At 5:42, the time elapsed is 42 of the 60 minutes in the 5 o'clock  hour. Since the minute hand will make the compele revolution during that hour, at 5:42 it has traveled \(\dfrac{42}{60}=\dfrac{7}{10}\) of the full 360 degrees, or\(\dfrac{7}{10}.360^o=252^o\).

     The hour hand makes \(\dfrac{1}{12}\) of a complete revolution every hour. So, from 12:00 to 5:00, it travels \(\dfrac{5}{12}\) of the full 360 degrees, or \(\dfrac{5}{12}.360^o=150^o\)

     By 5:42, it has traveled another \(\dfrac{7}{10}\) of \(\dfrac{1}{12}\) of the full degrees, or \(\dfrac{7}{10}.\dfrac{1}{12}=21^o\), for a total of 150 + 21 = 171 degrees.

     The measure of the angle between the hands at 5;42 is 252 - 171 = 81 degree

\(ANSWER:81^o\)