Find angle x, y, r

In triangle OEC,

angle OEC + angle OCE + angle EOC = 180

90 + 30 + angle EOC = 180

angle EOC = 60 ...(i)

 

In triangle OBC,

OB = OC                   radii of the same circle

angle OBC = angle OCB = 30     opposite angles of equal sides

In triangle OEC,

angle OEC + angle OBC + angle BOC = 180

90 + 30 + angle BOC = 180

angle BOC = 60 ...(ii)

 

AE is straight line. 

Hence, 

angle BOC + angle BOA = 180

60 + angle BOA = 180               from (ii)

angle BOA = 120

 

In triangle OAB,

OA = OB                     radii of the same circle

angle OAB = angle OBA = x       opposite angles of equal sides

In triangle OAB,

angle OAB + angle OBA + angle AOB = 180

x + x + 120 = 180

2x = 60

x = 30

 

2angle DC=arc DC             inscribed angle theorem

 

angle DOC = arc DC          central angle

Hence, angle DOC = 2angle DBC = 2y

 

We know that

angle AOD + angle DOC + angle COE = 180                 AE is diameter and straight line

90 + 2y + 60 = 180

2y = 60

y = 30