In a circle centered at point O, diameters AD and BC are drawn, and the angle ABO is 55 °. Find the ODC angle

Consider the triangle AOW. This triangle is isosceles, because OA and OB are the radii, therefore they are equal.
By the property of an isosceles triangle, ∠OAB = ∠OBA.
Consider the triangles AOW and COD. ∠DOC = ∠AOB, because they are vertical. СО = DO = OB = OA, because these are the radii of a circle.
Consequently, the triangles AOW and COD are equal (by the first sign). Therefore, ∠OBA = ∠OAB = ∠ODC = ∠OCD = 55 °
Answer: ∠OCD = 55 °

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