what is the sum of the exterior angle measures, one at each vertex, of a convex quadrilateral?

Explanation:

Step1: Recall the exterior angle sum theorem for polygons

The sum of the exterior angle measures of any convex polygon, one at each vertex, is always ( 360^\circ ). This is a fundamental theorem in geometry. For a convex quadrilateral (which is a 4 - sided convex polygon), we can also derive it by considering the relationship between interior and exterior angles. At each vertex, an interior angle and its corresponding exterior angle are supplementary (they add up to ( 180^\circ )). Let the interior angles of the quadrilateral be ( I_1, I_2, I_3, I_4 ) and the exterior angles be ( E_1, E_2, E_3, E_4 ). Then ( I_1 + E_1=180^\circ ), ( I_2 + E_2 = 180^\circ ), ( I_3+E_3 = 180^\circ ), ( I_4 + E_4=180^\circ ). Summing these equations: ( (I_1 + I_2+I_3 + I_4)+(E_1 + E_2+E_3 + E_4)=4\times180^\circ=720^\circ ). We know that the sum of the interior angles of a quadrilateral is ( (4 - 2)\times180^\circ=360^\circ ). Substituting this into the previous equation: ( 360^\circ+(E_1 + E_2+E_3 + E_4)=720^\circ ). Solving for the sum of exterior angles: ( E_1 + E_2+E_3 + E_4=720^\circ - 360^\circ = 360^\circ ).

Answer:

( 360 )