Euclid's WorkshopBook I
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Proposition 28 of 48 • Theorem

If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another.

Two more ways to prove lines are parallel using a transversal: (1) if a corresponding angle (exterior on one side) equals the interior angle on the same side of the other line, or (2) if the two interior angles on the same side (co-interior angles) add up to two right angles (180 degrees).

Before You Read

Proposition 27 showed that equal alternate interior angles imply parallelism. But in practice there are other angle pairs worth checking: the corresponding angle on the same side, or the two interior angles on the same side added together. Proposition 28 says these work too: a transversal that makes equal corresponding angles, or co-interior angles summing to two right angles (180°), guarantees the lines are parallel.

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A B C D G H E F

What Euclid Is Doing

Setup: A transversal GH crosses two lines AB and CD. We consider two conditions: (1) a corresponding angle equals the interior angle on the same side (∠EGB = ∠GHD), or (2) the co-interior angles are supplementary (∠BGH + ∠GHD = two right angles). Either condition should force the lines to be parallel.

Approach: Euclid reduces both conditions to the already-proved Proposition 27 (equal alternate angles imply parallel). For condition (1), vertical angles (Prop 15) convert equal corresponding angles into equal alternate angles. For condition (2), supplementary angle properties (Prop 13) do the same conversion.

Conclusion: Condition (1): ∠EGB = ∠GHD (corresponding angles equal). But ∠EGB = ∠AGH by vertical angles (Prop 15). So ∠AGH = ∠GHD — these are alternate interior angles. By Proposition 27, AB is parallel to CD. Condition (2): ∠BGH + ∠GHD = two right angles. But ∠AGH + ∠BGH = two right angles (Prop 13, supplementary angles on a straight line). So ∠AGH + ∠BGH = ∠BGH + ∠GHD. Subtract ∠BGH from both sides (Common Notion 3): ∠AGH = ∠GHD. Again, these are alternate interior angles. By Proposition 27, AB is parallel to CD. ✓

Key Moves

  1. Condition (1): Given ∠EGB = ∠GHD (corresponding angles equal).
  2. ∠EGB = ∠AGH by vertical angles (Proposition 15).
  3. So ∠AGH = ∠GHD — alternate interior angles are equal.
  4. By Proposition 27, AB is parallel to CD ✓
  5. Condition (2): Given ∠BGH + ∠GHD = two right angles (co-interior angles supplementary).
  6. ∠AGH + ∠BGH = two right angles (Proposition 13, angles on a straight line).
  7. So ∠AGH = ∠GHD (subtract ∠BGH from both equal sums).
  8. By Proposition 27, AB is parallel to CD ✓

Try It Yourself

Draw two clearly parallel lines and a transversal crossing both. Label all eight angles formed. Identify a pair of corresponding angles (one at each crossing, same position relative to the transversal) and measure them—they should be equal. Now add any two co-interior angles on the same side of the transversal. Do they sum to 180°? Try the same experiment with non-parallel lines and notice what changes.

Proof Challenge

Available Justifications

1.

Given: ∠EGB = ∠GHD (corresponding angles equal)

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2.

∠EGB = ∠AGH (vertical angles)

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3.

So ∠AGH = ∠GHD (alternate interior angles equal)

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4.

By Prop 27, lines AB and CD are parallel

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5.

Case 2: Given ∠BGH + ∠GHD = two right angles (co-interior angles supplementary)

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6.

∠AGH + ∠BGH = two right angles (angles on a straight line)

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7.

So ∠AGH + ∠BGH = ∠BGH + ∠GHD, therefore ∠AGH = ∠GHD (alternate interior angles equal)

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8.

By Prop 27, lines AB and CD are parallel

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 28.

  • ✓Lesson Plan (prop-28-lesson-plan.pdf)
  • ✓Student Worksheet (prop-28-worksheet.pdf)
  • ✓Answer Key (prop-28-answer-key.pdf)
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Why It Matters

This proposition gives two additional and very practical criteria for parallelism. In combination with Proposition 27, we now have three ways to detect parallel lines using a transversal: equal alternate angles, equal corresponding angles, or supplementary co-interior angles. These are the three standard parallelism tests used in all of geometry.

Modern connection: The corresponding angles test is perhaps the most intuitive of the three and is widely used in construction: a builder checks parallel framing members by verifying that a cross-piece makes the same angle on both sides. The co-interior (same-side interior) angle test is equally important in civil engineering when verifying that road grades or rail lines remain parallel.

Historical note: Like Proposition 27, this proposition does not use the parallel postulate. Euclid derives it purely from the properties of supplementary angles and vertical angles, reducing everything to Prop 27. This careful logical economy — reducing new results to previously proved ones — is a hallmark of Euclid's method.

Discussion Questions

  • Both conditions in this proposition reduce to Proposition 27 through simple angle manipulations. Why do you think Euclid stated them as a separate proposition rather than just noting them as corollaries?
  • The proof uses both Proposition 13 (supplementary angles) and Proposition 15 (vertical angles). How are these two propositions related to each other?
  • Can you think of a fourth condition involving angles at a transversal that would also guarantee parallelism? Would it reduce to one of the three known conditions?
Euclid's Original Proof
If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, 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 exterior angle EGB equal to the interior and opposite angle GHD, or the interior angles on the same side, namely BGH, GHD, equal to two right angles;

I say that AB is parallel to CD.

For, since the angle EGB is equal to the angle GHD,

while the angle EGB is equal to the angle AGH, [I.15]

the angle AGH is also equal to the angle GHD; [C.N. 1]

and they are alternate;

therefore AB is parallel to CD. [I.27]

Again, since the angles BGH, GHD are equal to two right angles,

and the angles AGH, BGH are also equal to two right angles, [I.13]

the angles AGH, BGH are equal to the angles BGH, GHD.

Let the angle BGH be subtracted from each;

therefore the remaining angle AGH is equal to the remaining angle GHD; [C.N. 3]

and they are alternate;

therefore AB is parallel to CD. [I.27]

Therefore etc.

Q.E.D.

What's Next

Propositions 27 and 28 both go in one direction: if certain angle conditions hold, then the lines are parallel. Proposition 29 reverses the arrow—starting from the assumption that two lines are parallel, it derives all three angle relationships at once. Crucially, this reversal is where Euclid finally invokes the parallel postulate for the first time in the entire book.