Euclid's WorkshopBook I
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Proposition 33 of 48 • Theorem

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

If two line segments are both equal in length and parallel, and you connect their endpoints on the same side, the connecting lines are also equal and parallel. This is the first step toward proving that parallelograms have nice properties.

Before You Read

Imagine two fence posts of equal height planted the same distance apart and perfectly upright—intuitively the tops should be as far apart as the bottoms, and the line joining them should be parallel to the ground. But can you prove it with only angles and triangles?

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What Euclid Is Doing

Setup: We have two line segments AB and CD that are equal in length and parallel. They are joined at corresponding ends: A to C and B to D ('in the same direction' — meaning AC and BD do not cross). We must prove that AC and BD are themselves equal and parallel.

Approach: Euclid draws the diagonal BC, creating two triangles ABC and BCD. Since AB is parallel to CD, the transversal BC produces equal alternate interior angles (Proposition 29). Combined with the shared side BC and the equal sides AB = CD, SAS congruence (Proposition 4) gives the result. The congruent triangles yield both the equal sides (AC = BD) and equal alternate angles that prove parallelism (Proposition 27).

Conclusion: Join BC (Postulate 1). AB is parallel to CD, and BC is a transversal. By Proposition 29, alternate interior angles ∠ABC = ∠BCD. Now in triangles ABC and DCB: AB = DC (given), BC = CB (common), and ∠ABC = ∠DCB (just proved). By SAS (Proposition 4), triangle ABC is congruent to triangle DCB. Therefore AC = DB (corresponding sides). Also ∠ACB = ∠DBC (corresponding angles). But these are alternate interior angles formed by transversal BC crossing lines AC and BD. By Proposition 27, AC is parallel to BD. ✓

Key Moves

  1. Given: AB = CD and AB is parallel to CD, joined 'in the same direction' by AC and BD.
  2. Draw diagonal BC (Postulate 1), creating triangles ABC and DCB.
  3. AB parallel to CD with transversal BC: alternate angles ∠ABC = ∠DCB (Proposition 29).
  4. In triangles ABC and DCB: AB = DC, BC = CB (common), ∠ABC = ∠DCB.
  5. By SAS (Proposition 4), triangle ABC is congruent to triangle DCB.
  6. Therefore AC = DB (corresponding sides) and ∠ACB = ∠DBC (corresponding angles).
  7. ∠ACB and ∠DBC are alternate interior angles for lines AC and BD with transversal BC.
  8. By Proposition 27, AC is parallel to BD ✓

Try It Yourself

Draw two equal parallel segments AB and CD on your paper, joining them on the same side with lines AC and BD. Draw the diagonal BC and see whether you can find two congruent triangles hiding inside the figure—the alternating angles on that diagonal are the key.

Proof Challenge

Available Justifications

1.

Given: AB = CD and AB ∥ CD.

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2.

Draw AC, BD, and diagonal BC.

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3.

∠ABC = ∠BCD (alternate angles, since AB ∥ CD with transversal BC)

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4.

In triangles ABC and DCB: AB = DC, BC = BC (common), ∠ABC = ∠DCB. So △ABC ≅ △DCB by SAS.

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5.

Therefore AC = DB and ∠ACB = ∠DBC (corresponding parts)

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6.

∠ACB = ∠DBC are alternate angles for lines AC and BD with transversal BC, so AC ∥ BD

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 33.

  • ✓Lesson Plan (prop-33-lesson-plan.pdf)
  • ✓Student Worksheet (prop-33-worksheet.pdf)
  • ✓Answer Key (prop-33-answer-key.pdf)
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Why It Matters

This proposition is the bridge between the theory of parallel lines and the theory of parallelograms. It shows that connecting equal, parallel segments 'in the same direction' automatically produces a parallelogram. This is the setup for Proposition 34, which fully characterizes parallelograms.

Modern connection: In vector geometry, this proposition says that if two vectors are equal (same magnitude and direction), then the line segments connecting their tails and tips are also equal and parallel. This is essentially the definition of vector equality — and it is the geometric foundation for the parallelogram law of vector addition used throughout physics.

Historical note: The phrase 'in the same direction' is crucial. If the joins crossed (connecting A to D and B to C), the result would not hold — you would get a crossed quadrilateral, not a parallelogram. Euclid is careful to specify the non-crossing condition, showing his awareness of this subtlety.

Discussion Questions

  • What goes wrong if the lines are joined 'in opposite directions' — that is, if AC and BD cross each other? Draw a picture and explain.
  • The proof uses a diagonal to create two congruent triangles. This 'diagonal argument' becomes a recurring technique. Why is the diagonal such a powerful tool for studying quadrilaterals?
  • This proposition establishes one direction: equal and parallel sides imply equal and parallel joins. Is the converse true? If AC = BD and AC is parallel to BD, must AB = CD and AB be parallel to CD?
Euclid's Original Proof
Straight lines which join the ends of equal and parallel straight lines (respectively) in the same directions are themselves also equal and parallel.

Let AB, CD be equal and parallel, and let the straight lines AC, BD join them at their extremities in the same directions;

I say that AC, BD are also equal and parallel.

Let BC be joined. [Post. 1]

Then, since AB is parallel to CD, and BC has fallen upon them, the alternate angles ABC, BCD are equal to one another. [I.29]

And, since AB is equal to CD, and BC is common,

the two sides AB, BC are equal to the two sides DC, CB;

and the angle ABC is equal to the angle BCD;

therefore the base AC is equal to the base BD,

and the triangle ABC is equal to the triangle DCB,

and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; [I.4]

therefore the angle ACB is equal to the angle CBD.

And, since the straight line BC falling on the two straight lines AC, BD has made the alternate angles equal to one another,

AC is parallel to BD. [I.27]

And it was also proved equal to it.

Therefore etc.

Q.E.D.

What's Next

Proposition 33 shows that connecting equal, parallel segments produces a parallelogram. The very next proposition, 34, unpacks everything that's true inside a parallelogram: opposite sides are equal, opposite angles are equal, and the diagonal cuts it into two equal triangles.