A middle school principal has 80 keys on her key chain to distribute to staff on the first day of school. Of these 80 keys, 60 open classroom doors, 40 open the door to the teachers' lounge, and 30 open classroom doors and the teachers' lounge. Using this information, answer each of the following questions.

Let CCC be the event that a randomly selected key opens a classroom and TTT be the event that a randomly selected key opens the teachers' lounge.

What is P(C)P(C)P, left parenthesis, C, right parenthesis, the probability that a key opens a classroom?

Which of the following is a justification used while proving the similarity of triangles lom and mon

Which of the following is the shape of a cross section of the figure shown below

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?