1. **Problem Statement:** Given two overlapping circles with points A, B on the larger circle and points D, O on the smaller circle, and given angles 90° at D, 42°, 30°, and 120°, find the sizes of angles CBD, ODB, BAD, ABC, and ODA. 2. **Key Information and Rules:** - Angle at D is 90° (right angle). - Angles around a point sum to 360°. - Angles in a triangle sum to 180°. - Angles subtended by the same chord in a circle are equal. 3. **Find angle CBD:** - Angle CBD is adjacent to angle 42° and 90° at D. - Since angle at D is 90°, and angle at the top right is 42°, angle CBD = 90° - 42° = 48°. 4. **Find angle ODB:** - Angle ODB is inside the smaller circle at point D. - Given angle at D is 90°, and angle near D inside smaller circle is 30°. - Using triangle sum: angle ODB = 180° - 90° - 30° = 60°. 5. **Find angle BAD:** - Angle BAD is on the larger circle. - Given angle between A and B is 120°. - Since angle BAD and 120° are subtended by the same chord, angle BAD = 120°. 6. **Find angle ABC:** - Angle ABC is on the larger circle. - Using triangle sum in triangle ABC: angles BAD + ABC + BAC = 180°. - Given BAD = 120°, and assuming BAC = 30° (from given 30° near D), then ABC = 180° - 120° - 30° = 30°. 7. **Find angle ODA:** - Angle ODA is at point D inside smaller circle. - Using triangle sum in triangle ODA: angles ODA + OAD + AOD = 180°. - Given OAD = 42°, AOD = 90°, then ODA = 180° - 42° - 90° = 48°. **Final answers:** - a) Angle CBD = $48^\circ$ - b) Angle ODB = $60^\circ$ - c) Angle BAD = $120^\circ$ - d) Angle ABC = $30^\circ$ - e) Angle ODA = $48^\circ$