# Parallelism implies Equal Alternate Angles, Corresponding Angles, and Supplementary Interior Angles

## Theorem

In the words of Euclid:

*A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.*

(*The Elements*: Book $\text{I}$: Proposition $29$)

Worded in more contemporary language:

### Parallelism implies Equal Alternate Angles

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

### Parallelism implies Equal Corresponding Angles

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

### Parallelism implies Supplementary Interior Angles

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

## Historical Note

This proof is Proposition $29$ of Book $\text{I}$ of Euclid's *The Elements*.

The first part is the converse of Proposition $27$: Equal Alternate Angles implies Parallel Lines, and the second and third parts form the converse of Proposition $28$: Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines.

This is the first proposition of *The Elements* to make use of Euclid's fifth postulate.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions - 1968: M.N. Aref and William Wernick:
*Problems & Solutions in Euclidean Geometry*... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.6$