Question

line ef is tangent to circle g at point a.
if the measure of $angle cae$ is $95^circ$, what is the measure of $overparen{cba}$?
$90^circ$
$95^circ$
$190^circ$
$195^circ$

Explanation:

Step1: Recall tangent-chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc.

Step2: Find arc $\overset{\frown}{CA}$

$\angle CAE = \frac{1}{2}m\overset{\frown}{CA}$
$m\overset{\frown}{CA} = 2 \times \angle CAE = 2 \times 95^\circ = 190^\circ$

Step3: Find total circle degrees

A full circle is $360^\circ$.

Step4: Calculate $\overset{\frown}{CBA}$

$m\overset{\frown}{CBA} = 360^\circ - m\overset{\frown}{CA} = 360^\circ - 190^\circ = 170^\circ$
Correction: Re-identify intercepted arc

Step1: Correct intercepted arc for $\angle CAE$

$\angle CAE$ intercepts the major arc $\overset{\frown}{CBA}$? No: The tangent-chord angle intercepts the arc that is not adjacent to the angle's exterior. The angle $\angle CAE$ is outside the circle relative to arc $\overset{\frown}{CDA}$, so it intercepts $\overset{\frown}{CBA}$? No, correct rule: The measure of tangent-chord angle equals half the measure of the intercepted major arc if the angle opens to the major arc, or half the minor arc if it opens to the minor arc.
$\angle CAE = 95^\circ$, which is greater than $90^\circ$, so it intercepts the major arc $\overset{\frown}{CBA}$.

Step2: Apply theorem correctly

$\angle CAE = \frac{1}{2}m\overset{\frown}{CBA}$
$m\overset{\frown}{CBA} = 2 \times 95^\circ = 190^\circ$

Answer:

190° (Option C: 190°)