Two angles of a quadrilateral measure \( 103^{\circ} \) and \( 170^{\circ} \). The other two angles are in a ratio of \( 13: 16 \). What are the measures of those two angles? Submit

To find the measures of the two unknown angles in the quadrilateral, we start by recalling that the sum of the interior angles of a quadrilateral is always \( 360^{\circ} \). You have two angles measuring \( 103^{\circ} \) and \( 170^{\circ} \). Adding these two gives: \[ 103^{\circ} + 170^{\circ} = 273^{\circ} \] Now, we can find the sum of the other two angles: \[ 360^{\circ} - 273^{\circ} = 87^{\circ} \] Let the measures of the two unknown angles be \( 13x \) and \( 16x \) according to the given ratio \( 13:16 \). Setting up the equation based on their sum, we have: \[ 13x + 16x = 87 \] Combining like terms yields: \[ 29x = 87 \] Solving for \( x \): \[ x = \frac{87}{29} = 3 \] Now we can find the measures of the two angles: \[ 13x = 13 \times 3 = 39^{\circ} \] \[ 16x = 16 \times 3 = 48^{\circ} \] Thus, the measures of the two unknown angles are \( 39^{\circ} \) and \( 48^{\circ} \).