The straight lines joining equal and parallel straight lines [at the extremities which are] in the same directions [respectively] are themselves also equal and parallel.
[Desmos graphs can be found here]
Let AB and CD be equal and parallel, and let the straight lines AC and BD join them at the extremities in the same directions as shown.
We will prove that AC and BD are equal and parallel.
Let BC be joined, creating triangles ABC and BCD.
BC is a straight line that falls on the parallel lines AB and CD, so the alternate angles ABC and BCD are equal by Proposition 29 (a straight line that falls on two parallel lines makes alternate angles that are equal). Side AB is equal to side CD by hypothesis, and side BC is common to the triangles ABC and BCD.
So, by Proposition 4 (side-angle-side), the base AC is equal to the base BD and triangle ABC is equal to the triangle DCB. Therefore the angle ACB is equal to the angle CBD.
Since BC falls on the two straight lines AC and BD and the alternate angles ACB and CBD are equal, then by Proposition 27 (if a straight line that falls on two straight lines makes the alternate angles equal then the two lines are parallel to each other) we have that AC is parallel to BD.
So, we have proved that AC is equal to BD, and AC is parallel to BD.
Q.E.D.
ABCD is a parallelogram: opposite sides are equal and parallel. In the next proposition we will prove that the opposite sides and opposite angles in a parallelogram are equal – George