1. **Problem Statement:** (a) Given a circle with arcs $\overset{\frown}{AB} = 170^\circ$ and $\overset{\frown}{AC} = 66^\circ$, find the measure of angle $\angle ADC$ formed by a tangent and a secant. (b) Given a circle with $\angle AEB = 46^\circ$ and arc $\overset{\frown}{AB} = 46^\circ$, find the measure of arc $\overset{\frown}{CD}$. 2. **Formulas and Rules:** - For (a), the angle formed by a tangent and a secant intersecting outside the circle is half the difference of the intercepted arcs: $$\angle ADC = \frac{1}{2} \left| m\overset{\frown}{AB} - m\overset{\frown}{AC} \right|$$ - For (b), the angle formed by two chords intersecting inside the circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle: $$\angle AEB = \frac{1}{2} (m\overset{\frown}{AB} + m\overset{\frown}{CD})$$ 3. **Calculations:** (a) Calculate $\angle ADC$: $$\angle ADC = \frac{1}{2} |170 - 66| = \frac{1}{2} \times 104 = 52^\circ$$ (b) Calculate $\overset{\frown}{CD}$: Start with the formula: $$46 = \frac{1}{2} (46 + m\overset{\frown}{CD})$$ Multiply both sides by 2: $$92 = 46 + m\overset{\frown}{CD}$$ Subtract 46 from both sides: $$m\overset{\frown}{CD} = 92 - 46 = 46^\circ$$ 4. **Final Answers:** - (a) $m\angle ADC = 52^\circ$ - (b) $m\overset{\frown}{CD} = 46^\circ$