Question

find ( mangle b ), ( mangle d ), and ( moverarc{cd} ).

Explanation:

Step1: Calculate $m\angle B$

An inscribed angle is half its intercepted arc.
$m\angle B = \frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{CD})$
First find $m\overset{\frown}{CD}$: total circle is $360^\circ$, so $m\overset{\frown}{CD}=360^\circ - 69^\circ - 49^\circ - 149^\circ=93^\circ$
Then $m\angle B = \frac{1}{2}(69^\circ + 93^\circ)=\frac{1}{2}(162^\circ)=81^\circ$

Step2: Calculate $m\angle D$

Inscribed angle intercepts $\overset{\frown}{AB} + \overset{\frown}{BC}$
$m\angle D = \frac{1}{2}(m\overset{\frown}{AB} + m\overset{\frown}{BC})$
$m\angle D = \frac{1}{2}(49^\circ + 149^\circ)=\frac{1}{2}(198^\circ)=99^\circ$

Step3: Confirm $m\overset{\frown}{CD}$

Total circle minus known arcs:
$m\overset{\frown}{CD}=360^\circ - 69^\circ - 49^\circ - 149^\circ=93^\circ$

Answer:

$m\angle B = 81^\circ$
$m\angle D = 99^\circ$
$m\overset{\frown}{CD} = 93^\circ$