I see you've uploaded a geometry exercise sheet. Let’s start by tackling one problem at a time. I’ll begin with the first problem and guide you through it step by step.

Problem 1:

Express the decimal fraction of the angle 12.33° in minutes and seconds.

Step-by-step solution:

1. Understand the problem: The angle given is 12.33°. We need to convert the decimal part (0.33°) into minutes and seconds.

2. Conversion of decimal degrees to minutes:

   • One degree is equal to 60 minutes.
   • To find the number of minutes in 0.33°, we multiply:
     $0.33 \times 60 = 19.8 \text{ minutes}$
   • This gives us 19 minutes, and the decimal part (0.8 minutes) needs to be converted to seconds.

3. Convert the decimal part of minutes to seconds:

   • One minute is equal to 60 seconds.
   • Multiply 0.8 minutes by 60 to convert to seconds: $0.8 \times 60 = 48 \text{ seconds}$

4. Final result:

   • The decimal part of 12.33° is equivalent to 19 minutes and 48 seconds.
   • Therefore, 12.33° = 12° 19' 48".

Final Answer:

12.33° = 12° 19' 48"

Would you like to go over the next problem or need further clarifications on this one?

Here are 5 related questions you might find helpful:

1. How do we convert from minutes and seconds back to decimal degrees?
2. What is the general rule for converting decimals into minutes and seconds for any angle?
3. How can we handle angles greater than 360° in these types of conversions?
4. What is the difference between degrees, minutes, and seconds vs decimal degrees?
5. How do we convert a fraction of a degree into minutes when dealing with angles in geometry problems?

Tip: When dealing with angles, keep in mind that 1° = 60 minutes and 1 minute = 60 seconds, which can help you quickly convert between the different units.