Let the measure of the angle be \( x \) degrees. The measure of its complementary angle, which together with \( x \) adds up to 90 degrees, can be expressed as \( 90 - x \) degrees.
According to the problem, the measure of the angle is eight times the measure of its complementary angle. This can be set up as the following equation:
\[ x = 8(90 - x) \]
Now, we can solve the equation for \( x \):
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Distribute the 8 on the right side: \[ x = 720 - 8x \]
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Add \( 8x \) to both sides: \[ x + 8x = 720 \] \[ 9x = 720 \]
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Divide both sides by 9: \[ x = \frac{720}{9} = 80 \]
Now that we have the measure of the angle \( x \), we can find its complementary angle by substituting \( x \) back:
\[ 90 - x = 90 - 80 = 10 \]
Thus, the measures of the angles are:
- The angle: \( 80 \) degrees
- The complementary angle: \( 10 \) degrees
In conclusion, the measures of the angles are \( 80 \) degrees and \( 10 \) degrees.