Question

$\overleftrightarrow{dy}$ is tangent to circle $o$ at point $d$, and $\overleftrightarrow{dp}$ is a secant line. if $\text{m}\overarc{dzp} = 238^{\circ}$, find $\text{m}\angle pdy$.

Explanation:

Step1: Find minor arc $\overset{\frown}{DP}$

The total degree measure of a circle is $360^\circ$. Subtract the given major arc measure from $360^\circ$.
$\text{m}\overset{\frown}{DP} = 360^\circ - 238^\circ = 122^\circ$

Step2: Apply tangent-secant angle theorem

The measure of an angle formed by a tangent and a secant outside a circle is half the measure of its intercepted arc.
$\text{m}\angle PDY = \frac{1}{2} \times \text{m}\overset{\frown}{DP}$

Step3: Calculate the angle measure

Substitute the value of $\text{m}\overset{\frown}{DP}$ into the formula.
$\text{m}\angle PDY = \frac{1}{2} \times 122^\circ = 61^\circ$

Answer:

$61^\circ$