Question

A quadrilateral circumscribed about a circle has angles of 80 90 94 and 96. find the measures of the four non overlapping arcs determined by the points of tangency

Answer 1

For the answer to the question above, the two lines from the 80º angle to the points of tangency:

Try to draw two lines from the points of tangency to the center of the circle:

a quadrilateral with two right angles is formed as you may see.

the angles of a quadrilateral add up to 360º
The center angle is 360º - (90º + 90º + 80º) =100º

Similarly, the angles at the center for other vertices of original quadrilaterals are: 90º, 86º, and 84º
It depends upon the site you measure your arc in, (usual radians) the arcs are π /4 (90º), 7π /30 (84º), 5π /18 (100º) and 43π /180 (86º)