PBOA is a quadrilateral and therefore the measures of its angles must add up to 360°.
If we recognize that a tangent segment is part of a tangent line, by the Perpendicular Tangent Theorem, we know that PB is perpendicular to OB. Likewise, PA is perpendicular to OA. Therefore, ∠PAO and ∠PBO are both right angles. PBOA is a quadrilateral, so the measures of all its angles must add up to 360°. Since each angle in PBOA other than ∠APB is a right angle, ∠APB must also be a right angle.