Question

If the measure of a tangent angle is 68 then what is the measure of the intercept arc inside the angle

Answer 1

For a surveyor, the answer is obviously 72 degrees. this is from the fact that total deflection angle for a circular highway curve is 1/2 of the included angle. the explanation is located below. 

if you have a tangent at each end of the chord you have an isosceles triangle with two 36 degree angles and one angle of 108 degrees.

Now draw two lines from the points of tangency to the center of the circle. By definition, these radii are perpendicular to the tangent so they make an angle of 90-36 = 54 degrees with the chord.

so now you have an isosceles triangle with two 54 degree angles so the include angle must be 180-54-54 = 72 degrees 

Hope this helps :)