Zenith Distance is Complement of Celestial Altitude

Theorem
Let $X$ be the position of a star (or other celestial body) on the celestial sphere.

The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$:


 * $z = 90 \degrees - a$

Proof
The vertical circle through $X$ is defined as the great circle that passes through $Z$.

By definition, the angle of the arc from $Z$ to the horizon is a right angle.

Hence $z + a = 90 \degrees$.

The result follows.