A quadrilateral has two angles that measure \( 44.8^{\circ} \) and \( 173.2^{\circ} \). The other two angles are in a ratio of \( 3: 17 \). What are the measures of those two angles?

The sum of the interior angles in a quadrilateral is \( 360^{\circ} \). Therefore, we can find the sum of the other two angles by subtracting the measures of the known angles from \( 360^{\circ} \): \[ 360^{\circ} - (44.8^{\circ} + 173.2^{\circ}) = 360^{\circ} - 218^{\circ} = 142^{\circ}. \] Let the measures of the other two angles be \( 3x \) and \( 17x \). According to the ratio given, we add these two angles to equal \( 142^{\circ} \): \[ 3x + 17x = 142^{\circ} \] This simplifies to: \[ 20x = 142^{\circ} \implies x = \frac{142^{\circ}}{20} = 7.1^{\circ}. \] Now we can find the measures of the two angles: \[ 3x = 3 \times 7.1^{\circ} = 21.3^{\circ}, \quad 17x = 17 \times 7.1^{\circ} = 120.7^{\circ}. \] Thus, the measures of the other two angles are \( 21.3^{\circ} \) and \( 120.7^{\circ} \).