The exterior angles of polygon MASON have the following measures: \( m \angle M=(5 r+23)^{\circ}, m \angle A=(4 r)^{\circ}, m \angle S=(7 r-20)^{\circ} \), \( m \angle O=(3 r+23)^{\circ} \), and \( m \angle N=(5 r-2)^{\circ} \). What is the measure of the largest exterior angle in this situation? Record your answer in the blank provided. Be sure to round your answer to the nearest hundredth.

To solve for the largest exterior angle of polygon MASON given the measures, we start by summarizing the angles: 1. \( m \angle M = (5r + 23)^\circ \) 2. \( m \angle A = (4r)^\circ \) 3. \( m \angle S = (7r - 20)^\circ \) 4. \( m \angle O = (3r + 23)^\circ \) 5. \( m \angle N = (5r - 2)^\circ \) The sum of the exterior angles of any polygon is always \( 360^\circ \). Therefore, we can write the equation: \[ (5r + 23) + (4r) + (7r - 20) + (3r + 23) + (5r - 2) = 360 \] Combining like terms: \[ (5r + 4r + 7r + 3r + 5r) + (23 - 20 + 23 - 2) = 360 \] \[ (24r) + (24) = 360 \] Now, subtract 24 from both sides: \[ 24r = 336 \] Next, divide by 24: \[ r = 14 \] Now, substituting \( r = 14 \) back into each expression to find the measure of each angle: 1. \( m \angle M = 5(14) + 23 = 70 + 23 = 93^\circ \) 2. \( m \angle A = 4(14) = 56^\circ \) 3. \( m \angle S = 7(14) - 20 = 98 - 20 = 78^\circ \) 4. \( m \angle O = 3(14) + 23 = 42 + 23 = 65^\circ \) 5. \( m \angle N = 5(14) - 2 = 70 - 2 = 68^\circ \) Now we compare these angles: - \( m \angle M = 93^\circ \) - \( m \angle A = 56^\circ \) - \( m \angle S = 78^\circ \) - \( m \angle O = 65^\circ \) - \( m \angle N = 68^\circ \) From this, the largest angle is \( m \angle M = 93^\circ \). Thus, the measure of the largest exterior angle is: \[ \boxed{93.00} \]