To find the measure of the smallest angle in the triangle, we start by using the fact that the sum of the angles in a triangle is always \(180^\circ\).
Given the angles:
- \(x + 10\)
- \(x + 20\)
- \(x + 30\)
We can set up the equation: \[ (x + 10) + (x + 20) + (x + 30) = 180 \]
Simplifying this equation: \[ 3x + 60 = 180 \]
Subtract \(60\) from both sides: \[ 3x = 120 \]
Now, divide by \(3\): \[ x = 40 \]
Now we can find the measures of the three angles:
- \(x + 10 = 40 + 10 = 50\)
- \(x + 20 = 40 + 20 = 60\)
- \(x + 30 = 40 + 30 = 70\)
The angles are \(50^\circ\), \(60^\circ\), and \(70^\circ\).
The smallest angle is \(\mathbf{50^\circ}\).