Question

line ef is tangent to circle g at point a. if the measure of ∠cae is 95°, what is the measure of cba? 90° 95° 190° 195°

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle GAF = 90^{\circ}$.

Step2: Use angle - addition property

We know that $\angle CAE=95^{\circ}$ and $\angle GAF = 90^{\circ}$. The sum of angles around point A is related to the arc measures. The measure of an inscribed - angle is half of the measure of its intercepted arc. The measure of the central angle is equal to the measure of its intercepted arc.
We know that $\angle CAE$ is an exterior angle with respect to the circle - related angles at point A. Since $\angle GAF = 90^{\circ}$ and $\angle CAE=95^{\circ}$, the measure of the arc $\overline{CBA}$ is $190^{\circ}$. This is because the measure of an angle formed by a tangent and a chord ($\angle CAE$) is half of the measure of the intercepted arc. Let the measure of arc $\overline{CBA}=x$. Then $\angle CAE=\frac{1}{2}x$ (angle between tangent and chord property). Given $\angle CAE = 95^{\circ}$, we have $x = 2\times95^{\circ}=190^{\circ}$.

Answer:

$190^{\circ}$