Find the measure of <ABC if the shaded are is 120 pi cm squared and the r=4?

To find the measure of ∠ABC, we need to use the information given about the shaded area and the radius of the circle.

First, let's recall some basic knowledge about angles in circles. In a circle, the measure of an angle at the center of the circle is twice the measure of an angle at the circumference (arc) intercepted by the same two rays.

In this case, the shaded area represents a sector of the circle. The area of a sector can be calculated using the formula:

Area of sector = (θ / 360°) × πr²

where θ is the measure of the central angle in degrees and r is the radius of the circle.

We are given that the shaded area is 120π cm² and the radius is 4 cm. Plugging these values into the area formula, we can solve for θ:

120π = (θ / 360°) × π(4²)
120π = (θ / 360°) × 16π
120 = (θ / 360°) × 16
120 = θ / 22.5

Now, let's solve for θ by multiplying both sides of the equation by 22.5:

120 × 22.5 = θ
2,700 = θ

Therefore, the measure of ∠ABC is 2700 degrees.

Alternatively, if you prefer using radians instead of degrees, you can convert the measure of the angle from degrees to radians by multiplying it by π / 180°. In this case, the measure of ∠ABC would be 2700 × π / 180 = 15π radians.