Answer:
A beautiful elm tree, the oldest one in town, stood in front of her house.
Apposition: "the oldest one in town"
Purpose: Elaborates on the subject by providing additional information about the beautiful elm tree.
Spotting the car accident ahead, the bike messenger turned to the right and rode up onto the sidewalk to avoid a crash.
Diazeugma: "turned to the right and rode up onto the sidewalk"
Purpose: Establishes a clear sequence of events by connecting the two actions of turning and riding up onto the sidewalk.
My mother, the media center director at my school, showed us how to find obscure information on the Internet.
Apposition: "the media center director at my school"
Purpose: Identifies the subject by providing additional information about my mother's job title.
The bear backed away from the dumpster, snarled at the hiker, and lumbered into the woods.
Diazeugma: "backed away from the dumpster, snarled at the hiker, and lumbered into the woods"
Purpose: Adds dimension to the verb by describing three separate actions the bear took in quick succession.
Our new gymnasium, the gift of generous donors, was outfitted with all the latest exercise equipment.
Apposition: "the gift of generous donors"
Purpose: Elaborates on the subject by providing additional information about the new gymnasium and how it was obtained.
Step-by-step explanation:
Use at your own discretion.