The problem asks for the measure of the inscribed angle ACB in a circle, given that angle ACB $= (7x + 16)^\circ$ and the intercepted arc APB measures $(18x - 32)^\circ$.

1. Problem Description

The problem asks for the measure of the inscribed angle ACB in a circle, given that angle ACB $= (7x + 16)^\circ$ and the intercepted arc APB measures $(18x - 32)^\circ$.

2. Solution Steps

The measure of an inscribed angle is half the measure of its intercepted arc. Therefore, we have:

$m\angle ACB = \frac{1}{2} \times m\text{arc }APB$

$(7x + 16) = \frac{1}{2} (18x - 32)$

Multiply both sides by 2:

$2(7x + 16) = 18x - 32$

$14x + 32 = 18x - 32$

Subtract $14x$ from both sides:

$32 = 4x - 32$

Add 32 to both sides:

$64 = 4x$

Divide by 4:

$x = 16$

Now we can find the measure of the inscribed angle ACB:

$m\angle ACB = (7x + 16)^\circ = (7(16) + 16)^\circ = (112 + 16)^\circ = 128^\circ$

3. Final Answer

The measure of the inscribed angle is 128 degrees.