The angles of a triangle sum to 180 degrees. Given the angles \( (x + 10)^\circ \), \( (x + 20)^\circ \), and \( (x + 30)^\circ \), we can set up the equation:
\[ (x + 10) + (x + 20) + (x + 30) = 180 \]
Now, simplify the equation:
\[ 3x + 60 = 180 \]
Next, subtract 60 from both sides:
\[ 3x = 120 \]
Now, divide by 3:
\[ x = 40 \]
Now we can find the measure of each angle:
- \( (x + 10) = 40 + 10 = 50^\circ \)
- \( (x + 20) = 40 + 20 = 60^\circ \)
- \( (x + 30) = 40 + 30 = 70^\circ \)
The angles are \( 50^\circ \), \( 60^\circ \), and \( 70^\circ \).
Therefore, the smallest angle is:
\[ \boxed{50^\circ} \]
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