The problem gives a circle with inscribed angles $\angle STV$ and $\angle SUV$. The measures of these angles are given as $m\angle STV = (3x-6)^\circ$ and $m\angle SUV = (2x+10)^\circ$. We are asked to find the measure of $\angle SUV$.

1. Problem Description

The problem gives a circle with inscribed angles $\angle STV$ and $\angle SUV$. The measures of these angles are given as $m\angle STV = (3x-6)^\circ$ and $m\angle SUV = (2x+10)^\circ$. We are asked to find the measure of $\angle SUV$.

2. Solution Steps

Since angles $\angle STV$ and $\angle SUV$ intercept the same arc, $\stackrel{\frown}{SV}$, the inscribed angle theorem states that they are equal in measure.

$m\angle STV = m\angle SUV$

Substitute the expressions for the angle measures:

$3x - 6 = 2x + 10$

Solve for $x$:

$3x - 2x = 10 + 6$

$x = 16$

Now substitute the value of $x$ into the expression for $m\angle SUV$:

$m\angle SUV = 2x + 10$

$m\angle SUV = 2(16) + 10$

$m\angle SUV = 32 + 10$

$m\angle SUV = 42$

3. Final Answer

42