Since angles \( \angle 1 \) and \( \angle 2 \) are supplementary, we have:
\[ m\angle 1 + m\angle 2 = 180^\circ \]
Given that:
\[ m\angle 1 = 3x + 15 \quad \text{and} \quad m\angle 2 = 2x + 30 \]
we can substitute these expressions into the equation for supplementary angles:
\[ (3x + 15) + (2x + 30) = 180 \]
Now, combine like terms:
\[ 3x + 2x + 15 + 30 = 180 \]
\[ 5x + 45 = 180 \]
Next, subtract 45 from both sides:
\[ 5x = 180 - 45 \]
\[ 5x = 135 \]
Now, divide by 5:
\[ x = \frac{135}{5} = 27 \]
Now that we have \( x \), we can find the measure of \( \angle 1 \):
\[ m\angle 1 = 3x + 15 = 3(27) + 15 \]
Calculating this gives:
\[ m\angle 1 = 81 + 15 = 96 \]
Thus, the measure of \( \angle 1 \) is:
\[ \boxed{96^\circ} \]