Heights and Distances

352. Up and Down Angles · elevation meets depression

Angle of elevation from C always equals angle of depression from A.

ABD∠ACB = 50°∠ACB = 50°∠DAC = 50°∠DAC = 50°C
The angle of elevation is measured upward from the horizontal to an object above; the angle of depression is measured downward to an object below. Because the two horizontals are parallel, the angle of elevation from C equals the angle of depression from A (alternate angles).

Stuck? Ask Guru

Selina ICSE: Heights and Distances

What this lesson covers

Try to break it

Drag C along the ground. The angle of elevation at C (looking up to the tower's top) and the angle of depression at the tower's top (looking down to C) always read the same value — they're alternate angles cut by the vertical between the two horizontals. Pull C closer and both grow; push C away and both shrink, always in lockstep.

The proof, step by step

Prove that the angle of elevation from C equals the angle of depression from A.

  • The ground BC and the horizontal line AD are both level, so BC is parallel to AD.
  • The line of sight AC acts as a transversal intersecting the parallel lines BC and AD.
  • By the alternate interior angles theorem, angle ACB (elevation) equals angle DAC (depression).

Worked example

A tower stands vertically on level ground. From a point C on the ground, the angle of elevation to the top of the tower is 45°. If the tower is 50 meters tall, what is the distance from point C to the base of the tower?

In right triangle ABC, tan(45°) = AB/BC. Since tan(45°) = 1, AB = BC. Given AB = 50m, the distance BC is also 50m.

  • 25 meters
  • 50 meters — correct
  • 70.7 meters
  • 100 meters
Hold to talk

Subscription Status