The figure below shows parallel lines cut by a transversal: a pair of parallel lines is shown with arrowheads on each end. a transversal cuts through these two lines. an angle formed between the top parallel line and the transversal on the inner left side is marked 1. another angle formed between the bottom parallel line and the transversal on the inner right side is marked 2. which statement is true about ∠1 and ∠2? ∠1 and ∠2 are congruent because they are a pair of adjacent angles. ∠1 and ∠2 are complementary because they are a pair of adjacent angles. ∠1 and ∠2 are congruent because they are a pair of alternate interior angles. ∠1 and ∠2 are complementary because they are a pair of alternate interior angles.

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.