : B : C)}=1: 3: 7}$

$\rm{\implies Sum\:of\: angles\:_{(A+B+C)}=242\degree}$

$\sf\large\underline\blue{To\: Find:}$

$\rm{\implies The\: angle\:at\:vertex\:(c)=?}$

$\rm{\implies Ratio\: of\: angles\:C\:to\:D=?}$

$\sf\large\underline\blue{Solution:}$

• To calculate the angle at vertex c and the ratio of angle c to d , at first we have to assume the angles of quarilateral ABCD be x then calculate the value of x after that we have find the angles of quarilateral separately after putting the value of X. As we know that sum of angles of quarilateral is equal to 360°, here at first we have to find out the angles of A , B , C with the help of given clue in the question:]

$\sf\large\underline\blue{Let,\:\angle\:A=x\:\:\angle\:B=3x\:\:\angle\:C=7x:}$

$\tt{\implies Sum\:of\:3\: angles=242\degree}$

$\tt{\implies x+3x+7x=242\degree}$

$\tt{\implies 11x=242\degree\implies x=22\degree}$

$\sf\small\underline\red{So,\:\angle\:A=22\:\:\angle\:B=3*22=66\:\:\angle\:C=7*22=154:}$

• Now calculate angle D here by applying sum angles property:]

$\rm{\implies Sum\:all\: angles\: quarilateral=360\degree}$

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

• Putting the value of angles here:]

$\rm{\implies 22+66+154+\angle\:D=360}$

$\rm{\implies 242+\angle\:D=360}$

$\rm{\implies \angle\:D=360-242}$

$\rm{\implies \angle\:D=118\degree}$

$\sf\large{Hence,}$

$\rm{\implies The\: angle\:at\:vertex\:(c)=154\degree}$

$\rm{\implies Ratio\: of\: angles\:C\:to\:D=154:118}$

$\rm{\implies Ratio\: of\: angles\:C\:to\:D=77:59}$

Answer 2

$\sf \red{Answer}$

$\sf \pink{Given - }$

$\bf \angle A :\angle B : \angle C = 1 : 3 : 7$

$\bf \angle A + \angle B + \angle C = 242 \degree$

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$\sf \pink{To \: find - }$

$\implies$Angle at vertex C $\longrightarrow$$\bf \angle C$

$\implies$Ratio of $\bf \angle C$ to $\bf \angle D$$\longrightarrow$ $\bf \angle C : \angle D$

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$\sf \pink{Solution -}$

As the ratio of $\bf \angle A :\angle B : \angle C$ equals to $\bf 1 : 3 : 7$

Let the $\bf \angle A = x$

Let the $\bf \angle B = 3x$

Let the $\bf \angle C = 7x$

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Sum of $\bf \angle A + \angle B + \angle C = 242 \degree$,

$\implies$$\bf x + 3x + 7x = 242 \degree$

$\implies$$\bf 11x = 242 \degree$

$\implies$$\bf x = 22 \degree$

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$\bf \angle A = x = 22 \degree$

$\bf \angle B = 3x = 3 \times 22 = 66 \degree$

$\bf \angle C = 7x = 7 \times 22 = 154 \degree$

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As the sum of all angles of quadrilateral is 360° -

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

$\implies$$\bf 22 \degree + 66 \degree + 154 \degree + \angle D = 360 \degree$

$\implies$$\bf 242 \degree + \angle D = 360 \degree$

$\implies$$\bf \angle D = 118 \degree$

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Angle at vertex C = $\bf 154 \degree$

Ratio of $\bf \angle C$ to $\bf \angle D$

$\implies$$\bf = 154 : 118$

$\implies$$\bf = 77 : 59$

$\bf \angle C : \angle D = 77 \degree : 59 \degree$

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