The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is \(180\) degrees. Given the angles of the triangle as \(2x\), \(3x\), and \(4x\), we can set up the following equation:
\[ 2x + 3x + 4x = 180 \]
Simplifying this, we get:
\[ 9x = 180 \]
To find \(x\), we divide both sides by \(9\):
\[ x = \frac{180}{9} = 20 \]
Now, we can find the measures of the angles:
- The first angle: \(2x = 2(20) = 40\) degrees
- The second angle: \(3x = 3(20) = 60\) degrees
- The third angle: \(4x = 4(20) = 80\) degrees
The smallest angle is \(40\) degrees.
Therefore, the measure of the smallest angle in this triangle is 40 degrees.
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