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Proposition 13 of 48 • Theorem

If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

When a line meets another line, the two angles formed on either side always add up to exactly 180 degrees (two right angles).

Before You Read

Imagine a stick leaning against a flat floor. It makes one angle on the left side and a different angle on the right. Can those two angles ever add up to something other than 180 degrees? Try tilting the stick more or less steeply in your mind—does the sum change?

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A B C D E

What Euclid Is Doing

Setup: Line CD meets line AB at point C, forming two angles: ∠DCA on the left and ∠DCB on the right. We want to show these two angles always add up to two right angles (180°).

Approach: There are two cases. If CD happens to be perpendicular to AB, then both angles are already right angles and we're done. If not, we construct a perpendicular CE to AB at C (using Prop 11) and use it as a reference to relate the angles.

Conclusion: Case 1: If ∠DCA = ∠DCB, they're both right angles by Definition 10, and two right angles is exactly what we need. Case 2: If they're unequal, draw CE perpendicular to AB (Prop 11). Now ∠ECA is a right angle and ∠ECB is a right angle. Notice that ∠DCA = ∠DCE + ∠ECA (the whole equals the sum of its parts). And ∠DCB + ∠DCE = ∠ECB (same reasoning). So ∠DCA + ∠DCB = (∠DCE + ∠ECA) + ∠DCB = ∠ECA + (∠DCE + ∠DCB) = ∠ECA + ∠ECB = right angle + right angle = two right angles. ✓

Key Moves

  1. Case 1: If ∠DCA = ∠DCB, both are right angles by Definition 10 — done
  2. Case 2: If ∠DCA ≠ ∠DCB, draw CE perpendicular to AB at C (Proposition 11)
  3. ∠ECA = right angle and ∠ECB = right angle (by construction)
  4. ∠DCA = ∠DCE + ∠ECA (Common Notion 2: whole = sum of parts)
  5. ∠DCB + ∠DCE = ∠ECB (Common Notion 2)
  6. ∠DCA + ∠DCB = ∠ECA + ∠ECB = two right angles (Common Notion 1) ✓

Try It Yourself

Draw a straight line and then draw another line meeting it from above at any angle you like. Measure both angles formed at the meeting point with a protractor. Add them together. Now try it with five different angles—steep, shallow, exactly 90 degrees. What do you always get?

Proof Challenge

Available Justifications

1.

If ∠DCA and ∠DCB are both right angles, we are done

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2.

Otherwise, draw CE perpendicular to AB at point C

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3.

∠DCA = ∠DCE + ∠ECA

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4.

∠DCA + ∠DCB = ∠DCE + ∠ECA + ∠DCB = ∠ECA + (∠DCE + ∠DCB)

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5.

∠ECA + ∠ECB are two right angles; ∠DCE + ∠DCB = ∠ECB; so the sum equals two right angles

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 13.

  • ✓Lesson Plan (prop-13-lesson-plan.pdf)
  • ✓Student Worksheet (prop-13-worksheet.pdf)
  • ✓Answer Key (prop-13-answer-key.pdf)
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Why It Matters

This is the supplementary angle theorem—one of the most heavily used results in all of geometry. Whenever two angles sit on opposite sides of a line meeting at a point, they sum to 180°. This fact is the backbone of angle reasoning throughout the Elements.

Modern connection: Supplementary angles appear everywhere: in physics (action-reaction forces along a line), in optics (angle of incidence and refraction), and in everyday design. Any time you fold a piece of paper, the angles on either side of the fold are supplementary.

Historical note: This proposition marks a shift in Book I. The first ten propositions were constructions—building things with compass and straightedge. Now Euclid begins proving theorems about angles, starting the transition toward the parallel postulate and triangle angle sums.

Discussion Questions

  • Why does Euclid handle the case where both angles are already right angles separately?
  • How does the auxiliary perpendicular CE help break down the angles?
  • Can you think of real-world situations where supplementary angles naturally arise?
Euclid's Original Proof
If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.

For let any straight line AB set up on the straight line CD make the angles CBA, ABD;

I say that the angles CBA, ABD are either two right angles or equal to two right angles.

Now, if the angle CBA is equal to the angle ABD, they are two right angles. [Def. 10]

But, if not, let BE be drawn from the point B at right angles to CD; [I.11]
therefore the angles CBE, EBD are two right angles.

Then, since the angle CBE is equal to the two angles CBA, ABE,
let the angle EBD be added to each; [C.N. 2]
therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD.

Again, since the angle DBA is equal to the two angles DBE, EBA,
let the angle ABC be added to each; [C.N. 2]
therefore the angles DBA, ABC are equal to the three angles DBE, EBA, ABC.

But the angles CBE, EBD were also proved equal to the same three angles;
and things which are equal to the same thing are also equal to one another; [C.N. 1]
therefore the angles CBE, EBD are also equal to the angles DBA, ABC.

But the angles CBE, EBD are two right angles;
therefore the angles DBA, ABC are also equal to two right angles.

Therefore if a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.

Q.E.D.

What's Next

Proposition 13 proves that a line standing on another makes angles summing to two right angles. Proposition 14 reverses the question: if you have two rays on opposite sides of a line making angles that sum to two right angles, must those two rays form a single straight line? That converse is far less obvious—and Euclid proves it by contradiction.