Euclid's WorkshopBook I
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Proposition 27 of 48 • Theorem

If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

If a line crosses two other lines and the alternate interior angles are equal, then those two lines are parallel — they will never meet, no matter how far they are extended.

Before You Read

A single line crossing two others creates a set of angle pairs—including the 'alternate interior angles' on opposite sides of the transversal, between the two lines. Proposition 27 makes a striking claim: if those alternate angles are equal, the two lines are guaranteed to be parallel and will never meet no matter how far they extend. Can you see why equal alternate angles would force the lines apart forever?

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A B C D E F G

What Euclid Is Doing

Setup: A transversal EF crosses two lines AB and CD, creating alternate interior angles ∠AEF and ∠EFD. We are given that these alternate angles are equal, and we need to prove that AB and CD are parallel (Definition 23: they never meet).

Approach: Euclid uses proof by contradiction. Assume the lines are not parallel — they meet at some point G on one side. This creates a triangle EFG in which one of the equal alternate angles is an exterior angle and the other is an interior opposite angle of the same triangle. But Proposition 16 says an exterior angle is always strictly greater than any interior opposite angle — contradiction.

Conclusion: Assume AB and CD meet at a point G on the B, D side. Then triangle EFG is formed. In this triangle, ∠AEF is an exterior angle at vertex E (since AE and EG are on opposite sides of the transversal at E). The angle ∠EFG = ∠EFD is an interior angle of the triangle at F. By Proposition 16, the exterior angle ∠AEF must be strictly greater than the remote interior angle ∠EFD. But we are given ∠AEF = ∠EFD — contradiction. Similarly, assuming they meet on the other side gives a symmetric contradiction. Therefore AB and CD do not meet on either side, so they are parallel by Definition 23. ✓

Key Moves

  1. Given: transversal EF crosses lines AB and CD with ∠AEF = ∠EFD (alternate interior angles equal).
  2. Assume for contradiction that AB and CD are not parallel — they meet at some point G.
  3. Then EFG is a triangle. ∠AEF is an exterior angle at E, and ∠EFD (= ∠EFG) is a remote interior angle at F.
  4. By Proposition 16, exterior angle ∠AEF > interior angle ∠EFG = ∠EFD.
  5. But ∠AEF = ∠EFD by hypothesis — contradiction.
  6. The same argument works if we assume they meet on the other side.
  7. Therefore AB is parallel to CD (Definition 23) ✓

Try It Yourself

Draw two lines and a transversal. Measure the alternate interior angles. If they are equal, the lines should be parallel—verify by extending them far and checking that the gap stays constant. Now deliberately make the alternate angles slightly unequal and extend the lines: do they converge on one side, exactly as the inequality predicts?

Proof Challenge

Available Justifications

1.

Assume lines AB and CD are NOT parallel — they meet at point G

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2.

Then A, B, G (or C, D, G) form a triangle

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3.

∠AEF is an exterior angle of this triangle, so ∠AEF > ∠EFD

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4.

But ∠AEF = ∠EFD (given as alternate angles) — contradiction. So lines are parallel.

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 27.

  • ✓Lesson Plan (prop-27-lesson-plan.pdf)
  • ✓Student Worksheet (prop-27-worksheet.pdf)
  • ✓Answer Key (prop-27-answer-key.pdf)
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Why It Matters

This is Euclid's first proposition about parallel lines, and the first sufficient condition for parallelism. It opens the door to the theory of parallels that dominates the second half of Book I. Crucially, it does not use the parallel postulate — it works in absolute geometry.

Modern connection: Alternate interior angles are one of the first things taught in any geometry course. Architects and engineers use this criterion constantly: to verify that structural beams are parallel, check that a cross-brace makes equal angles on both sides. The principle also underlies the Z-angle rule used in technical drawing.

Historical note: Proposition 27 is remarkable for what it does not need: the parallel postulate (Postulate 5). Euclid carefully separates results that can be proved without Postulate 5 (Props 27-28) from those that require it (Prop 29 onward). This distinction became the foundation of non-Euclidean geometry two millennia later.

Discussion Questions

  • The proof uses Proposition 16 (exterior angle theorem). Why is this the right tool? Could you use any other proposition instead?
  • Proposition 27 proves: equal alternate angles imply parallel lines. The converse — parallel lines imply equal alternate angles — is Proposition 29. Why can Euclid not prove the converse without the parallel postulate?
  • In hyperbolic (non-Euclidean) geometry, Proposition 27 still holds but Proposition 29 does not. What does this tell you about the logical relationship between the two statements?
Euclid's Original Proof
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

For let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another;

I say that AB is parallel to CD.

For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.

Let them be produced and meet, in the direction of B, D, at G.

Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG:

which is impossible. [I.16]

Therefore AB, CD when produced will not meet in the direction of B, D.

Similarly it can be proved that neither will they meet towards A, C.

But straight lines which do not meet in either direction are parallel; [Def. 23]

therefore AB is parallel to CD.

Therefore etc.

Q.E.D.

What's Next

Proposition 27 gives one way to prove lines parallel using alternate interior angles. But there are other useful angle relationships created by a transversal—corresponding angles and co-interior (same-side) angles. Proposition 28 covers both of these additional criteria, giving the complete toolkit for detecting parallel lines from angle measurements alone.