Question

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

Explanation:

Step1: Find arc BC

First, calculate the measure of arc BC by subtracting the given arcs from the total circle (360°).
$$m\overset{\frown}{BC} = 360^\circ - 149^\circ - 47^\circ - 85^\circ = 79^\circ$$

Step2: Calculate m∠B

∠B is an inscribed angle intercepting arcs AD and DC. The measure of an inscribed angle is half the sum of the intercepted arcs.
$$m\angle B = \frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{DC}) = \frac{1}{2}(47^\circ + 85^\circ) = \frac{1}{2}(132^\circ) = 66^\circ$$

Step3: Calculate m∠D

∠D is an inscribed angle intercepting arcs AB and BC. Use the inscribed angle theorem again.
$$m\angle D = \frac{1}{2}(m\overset{\frown}{AB} + m\overset{\frown}{BC}) = \frac{1}{2}(149^\circ + 79^\circ) = \frac{1}{2}(228^\circ) = 114^\circ$$

Answer:

$m\angle B = 66^\circ$
$m\angle D = 114^\circ$