Question

The sides of quadrilateral are in the ratio 2:3:5 and the 4th angel is 90 degres . find the measures of the other three angels

Answer 1

Appropriate Question :

• The $\sf{\pink{angles}}$ of of a quadrilateral are in the ratio 2:3:5 and the measure of 4th angle is 90°. Find the measure of the other three angles.

Given :

• The sides of quadrilateral are in the ratio 2:3:5.
• The fourth angle is 90°

To Find :

• Measure of the other three angles .

Solution :

Let x be the common multiple of the ratio.

So, the angles of quadrilateral are :

• $\sf{\red{1st\:angle\:=\:2x}}$
• $\sf{\purple{2nd\:angle\:=\:3x}}$
• $\sf{\blue{3rd\:angle\:=\:5x}}$

The fourth angle is already given in the question which measures 90°

$\underline{\sf{\red{Property:}}}$

• $\sf{All\:the\:angles\:of\:a\:quadrilateral\:adds\:up\:to\:360\:^\circ}$

Using the property, we can write,

$\purple{\longrightarrow}$$\sf{2x+3x+5x+90=360}$

$\purple{\longrightarrow}$ $\sf{2x+3x+5x=360-90}$

$\purple{\longrightarrow}$$\sf{5x+5x=270}$

$\purple{\longrightarrow}$$\sf{10x=270}$

$\purple{\longrightarrow}$$\sf{x=\cancel\dfrac{270}{10}}$

$\purple{\longrightarrow}$$\sf{x=27}$

Now, substituting x = 27 in the angle value of the quadrilateral we can figure out the measure of 1st, 2nd and 3rd angle of the quadrilateral.

$\large{\boxed{\sf{\red{1st\:angle\:=\:2x\:=\:2\:\times\:27\:=\:\green{54\:^\circ}}}}}$

$\large{\boxed{\sf{\purple{2nd\:angle\:=\:3x\:=\:3\:\times\:27\:=\:\:\green{81\:^\circ}}}}}$

$\large{\boxed{\sf{\blue{3rd\:angle\:=\:5x\:=\:5\:\times\:27\:=\:\green{135\:^\circ}}}}}$

Answer 2

Correct Question :

The angles of quadrilateral are in the ratio 2:3:5 and the 4th angel is 90 degres . find the measures of the other three angels.

Solution :

Let the common value of ratio of angles be x, Then Required 3 Angles of Quadrilateral will be 2x,3x and 5x

• Given 4th angle = 90°

We know that :

$\bigstar \boxed{ \sf \: Sum \: of \: all \: angles \: of \: Quadrilateral = {360}^{ \circ} }$

From Angle Sum Property :

$\longrightarrow \sf 2x + 3x + 5x + {90}^{ \circ} = {360}^{ \circ}$

$\longrightarrow \sf 10x+ {90}^{ \circ} = {360}^{ \circ}$

$\longrightarrow \sf 10x = {360}^{ \circ} - {90}^{ \circ}$

$\longrightarrow \sf 10x = {270}^{ \circ}$

$\longrightarrow \sf x = \cancel\dfrac{270}{ 10}$

$\longrightarrow { \sf x = {27}^{ \circ} }$

Hence,Common Value of ratio of angles = 27°

Let's put the value of x = 27° in our assumption

$\dag \: \boxed{ \sf \: First \: Angle = 2x = 2 \times {27}^{ \circ} = \sf{54}^{ \circ}}$

$\dag \: \boxed{ \sf \: Second \: Angle = 3x = 3 \times {27}^{ \circ} = \sf{81}^{ \circ}}$

$\dag \: \boxed{ \sf \: Third \: Angle = 5x = 5 \times {27}^{ \circ} = \sf{135}^{ \circ}}$

$\dag \: \boxed{ \sf \: Fourth \: Angle = \sf {90}^{ \circ}}$

Verification :

$\bigstar \boxed{ \sf \: Sum \: of \: all \: angles \: of \: Quadrilateral = {360}^{ \circ} }$

$\longrightarrow \sf \: {54}^{ \circ} + {81}^{ \circ} + {135}^{ \circ} + {90}^{ \circ} = {360}^{ \circ}$

$\longrightarrow \sf \: {135}^{ \circ} + {225}^{ \circ} = {360}^{ \circ}$

$\longrightarrow \sf \: {360}^{ \circ} = {360}^{ \circ}$

Hence,Verified!