Question

find the missing information for both parts below.
(a) in the figure below, $m\angle aeb=69^\circ$ and $m\overset{\frown}{cd}=79^\circ$. find $m\overset{\frown}{ab}$.
$m\overset{\frown}{ab} = \square^\circ$
(b) in the figure below, $m\overset{\frown}{ptr}=285^\circ$. find $m\angle psr$.
$m\angle psr = \square^\circ$

Explanation:

Step1: Use intersecting chords theorem

The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. So:
$$m\angle AEB = \frac{1}{2}(m\overset{\frown}{AB} + m\overset{\frown}{CD})$$

Step2: Solve for $m\overset{\frown}{AB}$

Rearrange the formula to isolate $m\overset{\frown}{AB}$:
$$m\overset{\frown}{AB} = 2(m\angle AEB) - m\overset{\frown}{CD}$$
Substitute $m\angle AEB=69^\circ$ and $m\overset{\frown}{CD}=79^\circ$:
$$m\overset{\frown}{AB} = 2(69^\circ) - 79^\circ = 138^\circ - 79^\circ = 59^\circ$$

Step3: Find minor arc $\overset{\frown}{PR}$

The total circumference of a circle is $360^\circ$. Subtract the major arc $\overset{\frown}{PTR}$ from $360^\circ$:
$$m\overset{\frown}{PR} = 360^\circ - 285^\circ = 75^\circ$$

Step4: Use tangent-secant angle theorem

The measure of an angle formed by two tangents outside a circle is half the difference of the measures of the intercepted arcs. The larger arc is $\overset{\frown}{PTR}$, the smaller is $\overset{\frown}{PR}$:
$$m\angle PSR = \frac{1}{2}(m\overset{\frown}{PTR} - m\overset{\frown}{PR})$$
Substitute values:
$$m\angle PSR = \frac{1}{2}(285^\circ - 75^\circ) = \frac{1}{2}(210^\circ) = 105^\circ$$

Answer:

(a) $m\overset{\frown}{AB} = 59^\circ$
(b) $m\angle PSR = 105^\circ$