In this figure, angles p and w are examples of a. vertical angles. b. alternate interior angles. c. alternate exterior angles. d. corresponding angles.

Since a figure is not given, it would be helpful to understand what the choices are. Vertical angles are the pair of angles that are opposite to each other when there are two lines that meet. Corresponding angles are angles that lies to a same relative position. They will have the same measurement of their angles.  Alternate interior and exterior angles are found when two lines are being crossed by a third line which is called the transversal line. The alternate interior are the angles on the opposite side of the transversal but are found inside the two lines. The exterior are the same but they lie outside the two lines.

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12