A quadrilateral has two angles that measure \( 317.6^{\circ} \) and \( 31.6^{\circ} \). The other two angles are in a ratio of \( 4: 5 \). What are the measures of those two angles?

To find the measures of the two unknown angles in the quadrilateral, we first need to determine the sum of the angles in a quadrilateral, which is \( 360^{\circ} \). Given angles: 1. \( 317.6^{\circ} \) 2. \( 31.6^{\circ} \) Now, we can find the total of the known angles: \[ 317.6 + 31.6 = 349.2^{\circ} \] Now, we subtract this sum from \( 360^{\circ} \) to find the total measure of the remaining two angles: \[ 360 - 349.2 = 10.8^{\circ} \] Let the measures of the two unknown angles be \( 4x \) and \( 5x \) since they are in the ratio \( 4:5 \). Therefore: \[ 4x + 5x = 10.8 \] This simplifies to: \[ 9x = 10.8 \] Now, we solve for \( x \): \[ x = \frac{10.8}{9} = 1.2 \] Now, we can find the measures of the two angles: 1. \( 4x = 4(1.2) = 4.8^{\circ} \) 2. \( 5x = 5(1.2) = 6.0^{\circ} \) Thus, the measures of the two angles are \( 4.8^{\circ} \) and \( 6.0^{\circ} \).