“A circle with centre O has diameter PQ. The line PQ is extended and meets the tangent at point A as R. If ∠RPQ = 27°, then what is the measure of ∠RQP?

Solution:

Since PQ is a diameter and is extended to point R, points P, Q, and R are collinear.

Given:
∠RPQ = 27°

This angle is the exterior angle at P of triangle PQR.
So, the interior angle at P is:
$180^\circ - 27^\circ = 153^\circ$

Now, since PQ is a diameter, the angle subtended by it at the circumference is a right angle:
∠PRQ = 90°

Now, in triangle PQR:
Sum of angles = 180°

So,
$27^\circ + 90^\circ + \angle RQP = 180^\circ$

$\angle RQP = 180^\circ - 117^\circ = 63^\circ$

Final Answer:

63° ✔️