Question

find ( mangle b ) and ( mangle c ).
( mangle b = square^circ )
( mangle c = square^circ )

Explanation:

Step1: Find arc AB measure

The total circle is $360^\circ$. Calculate arc $AB$:
$$m\overset{\frown}{AB} = 360^\circ - 129^\circ - 93^\circ - 83^\circ = 55^\circ$$

Step2: Calculate $m\angle B$

Use inscribed angle theorem: $\angle B$ intercepts arcs $AD$ and $DC$.
$$m\angle B = \frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{DC}) = \frac{1}{2}(129^\circ + 93^\circ) = \frac{1}{2}(222^\circ) = 111^\circ$$

Step3: Calculate $m\angle C$

Use inscribed angle theorem: $\angle C$ intercepts arcs $AB$ and $AD$.
$$m\angle C = \frac{1}{2}(m\overset{\frown}{AB} + m\overset{\frown}{AD}) = \frac{1}{2}(55^\circ + 129^\circ) = \frac{1}{2}(184^\circ) = 92^\circ$$

Answer:

$m\angle B = 111^\circ$
$m\angle C = 92^\circ$