Answer:
Here are the correct ways to join the sentences in each pair:
1. The bird is diving into the water. It appears graceful.
Correct: **Diving into the water, the bird appears graceful.**
2. You went on vacation. You forgot your everyday worries.
Correct: **Going on vacation, you forgot your everyday worries.**
3. Judy walked through the mall. She looked for the beauty salon.
Correct: **Walking through the mall, Judy looked for the beauty salon.**
4. Harrison seemed nervous before the audience. He sang a love ballad.
Correct: **Seeming nervous before the audience, Harrison sang a love ballad.**