Euclid's WorkshopBook I
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Proposition 31 of 48 • Construction

To draw a straight line through a given point parallel to a given straight line.

Given any point not on a line, you can construct a new line through that point that is parallel to the original line, using the technique of copying an angle from Proposition 23.

Before You Read

You need to draw a line through a floating point in space that runs perfectly parallel to a given line—but you can't measure angles with a protractor, only copy them. How would you even begin?

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What Euclid Is Doing

Setup: We are given a straight line BC and a point A not on it. We must construct a line through A that is parallel to BC.

Approach: Euclid picks any point D on BC and draws the line AD. He then uses Proposition 23 to construct an angle ∠DAE at point A equal to the angle ∠ADC, placing it on the opposite side of AD so that the two angles are alternate interior angles. The line EAF through these points is the desired parallel.

Conclusion: Take any point D on BC and join AD (Postulate 1). At point A on line AD, construct ∠DAE equal to ∠ADC on the opposite side (Proposition 23). Produce EA to F (Postulate 2). Now the transversal AD crosses lines EF and BC, and the alternate interior angles ∠DAE and ∠ADC are equal by construction. By Proposition 27 (equal alternate angles imply parallel lines), EF is parallel to BC. The line EAF passes through A and is parallel to BC. ✓

Key Moves

  1. Given: point A and line BC. Take any point D on BC.
  2. Join AD (Postulate 1) to create a transversal.
  3. At point A, construct ∠DAE equal to ∠ADC on the opposite side of AD (Proposition 23).
  4. Produce EA to F (Postulate 2), forming the line EAF through A.
  5. The alternate interior angles ∠DAE and ∠ADC are equal by construction.
  6. By Proposition 27, EF is parallel to BC ✓

Try It Yourself

Mark a point A above a line BC on your paper, then pick any point D on BC, draw the transversal AD, and use a compass to copy the angle at D up to point A on the other side—the line you draw through A is your parallel. Try it with a steep transversal and a shallow one and compare.

Proof Challenge

Available Justifications

1.

Pick an arbitrary point D on the given line BC

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2.

Join the given point A to D

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3.

At point A on line AD, construct ∠DAE equal to ∠ADC (alternate angle)

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4.

Extend EA to F, giving the full line EAF through point A

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5.

∠DAE = ∠ADC (alternate angles), so EAF ∥ BC

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 31.

  • ✓Lesson Plan (prop-31-lesson-plan.pdf)
  • ✓Student Worksheet (prop-31-worksheet.pdf)
  • ✓Answer Key (prop-31-answer-key.pdf)
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Why It Matters

This is the basic construction for producing parallel lines — the parallel counterpart of Proposition 12 (dropping a perpendicular). It is used immediately in Proposition 32 to prove the angle sum theorem and is essential for all subsequent area propositions. Without it, we could prove things about parallels that already exist but could never create new ones.

Modern connection: Every technical drawing that requires parallel lines uses this principle. In CAD software, the 'offset' or 'parallel line' tool automates exactly this construction. In architecture, producing parallel walls, beams, and reference lines is one of the most common operations.

Historical note: This is a construction proposition (Q.E.F. — 'that which was to be done'), not a theorem (Q.E.D.). Euclid is careful to show that the tools of straightedge and compass suffice to produce parallel lines. The construction relies on Proposition 23 (copying an angle), which in turn relies on Proposition 22 (constructing a triangle from three given sides).

Discussion Questions

  • The construction requires choosing a point D on BC and drawing the transversal AD. Does the choice of D matter? Will different choices of D produce different parallel lines through A, or the same one?
  • Euclid uses Proposition 27 (alternate angles imply parallel) to verify the construction. Could he have used Proposition 28 instead? What would that version of the construction look like?
  • This construction depends only on Props 23 and 27, neither of which uses the parallel postulate. So the ability to construct parallels does not require Postulate 5. Why is this significant for non-Euclidean geometry?
Euclid's Original Proof
To draw a straight line through a given point parallel to a given straight line.

Let A be the given point, and BC the given straight line;

thus it is required to draw a straight line through the point A parallel to the straight line BC.

Let a point D be taken at random on BC, and let AD be joined; [Post. 1]

on the straight line DA, and at the point A on it, let the angle DAE be made equal to the angle ADC [I.23]; and let the straight line AF be produced in a straight line with EA. [Post. 2]

Then, since the straight line AD falling on the two straight lines BC, EF has made the alternate angles EAD, ADC equal to one another,

therefore EAF is parallel to BC. [I.27]

Therefore the straight line EAF has been drawn through the given point A parallel to the given straight line BC.

Q.E.F.

What's Next

Now that we can construct parallel lines through any point, we have everything needed to prove the most famous theorem in elementary geometry: that the three interior angles of any triangle sum to exactly two right angles. That's Proposition 32.