Question

draw the following diagram: quadrilateral (abcd) inscribed in circle (e). the (moverarc{ab} = 54^circ), (moverarc{bc} = 115^circ), and (moverarc{cd} = 86^circ). select the true statement. (1 point)

(circ) (moverarc{adc} = 84.5^circ)
(circ) (mangle b = 95.5^circ)
(circ) (moverarc{ad} = 115^circ)
(circ) (mangle d = 169^circ)

Explanation:

Step1: Find $m\widehat{AD}$

The total degree of a circle is $360^\circ$. Subtract the given arc measures:
$$m\widehat{AD}=360^\circ - 54^\circ - 115^\circ - 86^\circ=105^\circ$$

Step2: Check $m\widehat{ADC}$

Add $m\widehat{AD}$ and $m\widehat{CD}$:
$$m\widehat{ADC}=105^\circ + 86^\circ=191^\circ$$

Step3: Calculate $m\angle B$

For inscribed quadrilaterals, $\angle B$ is half the measure of its intercepted arc $\widehat{ADC}$:
$$m\angle B=\frac{1}{2}m\widehat{ADC}=\frac{1}{2}\times191^\circ=95.5^\circ$$

Step4: Calculate $m\angle D$

$\angle D$ is half the measure of its intercepted arc $\widehat{ABC}$:
$$m\widehat{ABC}=54^\circ + 115^\circ=169^\circ$$
$$m\angle D=\frac{1}{2}\times169^\circ=84.5^\circ$$

Answer:

$\boldsymbol{m\angle B = 95.5^\circ}$