This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Find the angle between the tangent and chord CD. The exterior angle formed by the tangent and chord CD is 146^. The angle inside the circle, formed by the tangent and chord CD, is supplementary to this exterior angle. Angle (tangent at D, chord CD) = 180^ - 146^ = 34^ Step 2: Apply the Alternate Segment Theorem to find angle b. The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. The angle between the tangent at D and chord CD is 34^. The angle in the alternate segment subtended by chord CD is CAD. Therefore, CAD = 34^. Since angle b is CAD, we have: b = 34^ Step 3: Use the property of equal chords to find angle c. The chords AD and CD are marked with single dashes, indicating they are equal in length. This means that ADC is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So, ACD = CAD. Since CAD = b = 34^, then ACD = 34^. Since angle c is ACD, we have: c = 34^ Step 4: Relate the central angle a to the inscribed angle c. Angle a is the central angle AOD. Angle c is the inscribed angle ACD. Both angles subtend the same arc AD. The central angle subtended by an arc is twice the inscribed angle subtended by the same arc. So, a = 2 × ACD. a = 2 × c a = 2 × 34^ a = 68^ The angles are: a = 68^ b = 34^ c = 34^