To find the measure of the smallest angle in the triangle with angles \((x+10)^\circ\), \((x+20)^\circ\), and \((x+30)^\circ\), we first use the fact that the sum of the angles in a triangle is always \(180^\circ\).
Setting up the equation:
\[ (x + 10) + (x + 20) + (x + 30) = 180 \]
Combine like terms:
\[ 3x + 60 = 180 \]
Now, solve for \(x\):
\[ 3x = 180 - 60 \] \[ 3x = 120 \] \[ x = 40 \]
Now that we have the value of \(x\), we can find the measures of the angles:
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First angle: \[ x + 10 = 40 + 10 = 50^\circ \]
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Second angle: \[ x + 20 = 40 + 20 = 60^\circ \]
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Third angle: \[ x + 30 = 40 + 30 = 70^\circ \]
The angles of the triangle are \(50^\circ\), \(60^\circ\), and \(70^\circ\).
The smallest angle is:
\[ \boxed{50^\circ} \]