If O is the center and AB parallel to CD, then x=

If O is the center and AB parallel to CD, then x=

(A) 40
(B) 50
(C) 60
(D) 70
(E) 80

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Attachment:

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Solution:

• As shown in the below diagram, x° is an interior angle of triangle CFD.

o x° + ∠FCD + ∠FDC = 180°
o Or, \(x° + y° + z° = 180° ……….Eq.(i)\)

• Now, Arc BD subtends ∠BOD at centre and ∠BAD at a circumference point.

o \(∠BOD = 2*∠BAD = 20° ⟹ ∠BAD = 10° …….Eq.(ii)\)
• Since line AB is parallel to CD,

o ∠ADC or y = ∠BAD (Alternate interior angles)
o Substituting the value ∠BAD from Eq.(ii into the above equation, we get

 \(y = 10° …………Eq.(iii)\)
• Now, triangle CDE is a tringle circumscribed in half circle and has one of its side as diameter of the circle.

o Therefore, triangle CDE is a right-angled triangle, right angled at C.
o Or, ∠ECD or \(z = 90° …….Eq.(iv)\)
• Substituting the value of y and z from Eq.(iii) and (iv) into Eq.(i), we get

o \(x° + 10° + 90° = 180°\)
o Thus, \(x = 80\)