The Triangle Angle Sum Theorem states that the sum of the angles in a triangle must equal \(180^\circ\).
Given the angles of the triangle are \(2x\), \(3x\), and \(4x\), we can write the equation:
\[ 2x + 3x + 4x = 180^\circ \]
Combining the terms on the left side gives:
\[ 9x = 180^\circ \]
Now, we can solve for \(x\) by dividing both sides by 9:
\[ x = \frac{180^\circ}{9} = 20^\circ \]
Now, we can find the measures of the angles:
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The first angle is: \[ 2x = 2(20^\circ) = 40^\circ \]
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The second angle is: \[ 3x = 3(20^\circ) = 60^\circ \]
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The third angle is: \[ 4x = 4(20^\circ) = 80^\circ \]
To find the smallest angle, we compare the calculated angles: \(40^\circ\), \(60^\circ\), and \(80^\circ\). The smallest angle is:
\[ \boxed{40^\circ} \]
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