In the diagram, which pairs of angles are complementary? Which pairs of angles are supplementary?

The figure is given.

Following properties are used here,

1. In a circle, any line from center of circle to the point of contact of tangent is always perpendicular to tangent.
2. The line passing from the center and intersecting point of two tangents to its circle, always bisect the two angles, forming on the center and intersecting point.
3. In a right triangle, remaining two angles, except right angle, are complementary angles.
   Also, if the sum of two angles is $90°$, so these are complementary and if their sum is $180°$, so these angles are supplementary.

So, in given figure as $OA\perp AP,$ so $\angle PAO=90°$and thus $▵APO$is a right triangle.

Similarly as $OB\perp BP,$ so $\angle PBO=90°.$ Hence, $▵BPO$ is a right triangle. Thus, using first property, $\angle PAO=90°.$ So, these two angles are congruent.

Also, the sum of these two angles will be $180°,$ so these two angles are supplementary also.

Further, in right triangle $▵APO,$ as angle A is $90°,$ so by angle sum property of triangle,

$\angle POA+\angle APO=90°,$ Hence these two angles are complementary to each other. Similarly in right triangle $▵BPO,\angle POB+\angle BPO=90°.$ So, these two are also complementary angles. Also,

Using third property, another sets of congruent angles are $\angle POB=\angle POA$and $\angle APO=\angle BPO.$

Therefore, it is concluded that pairs of congruent angle are

$\left(i\right)\angle APO,\angle BPO\left(ii\right)\angle POB,\angle POA\left(iii\right)\angle PBO,\angle PAO.$

Pairs of complementary angles are

$\left(i\right)\angle APO,\angle POA\left(ii\right)\angle POB,\angle BPO.$

Further only pair of supplementary angles:

$\angle PBO,\angle PAO.$