Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines

Part 1
Given two infinite straight lines which are cut by a transversal, if the corresponding angles are equal, then the lines are parallel.

Part 2
Given two infinite straight lines which are cut by a transversal, if the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Part 1
Let $$AB$$ and $$CD$$ be infinite straight lines, and let $$EF$$ be a transversal that cuts them. Let at least one pair of corresponding angles, WLOG $$\angle EGB$$ and $$\angle GHD$$, be equal.

$$\angle GHD = \angle EGB = \angle AGH$$ by the Vertical Angle Theorem.

Thus $$AB \parallel CD$$ by Equal Alternate Interior Angles Implies Parallel.

Part 2
Let $$AB$$ and $$CD$$ be infinite straight lines, and let $$EF$$ be a transversal that cuts them. Let at least one pair of interior angles on the same side of the transversal, WLOG $$\angle BGH$$ and $$\angle DHG$$ be supplementary, so by definition $$\angle DHG + \angle BGH$$ equals two right angles.

$$\angle AGH + \angle BGH$$ equals two right angles.

Then from Euclid's first and third common notion and Euclid's fourth postulate, $$\angle AGH = \angle DHG$$.

Finally, $$AB \parallel CD$$ by Equal Alternate Interior Angles Implies Parallel.