Plan for Proof: Show that △APQ≅△BPQ by the SSS Congruence Theorem. Then show that △APM≅△BPM using the SAS Congruence Theorem. Use corresponding parts of congruent triangles to show that ∠AMP and ∠BMP are right angles. STATEMENTS REASONS Response area 1. Given Response area 2. Reflexive Property of Congruence Response area 3. SSS Congruence Theorem Response area 4. Corresponding parts of congruent triangles are congruent. Response area 5. Reflexive Property of Congruence Response area 6. SAS Congruence Theorem Response area 7. Corresponding parts of congruent triangles are congruent. Response area 8. Definition of a linear pair Response area 9. Linear Pair Perpendicular Theorem Response area 10. Definition of perpendicular lines

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.