angles of elevation and depression
Stand on flat ground and look up at the top of a building: the angle between your level line of sight and your tilted-up line of sight is the angle of elevation. Lean over a cliff and look down at a boat: the angle below the horizontal is the angle of depression. They are the everyday handles by which trigonometry grabs hold of heights and distances you cannot reach with a tape measure.
Both angles are always measured from the horizontal — a level line drawn through the observer's eye — not from the vertical and not from the ground at the object. The angle of elevation looks upward and the angle of depression looks downward. A key fact saves work: because the observer's horizontal and the object's horizontal are parallel lines cut by the line of sight, the angle of elevation from the lower point equals the angle of depression from the higher point (they are alternate interior angles). To use either, draw the right triangle whose vertical leg is the height difference and whose horizontal leg is the ground distance, then apply a trig ratio, usually tangent.
These angles turn a sighting instrument and a known distance into an unknown height, which is why they run through surveying, navigation, astronomy, and even aiming problems. The single most common error is measuring the depression angle from the vertical or from the downward slope rather than from the horizontal.
From a point 50 metres from the foot of a tower, the angle of elevation to the top is 40 degrees. The height equals the opposite side over the adjacent side times the distance: height = 50 x tan 40 degrees ≈ 50 x 0.839 = 42.0 metres.
The tangent links the angle you measured to the height you want over the distance you know.
Elevation and depression are both measured from the horizontal, never from the vertical; the elevation from below and the depression from above are equal because they are alternate interior angles between parallel horizontals.