The angles P, Q, R and S of a quadrilateral are in the ratio 1 : 3 : 7 : 9. Then

Question

The angles P, Q, R and S of a quadrilateral are in the ratio 1 : 3 : 7 : 9. Then, PQRS is a
a) Parallelogram
b) Trapezium with

P Q ∥ R S

c) Trapezium with

Q R ∥ P S

d) Kite

Answer:
:

Let the angles be x, 3x, 7x and 9x, then,

x + 3 x + 7 x + 9 x = 360^{\circ}

[

∵

sum of angles in any quadrilateral is

360^{\circ}

]

\Rightarrow 20 x = 360^{\circ} \Rightarrow x = \frac{360^{\circ}}{20} \Rightarrow x = 18^{\circ}

Then, the angles P, Q, R and S are

18^{\circ}, 54^{\circ}, 126^{\circ}

and

162^{\circ}

, respectively.
Since,

∠ P + ∠ S = 18^{\circ} + 162^{\circ} = 180^{\circ}

and

∠ Q + ∠ R = 54^{\circ} + 126^{\circ} = 180^{\circ}

S o, P Q ∥ S R

∴

The quadrilateral PQRS is a trapezium with

P Q ∥ S R

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