Euclid's WorkshopBook I
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Proposition 15 of 48 • Theorem

If two straight lines cut one another, then they make the vertical angles equal to one another.

When two straight lines cross, the opposite (vertical) angles are equal.

Before You Read

Two straight lines cross each other, forming four angles at the intersection. You probably assume the two opposite angles (the ones that share only the crossing point, not a side) are equal—but can you prove it? It seems obvious, yet 'obvious' is not a proof.

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A B C D E

What Euclid Is Doing

Setup: Two straight lines AB and CD cross at point E, creating four angles. The 'vertical' (or 'vertically opposite') angles are the ones across from each other: ∠AEC is opposite ∠DEB, and ∠AED is opposite ∠CEB. We want to prove each pair is equal.

Approach: The proof is elegant and short. Since AEB is a straight line, ∠AEC + ∠AED = two right angles (by Prop 13). Since CED is also a straight line, ∠AED + ∠DEB = two right angles (by Prop 13). Both expressions equal two right angles, so they equal each other.

Conclusion: Here it is: Line AEB is straight, so ∠AEC + ∠AED = two right angles (Prop 13). Line CED is straight, so ∠AED + ∠DEB = two right angles (Prop 13). Both sums equal two right angles, so ∠AEC + ∠AED = ∠AED + ∠DEB (Common Notion 1). Subtract ∠AED from both sides (Common Notion 3): ∠AEC = ∠DEB. By the same reasoning, ∠AED = ∠CEB. Vertical angles are equal. ✓

Key Moves

  1. AEB is a straight line, so ∠AEC + ∠AED = two right angles (Proposition 13)
  2. CED is a straight line, so ∠AED + ∠DEB = two right angles (Proposition 13)
  3. Both sums equal two right angles, so they equal each other (Common Notion 1)
  4. ∠AEC + ∠AED = ∠AED + ∠DEB
  5. Subtract ∠AED from both sides (Common Notion 3)
  6. Therefore ∠AEC = ∠DEB — the first pair of vertical angles are equal
  7. Similarly, ∠AED = ∠CEB — the second pair of vertical angles are equal ✓

Try It Yourself

Draw two straight lines crossing at any angle you choose. Measure all four angles at the intersection. Which pairs are equal? Now try it with lines crossing at 30 degrees, 60 degrees, and 80 degrees. Do the vertical angle pairs always match? Can you construct an argument using Proposition 13 as your tool?

Proof Challenge

Available Justifications

1.

Lines AB and CD cross at E; AE stands on line CD

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2.

∠AEC + ∠AED = two right angles

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3.

∠AED + ∠DEB = two right angles

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4.

Since ∠AEC + ∠AED = ∠AED + ∠DEB, therefore ∠AEC = ∠DEB

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5.

∠AED + ∠AEC = two right angles (BE stands on line CD)

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6.

∠AEC + ∠CEB = two right angles (AE stands on line CD)

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7.

Since ∠AED + ∠AEC = ∠AEC + ∠CEB, therefore ∠AED = ∠CEB. Both pairs of vertical angles are equal.

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 15.

  • ✓Lesson Plan (prop-15-lesson-plan.pdf)
  • ✓Student Worksheet (prop-15-worksheet.pdf)
  • ✓Answer Key (prop-15-answer-key.pdf)
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Why It Matters

The vertical angles theorem is one of the most frequently used facts in geometry. Whenever two lines cross, you immediately know that opposite angles are equal. This shows up constantly in proofs about parallel lines, triangles, and polygons.

Modern connection: Vertical angles appear in scissors, X-shaped supports, road intersections, and anywhere two linear elements cross. In physics, when light reflects off a surface, the angle of incidence equals the angle of reflection—a consequence of the same symmetry principle.

Historical note: Euclid notes that Thales of Miletus (c. 624-546 BC) was credited with discovering this theorem, making it one of the oldest known geometric results. The proof is beautifully simple—just two applications of Proposition 13 and one subtraction.

Discussion Questions

  • Why are they called 'vertical' angles when they're not necessarily up-and-down?
  • Can you prove vertical angles are equal without using Proposition 13?
  • If two lines cross and one pair of vertical angles is 60°, what are all four angles? How do you know?
Euclid's Original Proof
If two straight lines cut one another, they make the vertical angles equal to one another.

For let the straight lines AB, CD cut one another at the point E;

I say that the angle AEC is equal to the angle DEB, and the angle CEB to the angle AED.

For, since the straight line AE stands on the straight line CD, making the angles CEA, AED,
the angles CEA, AED are equal to two right angles. [I.13]

Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB,
the angles AED, DEB are equal to two right angles. [I.13]

But the angles CEA, AED were also proved equal to two right angles;
therefore the angles CEA, AED are equal to the angles AED, DEB. [Post. 4 and C.N. 1]

Let the angle AED be subtracted from each;
therefore the remaining angle CEA is equal to the remaining angle BED. [C.N. 3]

Similarly it can be proved that the angles CEB, DEA are also equal.

Therefore if two straight lines cut one another, they make the vertical angles equal to one another.

Q.E.D.

What's Next

Vertical angles give us a powerful equality we can reach for anytime two lines cross. Next Euclid turns to triangles and their exterior angles: Proposition 16 proves that the exterior angle of a triangle is strictly greater than either non-adjacent interior angle—a result with surprising consequences for parallel lines.