Apply angle relationships (as tested) - TACHS Math

Refer to the figure. Two chords $\overline{AB}$ and $\overline{CD}$ of a circle intersect inside the circle at point $P$ . Arc $AC$ (not containing $B$ or $D$ ) measures $84°$ and arc $BD$ (not containing $A$ or $C$ ) measures $46°$ . What is the measure of $\angle APC$ ?
Answer: $65°$ The measure of an angle formed by two chords intersecting inside a circle equals half the sum of the intercepted arcs: $\angle APC = \frac{1}{2}(84°+46°) = \frac{1}{2}(130°) = 65°$ . (A) $19°$ uses the external-angle formula (half the difference). (B) $38°$ is half of $76°$ , a miscalculation. (D) $130°$ is the sum of arcs without halving.
In the figure, $\overline{PA}$ is tangent to the circle at $A$ , and $\overline{PBC}$ is a secant passing through the circle, with $B$ closer to $P$ . If arc $AC$ (far arc, not containing $B$ ) $= 148°$ and arc $AB$ (near arc, not containing $C$ ) $= 62°$ , what is the measure of $\angle APC$ ?
Answer: $43°$ The angle formed by a tangent and a secant from an external point equals half the difference of the intercepted arcs: $\angle APC = \frac{1}{2}(\text{far arc} - \text{near arc}) = \frac{1}{2}(148°-62°) = \frac{1}{2}(86°) = 43°$ . (B) $105°$ is half the sum. (C) $86°$ is the difference without halving. (D) $74°$ is half of $148°$ .