Words starting with "zig":
* Ziggurat (a pyramid-shaped temple)
* Zigzagging (moving in a zigzag pattern)
* Zigzagged (past tense of zigzagging)
Words starting with "zi":
* Zip (to close quickly)
* Zipper (a fastening device)
* Zillion (a very large number)
Let me know if you'd like more examples!