To find the angles of the triangle, we start with the given measures:
- Angle 1: \((x + 10)^\circ\)
- Angle 2: \((x + 20)^\circ\)
- Angle 3: \((x + 30)^\circ\)
The sum of the angles in a triangle is always \(180^\circ\). Therefore, we can set up the equation:
\[ (x + 10) + (x + 20) + (x + 30) = 180 \]
Combining like terms gives us:
\[ 3x + 60 = 180 \]
Next, we can solve for \(x\):
\[ 3x = 180 - 60 \] \[ 3x = 120 \] \[ x = 40 \]
Now, we can substitute \(x\) back into the expressions for the angles:
- Angle 1: \(x + 10 = 40 + 10 = 50^\circ\)
- Angle 2: \(x + 20 = 40 + 20 = 60^\circ\)
- Angle 3: \(x + 30 = 40 + 30 = 70^\circ\)
The angles of the triangle are \(50^\circ\), \(60^\circ\), and \(70^\circ\).
The smallest angle is \(50^\circ\).
So, the measure of the smallest angle is \( \boxed{50^\circ} \).