Lower your head

The idea

The angle of depression of a point viewed is the angle formed by the line of sight with the horizontal when the point is below the horizontal level, as when we lower our head to look at it.

The girl on the balcony

A girl sits on the balcony of a building. She looks straight ahead: her line of sight is horizontal. Then she lowers her head to look at a flower pot on the ground.

Now her line of sight makes an angle with the horizontal again, but this time it points down.

The pot is moved farther away from the building. What happens to the angle between her line of sight and the horizontal?

• It gets bigger
• It gets smaller
• It stays the same

Move the pot

Drag the pot along the ground, or tap the ground. The dashed line is the horizontal at her eye level. The purple arc is the angle she lowers her head.

Down from the horizontal

The angle of depression of a point viewed is the angle formed by the line of sight with the horizontal, when the point is below the horizontal level: the case when we lower our head to look at the point.

It is measured from the horizontal going down. In the toy the farther the pot, the smaller the angle. When you looked back from the pot, the angle of elevation of the girl was the same size: the two horizontals are parallel and the line of sight crosses them, so the angles are alternate angles.

Notes

The angle of depression is the angle formed by the line of sight with the horizontal when the point viewed is below the horizontal level, as when we lower our head.

Check yourself

A girl on a balcony lowers her head to look at a pot on the ground. From which line is the angle of depression measured?

From the top of a cliff, the angle of depression of a boat is 35°. What angle does the line of sight make with the vertical cliff, going down (in degrees)?

Answer: 55

The horizontal and the vertical make 90°, and 35° of it lies between the horizontal and the line of sight. The rest is 90° − 35° = 55°.

A man at the top of a tower sees a car on the road at an angle of depression of 40°. The driver looks up at the man. What is the driver's angle of elevation, in degrees?

Answer: 40

The horizontal at the man is parallel to the road, so the two angles are alternate angles and are equal: 40°.

From a bridge you look at a bird above the level of your eyes and at a fish below it. Which is correct?

• The vertical wall of the building. The angle of depression is measured from the horizontal, not from the wall.
• The horizontal through her eye — correct. Yes! The angle is formed by the line of sight and the horizontal at her eye level.
• The ground. The ground is below her. The angle is measured from the horizontal through her eye.
• The bird gives an angle of elevation, the fish gives an angle of depression. — correct. Yes! We raise our head for the bird and lower it for the fish.
• The bird gives an angle of depression, the fish gives an angle of elevation.. It is the other way round. We raise our head for the bird.
• Both are angles of elevation.. The fish is below the horizontal level, so its angle is an angle of depression.