Two circles intersect in $A$ and $B$ . Quadrilaterals $P C B A$ and $A B D E$ are inscribed in these circles such that $P A E$ and $C B D$ are line segments. If $∠ P = 95^{o}$ and $∠ C = 40^{o}$ . Also, $∠ A E D = z .$ Then the value of $z$ is:

65^{o}

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105^{o}

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95^{o}

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85^{o}

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Solution

The correct option is D $85^{o}$
$G i v e n - T w o c i r c l e s i n t e r s e c t a t A & B . T h e q u a d r i l a t e r a l P C B A i s i n s c r i b e d i n o n e c i r c l e a n d t h e q u a d r i l a t e r a l E D B A i s i n s c r i b e d i n a n o t h e r c i r c l e s u c h t h a t t h e p o i n t s P, A & B a r e c o l l i n e a r a n d C, B & D a r e c o l l i n e a r . ∠ A E D = z, ∠ A P C = 95^{o} & ∠ P C B = 40^{o} . T o f i n d o u t - z = ? s o l u t i o n - ∠ A P C + ∠ A B C = 180^{o} (s i n c e t h e s u m o f t h e o p p o s i t e a n g l e s o f a c y c l i c q u a d r i l a t e r a l i s 180^{o}) . ⟹ ∠ A B C = 180^{o} - ∠ A P C = 180^{o} - 95^{o} = 85^{o} . A g a i n ∠ A B C + ∠ A B D = 180^{o} (l i n e a r p a i r) . ∴ ∠ A B D = 180^{o} - ∠ A B C = 180^{o} - 85^{o} = 95^{o} . N o w ∠ A B D + ∠ A E D = 180^{o} (s i n c e t h e s u m o f t h e o p p o s i t e a n g l e s o f a c y c l i c q u a d r i l a t e r a l i s 180^{o}) . ∴ ∠ A E D = z = 180^{o} - ∠ A B D = 180^{o} - 95^{o} = 85^{o} . H e n c e, o p t i o n D i s c o r r e c t .$

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Similar questions

Q. Two circle intersect at

A

and

B

. Quadrilaterals

P C B A

and

A B D E

are inscribed in these circles such that

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are line segments. Also,

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^{o}

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Q. Two circles intersect in A and B. Quadrilaterals PCBA and ABDE are inscribed in these circles such that PAE and CBD are line segments. If

∠ P = 95^{\circ} a n d ∠ C = 40^{\circ}

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Q. Two circles intersect in A and B. Quadrilaterals PCBA and ABDE are inscribed in these circles such that PAE and CBD are line segments. If

∠ P = 95^{\circ} a n d ∠ C = 40^{\circ}

. Find the value of

∠ D E A

$T w o c i r c l e s i n t e r s e c t i n A a n d B . Q u a d r i l a t e r a l s P C B A a n d A B D E a r e i n s c r i b e d i n t h e s e c i r c l e s s u c h t h a t P A E a n d C B D a r e l i n e s e g m e n t s . ∠ P = 95^{0} a n d ∠ C = 40^{0} . F i n d t h e v a l u e o f Z .$