To solve the problem, we need to understand the relationship between the tangent line, the radius of the circle, and the measures of angles and arcs in a circle.
In this scenario, line EF is tangent to circle G at point A. By the properties of circles, we know that a tangent line at a point on the circle forms a right angle with the radius of the circle that intersects at that same point. This means that the angle between the radius (GA) and the tangent line (EF) at point A is 90 degrees.
Now, we are given that angle CAE measures 95 degrees. Angle CAE is formed by rays CA and EA, where:
- CA is a line going from point C to point A, and
- EA is part of the tangent line EF at point A.
To find the measure of arc CBA, we will use the fact that the measure of an arc between two points in a circle is related to the angles formed by those points and the tangent line. Specifically, the measure of the arc (arc CBA) is equal to twice the angle formed by the radius and the tangent line at the point of tangency. In this case:
- The measure of angle CAE is 95 degrees.
- The angle CAE does not measure the same as the angle formed at the center of the circle, but is related to it.
- The angle at point C from the center of circle G to point A is 90 degrees (the right angle due to the tangent).
Therefore, we can calculate the measure of arc CBA:
Measure of arc CBA = 2 * (measure of angle CAE)
Thus:
Measure of arc CBA = 2 * 95 = 190 degrees.
In conclusion, the measure of arc CBA is 190 degrees.