The choices are:

first part:
by the linear pair theorem
by the vertical angles theorem
because they are corresponding angles for parallel lines cut by a transversal

second part:
aeb
bed
ecd
eba

Both part of this problem uses the vertical angles theorem since we have two pairs of vertical angles. So you'll have "vertical angles theorem" for the first drop-down box.

The second answer is angle BED (which is the same as angle DEB). This is the other pair of vertical angles.

Vertical angles are the opposite angles formed when you cross two lines, rays, or line segments. 

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Ithink the answer is c just trying to others