Concept:
The verbs hang and hung are used when an object is suspended from a hook, peg, or support. The word hanged is generally reserved for execution by hanging.
Step 1: Understand the sentence.
The sentence talks about placing a shirt on a peg:
\[
\text{"He \_\_\_\_\_ his shirt on the peg."}
\]
Step 2: Evaluate the options.
• Hanged -- used mainly for execution.
• Hung -- correct past tense when referring to objects.
• Perched -- means sat or rested on something.
• Sailed -- related to travel by water.
Step 3: Choose the appropriate word.
Since a shirt is being suspended on a peg, the correct word is hung.
\[
\boxed{\text{Hung}}
\]