Question

1. $overline{ab} parallel overline{cd}$

$p = \frac{?}{}$
$q = \frac{?}{}$

Explanation:

Step1: Relate arc to central angle

The measure of arc $\widehat{AB}$ equals the central angle over it:
$\text{Measure of } \widehat{AB} = 120^\circ$

Step2: Find arc $\widehat{CD}$

Since $\overline{AB} \parallel \overline{CD}$, the arcs between the parallel chords are supplementary to the central angles. The central angle over $\widehat{CD}$ is $98^\circ$, so:
$\text{Measure of } \widehat{CD} = 98^\circ$

Step3: Calculate $p$ (arc $\widehat{AC}$)

The total circumference of a circle is $360^\circ$. The remaining arcs $\widehat{AC}$ and $\widehat{BD}$ are equal (because $\overline{AB} \parallel \overline{CD}$), so:
$p = \frac{360^\circ - 120^\circ - 98^\circ}{2} = \frac{142^\circ}{2} = 71^\circ$

Step4: Calculate $q$ (inscribed angle)

The inscribed angle $q$ subtends arc $\widehat{AC}$, so its measure is half the measure of the arc:
$q = \frac{1}{2}p = \frac{1}{2} \times 71^\circ = 35.5^\circ$

Answer:

$p = 71^\circ$, $q = 35.5^\circ$