Complementary angles are two angles whose measures add up to 90 degrees.
Given the angles \( (x + 3)^\circ \) and \( (4x - 2)^\circ \), we can set up the following equation:
\[ (x + 3) + (4x - 2) = 90 \]
Now, let's simplify and solve the equation:
\[ x + 3 + 4x - 2 = 90 \]
Combine like terms:
\[ 5x + 1 = 90 \]
Next, subtract 1 from both sides:
\[ 5x = 89 \]
Now, divide both sides by 5:
\[ x = \frac{89}{5} \]
So, \( x = 17.8 \) degrees.
Now, we can find the measures of the angles:
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For \( (x + 3)^\circ \): \[ x + 3 = 17.8 + 3 = 20.8^\circ \]
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For \( (4x - 2)^\circ \): \[ 4x - 2 = 4(17.8) - 2 = 71.2 - 2 = 69.2^\circ \]
Thus, the measures of the two complementary angles are \( 20.8^\circ \) and \( 69.2^\circ \).