To solve for \( x \) in the equation given by the Triangle Angle Sum Theorem:
\[ 2x + 3x + 4x = 180 \]
Combine like terms:
\[ 9x = 180 \]
Now, divide both sides by 9:
\[ x = \frac{180}{9} = 20 \]
Next, we can find the measurements of each angle:
- The first angle is \( 2x = 2(20) = 40 \) degrees.
- The second angle is \( 3x = 3(20) = 60 \) degrees.
- The third angle is \( 4x = 4(20) = 80 \) degrees.
The smallest angle among the three is \( 40 \) degrees.
Thus, the final answers are:
- The value of \( x \) is \( 20 \).
- The measurement of the smallest angle is \( 40 \) degrees.