Angles 1 and 2 are vertical angles, as are angles 3 and 4. This has nothing to do with their arrangement one on top of the other (vertically). Rather it has to do with them being on opposite sides of the point where lines intersect. It is a defnition (vocabulary) thing. The "vertical angle theorem" tells you vertical angles are congruent.
Here, you have angles that are equal to angles that are equal and you're trying to show they're equal to each other. It is done using two different properties of equality: substitution and the transitive property. The use of these is a matter of choice by the author of the proof. Your role is to understand what they are doing.
The fill-ins are shown in bold text below.
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By the vertical angle theorem, ∠2 ≅ ∠1. Therefore ∠2 ≅ ∠4 by the substitution property.
By the vertical angle theorem, ∠4 ≅ ∠3. So ∠2 ≅ ∠3 by the transitive property of congruence.
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The substitution property says I can subtitute anything for its equal. If A=B and B=C, then I can put A where B is in the second equation to get A=C.
The transitive property says that if two things are equal to the same thing, they are equal to each other. If A=B=C, then A=C.
Here, either of these properties could be used exclusively, but the author of the proof chose to mix them.