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Explanation:
Label the center point of the circle as point G.
Also, label the top and bottom tangent points as H and J respectively. This means that H is on line segment ED, while J is on line segment EF.
The given arc measure of 100 degrees directly leads to the central angle HGJ to be 100 degrees.
Focus on quadrilateral EHGJ. It has two right angles at points H and J since they are tangent points. Angle G of this quadrilateral was found to be 100. Let's find angle E
E+H+G+J = 360 ..... any four angles of a quadrilateral add to 360
E+90+100+90 = 360
E+280 = 360
E = 360-280
E = 80
Or as a shortcut, we can note that angles E and G must add to 180, since H+J = 90+90 = 180, making up half of 360.
So, E+G = 180 leads to E+100 = 180 which solves to E = 80.
Once we know that E = 80, this leads to angle DEF to be 80 degrees.
This is an inscribed angle that subtends arc DF. By the inscribed angle theorem, we double the inscribed angle value to get the arc measure
arc DF = 2*(inscribed angle DEF)
arc DF = 2*(80)
arc DF = 160 degrees