(How) Reference one line of an ENTIRE shape to be vertical?

Misunderstood the request.

You are welcome, Tyler.

Use the rotate tool and turn on Smart Guides and then like this: https://www.dropbox.com/s/tyfy7801g3yiv4w/vertical.mov?dl=0

Tyler,

The Line Segment Tool and Smart Guides are your friends.

0) If not already, select everything to be rotated and Ctrl/Cld+G to Group;

1) ShiftClickDrag vertically upwards from the lower end Anchor Point of the path segment to be vertical past the upper end of the segment;

2) Select the Rotate Tool, and Click the lower coinciding end Anchor Point of the line and the path segment to set it as the centre;

3) Zoom in on the top end, then ClickDrag the top end Anchor Point of the path segment to snap to the vertical line, and you are there (Smart Guides say path when you are within snapping distance).

Edit: This works without your needing to know the angle.

If you know the rotation degree, you can select the rotation tool. In your example hold the alt key to move the rotation point to the upper right corner point (set snap to point in View). Release and then enter the rotation degree.