We are given a circle with diameter $AD$. The measure of arc $AC$ is $132^\circ$. We need to find the measure of angle $OCD$ and the measure of arc $CD$.

1. Problem Description

We are given a circle with diameter $AD$. The measure of arc $AC$ is $132^\circ$. We need to find the measure of angle $OCD$ and the measure of arc $CD$.

2. Solution Steps

a) Finding $m(\angle OCD)$:

Since $AD$ is a diameter, $m(\text{arc } ACD) = 180^\circ$.

We are given $m(\text{arc } AC) = 132^\circ$.

Then, $m(\text{arc } CD) = m(\text{arc } AD) - m(\text{arc } AC) = 180^\circ - 132^\circ = 48^\circ$.

Since $OC$ and $OD$ are radii of the circle, $OC = OD$. Therefore, triangle $OCD$ is an isosceles triangle with $OC = OD$.

Thus, $\angle OCD = \angle ODC$.

The measure of the central angle $\angle COD$ is equal to the measure of the arc $CD$, so $m(\angle COD) = 48^\circ$.

The sum of the angles in triangle $OCD$ is $180^\circ$.

So, $m(\angle OCD) + m(\angle ODC) + m(\angle COD) = 180^\circ$.

Since $m(\angle OCD) = m(\angle ODC)$, we have $2 \cdot m(\angle OCD) + 48^\circ = 180^\circ$.

$2 \cdot m(\angle OCD) = 180^\circ - 48^\circ = 132^\circ$.

$m(\angle OCD) = \frac{132^\circ}{2} = 66^\circ$.

b) Finding $m(\text{arc } CD)$:

As calculated in part (a), $m(\text{arc } CD) = 180^\circ - 132^\circ = 48^\circ$.

3. Final Answer

a) $m(\angle OCD) = 66^\circ$

b) $m(\text{arc } CD) = 48^\circ$