The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is always \(180^\circ\). Given the angles of the triangle are \(2x\), \(3x\), and \(4x\), we can set up the equation:
\[ 2x + 3x + 4x = 180^\circ \]
Combining the terms on the left side:
\[ 9x = 180^\circ \]
Now, we can solve for \(x\):
\[ x = \frac{180^\circ}{9} = 20^\circ \]
Now we can find the measures of the angles:
- The first angle is \(2x = 2(20^\circ) = 40^\circ\)
- The second angle is \(3x = 3(20^\circ) = 60^\circ\)
- The third angle is \(4x = 4(20^\circ) = 80^\circ\)
The smallest angle is \(40^\circ\).
Thus, the measure of the smallest angle in this triangle is:
\[ \boxed{40^\circ} \]
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