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Question

The angles of a quadrilateral are in the ratio $3 : 4 : 5 : 6$ . Then the quadrilateral is a trapezium.

A

True

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B

False

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Solution

The correct option is A True
$L e t ∠ A, ∠ B, ∠ C a n d ∠ D b e t h e a n g l e s o f q u a d i l a t e r a l ∠ A = 3 x, ∠ B = 4 x, ∠ C = 5 x a n d ∠ D = 6 x, w h e r e x i s a c o n s t a n t i n q u a d i l a t e r a l A B C D ∠ A + ∠ B + ∠ C + ∠ D = 360^{o} (A n g l e s u m p r o p e r t y) 3 x + 4 x + 5 x + 6 x = 360 18 x = 360^{o} x = 20^{o} ∠ A = 3 x = 3 \times 20 = 60^{o} ∠ B = 4 x = 4 \times 20 = 80^{o} ∠ C = 5 x = 5 \times 20 = 100^{o} ∠ D = 6 x = 6 \times 20 = 120^{o}$
$I n q u a d i l a t e r a l A B C D ∠ A + ∠ D = 60 + 120 = 180^{o} ∠ B + ∠ C = 80 + 100 = 180^{o} ∴ A B ∥ C D (S u m o f c o n s e c u t i v e i n t e r i o r a n g l e i s s u p p l e m e n t a r y) B u t ∠ A + ∠ B = 60 + 80 \neq 180^{o} ∴ A D i s n o t p a r a l l e l t o B C H e n c e q u a d i l a t e r a l i s a t r a p z i u m (a s i t h a s o n l y o n e p a i r o f p a r a l l e l s i d e)$ \\

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