Complementary angles are two angles whose measures add up to 90 degrees. Given that the two complementary angles have measures \( (2x)^\circ \) and \( (3x)^\circ \), we can set up the equation:
\[ 2x + 3x = 90 \]
Combining like terms, we have:
\[ 5x = 90 \]
Now, solve for \( x \):
\[ x = \frac{90}{5} = 18 \]
To find the measures of the two angles, we substitute \( x \) back into the expressions for the angles:
- First angle: \( 2x = 2(18) = 36^\circ \)
- Second angle: \( 3x = 3(18) = 54^\circ \)
So, the value of \( x \) is \( 18 \), and the measures of the two angles are \( 36^\circ \) and \( 54^\circ \).
Now, for the second part concerning the angles \( m\angle ABC = 105^\circ \) and \( m\angle CBD = (3x - 24)^\circ \):
To set up the equation, we note that the angles \( \angle ABC \) and \( \angle CBD \) are supplementary if they add up to 180 degrees:
\[ 105 + (3x - 24) = 180 \]
To solve for \( x \):
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Simplify the left side: \[ 105 + 3x - 24 = 180 \] \[ 81 + 3x = 180 \]
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Subtract 81 from both sides: \[ 3x = 180 - 81 \] \[ 3x = 99 \]
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Divide by 3: \[ x = \frac{99}{3} = 33 \]
Thus, the equation to solve for \( x \) is:
\[ 105 + (3x - 24) = 180 \]
And solving that gives \( x = 33 \).