The missing pieces are ...
3. Vertical angles are congruent.
6. AB║CD
10. BC║DA
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The Alternate Interior Angles theorem tells you alternate interior angles are congruent when they are formed by a transversal to parallel lines. The converse of that tells you the lines are parallel when the angles are congruent. Showing opposite sides are parallel is the point of the proof, as it allows you to invoke the definition of a parallelogram (the last step).