What is the relationship between angle of elevation and depression?

The angle of elevation and the angle of depression are equal when measured from the same horizontal line.

In more detail, the angle of elevation is the angle formed between the horizontal line and the line of sight when looking up at an object. Conversely, the angle of depression is the angle formed between the horizontal line and the line of sight when looking down at an object. These angles are always measured from the observer's eye level or a horizontal plane.

Imagine you are standing on a hill and looking at a tree at a lower elevation. The angle your line of sight makes with the horizontal line as you look down at the tree is the angle of depression. Now, if someone were standing at the base of the tree and looking up at you, the angle their line of sight makes with the horizontal line is the angle of elevation. These two angles are equal because they are alternate interior angles formed by a transversal intersecting two parallel lines (the horizontal lines at your eye level and the base of the tree).

This relationship is crucial in solving problems involving heights and distances, as it allows you to use trigonometric ratios to find unknown lengths or angles. For example, if you know the height of a building and the distance from the observer to the base of the building, you can use the angle of elevation to calculate the height of the building using trigonometric functions like tangent. Understanding this relationship simplifies many real-world problems and is a fundamental concept in trigonometry.