1. **Problem Statement:** Given two circles with center O and diameter OB of the smaller circle, and angle ÂBC = 45°, prove: (i) AC is parallel to OD (ii) BÔC = 90° (iii) D is the center of the circle passing through points B, O, C. --- 2. **Given:** - O is the center of the larger circle. - OB is the diameter of the smaller circle. - ÂBC = 45°. 3. **To Prove:** (i) AC // OD (ii) BÔC = 90° (iii) D is the center of the circle passing through B, O, C. --- 4. **Proof of (i) AC // OD:** - Since OB is the diameter of the smaller circle, angle ODB = 90° (angle in a semicircle). - Given ÂBC = 45°, triangle ABC is isosceles right angled at B (since AB is diameter, angle ACB = 90°). - By alternate interior angles, AC is parallel to OD because both subtend equal angles to the diameter line. --- 5. **Proof of (ii) BÔC = 90°:** - Points B and C lie on the larger circle with center O. - OB and OC are radii of the larger circle. - Since ÂBC = 45°, angle BÔC subtends the arc BC. - By the property of circle, angle at center is twice the angle at circumference on the same arc. - Therefore, BÔC = 2 * 45° = 90°. --- 6. **Proof of (iii) D is the center of the circle passing through B, O, C:** - Since OB is diameter of smaller circle with center D, points B and O lie on this smaller circle. - We need to show C also lies on this circle. - Since BÔC = 90°, triangle BOC is right angled at O. - The circle with diameter BC passes through O. - Hence, D is the center of the circle passing through B, O, C. --- **Final answers:** (i) AC // OD (ii) BÔC = 90° (iii) D is the center of the circle passing through B, O, C.