Question

find each angle or arc measure.

Explanation:

Step1: Solve top figure: Identify inscribed angle rule

The inscribed angle $\angle YZL$ intercepts arc $XY$. The measure of an inscribed angle is half the measure of its intercepted arc.
$\angle YZL = \frac{1}{2} \times m\overset{\frown}{XY}$

Step2: Calculate top unknown angle

Substitute $m\overset{\frown}{XY}=104^\circ$:
$\angle YZL = \frac{1}{2} \times 104^\circ = 52^\circ$

Step3: Solve bottom figure: Identify inscribed angle rule

The inscribed angle $\angle BCJ$ intercepts arc $BJ$. First find $m\overset{\frown}{BJ}$: the sum of arcs in a circle is $360^\circ$, so $m\overset{\frown}{BJ}=360^\circ - 64^\circ - 68^\circ - m\overset{\frown}{AJ}$. $\angle AJJ$ (correction: $\angle AJC$) intercepts arc $AC$, so $m\overset{\frown}{AC}=2\times71^\circ=142^\circ$, which is $m\overset{\frown}{AJ}+m\overset{\frown}{JC}$. Since $m\overset{\frown}{JC}=64^\circ$, $m\overset{\frown}{AJ}=142^\circ-64^\circ=78^\circ$. Now $m\overset{\frown}{BJ}=360^\circ-64^\circ-68^\circ-78^\circ=150^\circ$. The inscribed angle $\angle BCJ$ intercepts arc $BJ$, so:
$\angle BCJ = \frac{1}{2} \times m\overset{\frown}{BJ}$

Step4: Calculate bottom unknown angle

Substitute $m\overset{\frown}{BJ}=150^\circ$:
$\angle BCJ = \frac{1}{2} \times 150^\circ = 75^\circ$

Answer:

Top unknown angle: $52^\circ$
Bottom unknown angle: $75^\circ$