To find the measure of the smallest angle in the triangle, we first need to establish an equation using the fact that the sum of the angles in a triangle is \(180^\circ\).
Given the angles are:
- \( (6x+3)^\circ \)
- \( (2x+7)^\circ \)
- \( 90^\circ \)
We can write the equation:
\[ (6x + 3) + (2x + 7) + 90 = 180 \]
Now, combine like terms:
\[ 6x + 2x + 3 + 7 + 90 = 180 \]
This simplifies to:
\[ 8x + 100 = 180 \]
Next, we solve for \(x\):
\[ 8x = 180 - 100 \] \[ 8x = 80 \] \[ x = 10 \]
Now that we have \(x\), we can find the measures of the angles:
- \( (6x + 3)^\circ = 6(10) + 3 = 60 + 3 = 63^\circ \)
- \( (2x + 7)^\circ = 2(10) + 7 = 20 + 7 = 27^\circ \)
- \( 90^\circ \)
Now we compare the angles \( 63^\circ \), \( 27^\circ \), and \( 90^\circ \) to determine the smallest angle.
The smallest angle is:
\[ \boxed{27^\circ} \]