In the figure given, AB is a diameter of the circle with centre O and CD || BA. If ∠CAB = x, find the value of (i) ∠COB (ii) ∠DOC

Question

In the figure given, AB is the diameter of the circle with center O and CD || BA. If ∠CAB = x, find the value of 

(i) ∠COB 

(ii) ∠DOC 

(iii) ∠DAC 

(iv) ∠ADC.

Answer 1

(i) ∠COB = 2 ∠CAB = 2x° (angle at the centre = 2 × angle at the remaining part of the circumference) 

(ii) ∠OCD = ∠COB = 2x° (alternate ∠s, DC || AB) 

OD = OC (radii of the same circle) 

⇒ ∠OCD = ∠ODC 

⇒ ∠ODC = 2x° 

∴ In ∆DOC, ∠DOC = 180° – (2x° + 2x°) 

= 180° – 4x°  (∠sum prop. of a ∆) 

(iii) ∠DAC = \(\frac{1}{2}\)∠DOC = \(\frac{1}{2}\) (180 − 4x)°  

(angle made by arc DC at the centre = Twice the angle at the remaining part of the circumference) 

= (90 – 2x)° 

(iv) In ∆ADC, ∠ACD = ∠CAB = x° (alt ∠s; DC || AB) 

∴ ∠ADC = 180° – (x° + 90° – 2x°) 

= (90 + x)°.         (∠sum prop. of a ∆)