# Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram

### (Proposition 33 from Book 1 of Euclid's “Elements”)

Straight-lines joining equal and parallel (straight lines) on the same sides are themselves also equal and parallel. * Let $AB$ and $CD$ be equal and parallel (straight lines), and let the straight lines $AC$ and $BD$ join them on the same sides. * I say that $AC$ and $BD$ are also equal and parallel.

### Modern Formulation

Let $$\overline{AB}$$, $$\overline{CD}$$ be two equal parallel segments. Then the segments joining their adjacent endpoints, ($$\overline{AC}, \overline{BD}$$) are themselves parallel and equal in length: $$\overline{AC}=\overline{BD}$$ and $$\overline{AC}\parallel\overline{BD}$$. In other words, the quadrilateral $${ABCD}$$ forms a parallelogram.

Proofs: 1

Proofs: 1 2 3 4 5 6

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014