I'll help with number 5 since number 4 is a bit too small and squished to read.
Check out the attached image below for the full two-column proof. The added entries are in red.
The given statement is basically what the teacher shows at the top of the problem, which is the fact that angle 5 and angle 2 are congruent. You just repeat this statement.
The reason for statement 2 is that angle 2 and angle 4 are vertical angles. They are opposite angles formed by a pair of intersecting lines. This is the vertical angle theorem at work. We'll use the vertical angle theorem again for statement 4 when we say that angle 5 and angle 8 are the same measure.
The transitive property comes into play when we connect angles 2, 5, and 4 together, which helps us get line 3. We also connect angles 4, 5 and 8 together to get the last line, which is why the reason for statement 5 is "transitive property"