Complementary angles are two angles that add up to 90 degrees. Given the two angles are \( (4x)^\circ \) and \( (2x)^\circ \), we can set up the following equation:
\[ 4x + 2x = 90 \]
Combining like terms:
\[ 6x = 90 \]
Now, we can solve for \( x \) by dividing both sides by 6:
\[ x = \frac{90}{6} = 15 \]
Now that we have the value of \( x \), we can find the measures of the two angles:
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The first angle: \[ 4x = 4(15) = 60^\circ \]
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The second angle: \[ 2x = 2(15) = 30^\circ \]
Thus, the value of \( x \) is \( 15 \), and the measures of the two angles are \( 60^\circ \) and \( 30^\circ \).