line ef is tangent to circle g at point a. if the measure of ∠cae is 95°, what is the measure of ⌢cba? 90°…

Explanation:

Step1: Recall tangent - chord relationship

The measure of an inscribed angle is half the measure of its intercepted arc. The measure of an angle formed by a tangent and a chord is equal to the measure of the inscribed angle subtended by the same arc.

Step2: Identify the intercepted arc

The angle $\angle CAE$ is an angle formed by the tangent $EF$ and the chord $AC$. The arc $\overset{\frown}{CBA}$ is the intercepted arc for $\angle CAE$.

Step3: Apply the tangent - chord angle theorem

The measure of the angle formed by a tangent and a chord is equal to the measure of the inscribed angle subtended by the same arc. Also, the measure of the arc is twice the measure of the inscribed - angle subtended by it. Since $\angle CAE = 95^{\circ}$, the measure of the arc $\overset{\frown}{CBA}$ is $190^{\circ}$ because the measure of an angle formed by a tangent and a chord is half of the measure of the intercepted arc. Mathematically, if $\theta$ is the angle formed by the tangent and the chord and $m\overset{\frown}{s}$ is the measure of the intercepted arc, then $\theta=\frac{1}{2}m\overset{\frown}{s}$, so $m\overset{\frown}{s} = 2\theta$. Here, $\theta = 95^{\circ}$, so $m\overset{\frown}{s}=190^{\circ}$.

Answer:

$190^{\circ}$