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

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Parallelogram

Trapezium

Kite

None of these

answer is B.

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Detailed Solution

Given, the angles P, Q, R and S of a quadrilateral are in the ratio 1:3:7:9.
The angle sum property of a quadrilateral states that sum of all the interior angles of a quadrilateral is

360 ° .

Using this property, we will determine the angle measures of the quadrilateral PQRS and accordingly state the specific name for it.
So, we have, by the angle sum property of a quadrilateral,

∠ P + ∠ Q + ∠ R + ∠ S = 360 °

Now, based on the ratios given for the angles let us assume that

∠ P

is x,

∠ Q

is 3x,

∠ R

is 7x and

∠ S

is 9x. Thus, we can rewrite the above equation as follows,

x + 3 x + 7 x + 9 x = 360 ° 20 x = 360 ° x = 18 °

So, we have, the angle measures as follows,

∠ P = 18 ° ∠ Q = 3 × 18 ° = 54 ° ∠ R = 7 × 18 ° = 126 ° ∠ S = 9 × 18 ° = 162 °

Now, we can observe the following,

∠ P + ∠ S = 18 ° + 162 ° = 180 ° ∠ Q + ∠ R = 54 ° + 126 ° = 180 °

This means that the pairs of adjacent angles are supplementary.
Also, it can be observed that none of the angles are equal in measure.
So, of all the known quadrilaterals, it is concluded that PQRS is a trapezium.
This is because it is the property of a trapezium that its adjacent angle pairs are supplementary and no any pairs of angles are equal in measure.
Hence, the correct option is (2).

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