Three motorcyclists ride around a circular course starting at the same place and the same time. The first passes the starting point every ​min, the second every ​min, and the third every min. After how many minutes will all three pass the starting point again at the same​ time? Explain your reasoning.

What is the missing reason in the proof? segment addition congruent segments theorem transitive property of equality subtraction property of equality?

Tim has obtained a 3/27 balloon mortgage. after the initial period, he decided to refinance the balloon payment with a new 30-year mortgage. how many years will he be paying for his mortgage in total?

"curse these trig-loving pirates! " devora mutters as she looks at the map. then, after thinking for a moment, she walks back to the entrance of the secret cave, measuring 48 meters along the way. after performing a calculation, devora walks back to the empty treasure, faces the entrance, and turns a certain number of degrees to her left before walking 89 meters to the treasure. supposing that the angle at the cave entrance is acute and that devora's calculations were correct, how many degrees did she turn?