A tangent drawn from the external point P touches a circle at T. A second line through P cuts the same circle at A and then at B. The arc TB that does not contain A measures 146°, and the arc TA that does not contain B measures 54°. What is the measure of ∠TPA?

A tangent and a secant meeting outside the circle behave exactly like two secants: the angle is half the difference of the far arc and the near arc. Here ∠TPA = ½(146° − 54°) = ½ × 92° = 46°. (The three arcs check out: 54° + 146° + arc AB = 360°, so arc AB = 160°.) The trap 92° stops at the difference without halving, 100° halves the sum instead of the difference, and 73° halves only the far arc.