Question

is the measure of an angle depent upon the unit of length

Answer 1

As long as the units are the same in each problem (i.e. you are not using feet and meters), then the angle will be independent of the unit you use to measure teh lengths. this is because the angle is effectively a ratio between two lengths, suhc that the following trigonometric identities are obeyed \[\sin\theta=\frac{O}{H}\]\[\cos\theta=\frac{A}{H}\]\[\tan\theta=\frac{O}{A}\], where the terms O, A, and H denote the opposite, adjacent and hypotenuse of a right angle triangle (though the ratios still hold for irregular triangles). As you can see, the angle will depend upon the ratio of any two lengths, meaning that it is dimensionless as long as the same units are used for the lengths.