Perpendicular Proofs

What is the shortest distance between point C and Line AB? Round answers to the nearest hundredth. Prove that it's the shortest distance possible.

What are the measures of angles 1, 2, 3, and 4? Prove each measurement using theorems we've covered in class and proper geometric vocabulary.

Two lines, a and b, are perpendicular to line c. Line d is parallel to line c. The distance between lines a and b is x meters. The distance between lines c and d is y meters. What shape is formed by the segments between the points of intersection of the four lines? How do you know?

You are building a new room in a house (you may choose whether this is your house, you are a contractor, building a model for a project, or creating a set peace for a TV show). You need the walls to all meet at right angles, but someone has ruined you square (a measuring tool for making right angles). Someone is coming to inspect your work today and you're not sure when they'll arrive. So, there's no time in the moment to run to the store. How could you ensure that the wall you're about to build is in fact perpendicular to the previous wall?