PQ is a tangent at a point C to a circle with center O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

In this problem, you have a circle with center O, a diameter AB, and a tangent PQ at point C. You're asked to find the measure of angle PCA.

Here's how you can solve it:

Since AB is a diameter, it passes through the center O of the circle. This means that angle OAB is a right angle, and its measure is 90 degrees.

Since PQ is a tangent to the circle at point C, angle OCQ is also a right angle because the radius drawn to the point of tangency is perpendicular to the tangent line.

Now, consider triangle OCA. You have angle OAC equal to 30 degrees, angle OCA equal to 90 degrees (as mentioned above), and you want to find angle PCA.

The sum of angles in a triangle is 180 degrees. So, you can use the following equation to find angle PCA:

Angle PCA + Angle OAC + Angle OCA = 180 degrees

PCA + 30 degrees + 90 degrees = 180 degrees

Now, solve for PCA:

PCA = 180 degrees - 30 degrees - 90 degrees
PCA = 60 degrees

So, angle PCA is 60 degrees.