Question

◻PQRS is a parallelogram. The ratio of ∠P and ∠Q of this parallelogram is 4:2 Find ∠S.

Answer 1

$\Large {\underline { \sf \orange{Clarification :}}}$

Here, we are given that PQRS is a parallelogram and the ratio of ∠P and ∠Q of this parallelogram is 4:2 . We have to find ∠S.

In order to tackle this question, we'll use properties of the parallelogram.

Properties we have to use here :

• Angle sum property of a parallelogram

$\longmapsto$ As a parallelogram is a quadrilateral, so sum of all measure of the angles is 360°.

• Opposite angles of a parallelogram are equal.

$\Large {\underline { \sf \orange{ Explication \: of \: steps :}}}$

$\underline{\pmb{ \sf {\maltese \; \; \; Given \: Information \: : \; \; \; }}}$

• PQRS is a parallelogram.

• The ratio of ∠P and ∠Q of this parallelogram is 4:2.

$\underline{\pmb{ \sf {\maltese \; \; \; To \: Calculate \: : \; \; \; }}}$

• Measure of ∠S.

$\underline{\pmb{ \sf {\maltese \; \; \; \: Solution : \; \; \; }}}$

As we know that,

★ Opposite angles of a parallelogram are equal.

So,

$\underline{ \sf {\maltese \; \; \; According \: to \:the \: question : \; \; \; }}$

• ∠S = ∠Q
• ∠P = ∠R

Now, as ∠P and ∠Q of this parallelogram are in ratio of 4:2, so let ∠P be 4x and ∠Q be 2x.

So,

• ∠S = 2x
• ∠P = 4x
• ∠Q = 2x
• ∠R = 4x

$\underline{\small \sf {\maltese \; \; \; Finding \: value \: of \: x : \; \; \; }}$

We know that,

★ Sum of the angles of parallelogram = 360°

$\longrightarrow \sf { (4x)^{\circ} + (2x)^{\circ} + (2x)^{\circ} + (4x)^{\circ} = {360}^{\circ} }$

$\longrightarrow \sf { (12x)^{\circ} = {360}^{\circ} }$

$\longrightarrow \sf { x^{\circ} = \dfrac{{360}^{\circ}}{12} }$

$\longrightarrow \sf { x^{\circ} = {30}^{\circ} }$

$\underline{\small \sf {\maltese \; \; \; Finding \: measure \:of \: \angle S : \; \; \; }}$

Now, as per the properties of a parallelogram,

★ Opposite angles of a parallelogram are equal.

$\longrightarrow$ ∠S = ∠Q

$\longrightarrow$ ∠S = (2x)°

$\longrightarrow$ ∠S = (2 × 30)°

$\longrightarrow$ ❝ $\boxed{ \sf \orange {\angle S = 60^{\circ} }}$ ❞

❝ Therefore, measure of ∠S is 360°. ❞