Euclid's WorkshopBook I
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Proposition 23 of 48 • Construction

To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it.

Given an angle and a line with a marked point, copy that exact angle onto the line at that point.

Before You Read

You can copy a length to a new location (Proposition 2) and build a triangle from given lengths (Proposition 22). Now here is the next challenge: given an angle sitting somewhere on the page, how do you reproduce that exact same opening at a completely different point on a completely different line—using only compass and straightedge?

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C D E A B F G

What Euclid Is Doing

Setup: We are given a rectilinear angle ∠DCE and a straight line AB with a point A on it. We must construct an angle at A on line AB that is equal to ∠DCE.

Approach: Euclid's strategy is to convert the angle-copying problem into a triangle-copying problem. He picks arbitrary points D and E on the rays of the given angle, forming triangle CDE. Then he constructs a triangle with the same three side lengths on ray AB using Proposition 22. Since the triangles have equal sides, SSS congruence (Prop 8) guarantees the angles match.

Conclusion: Take arbitrary points D on ray CD and E on ray CE, forming triangle CDE (Postulate 1). By Proposition 22, construct triangle AFG on line AB at point A such that AF = CD, AG = CE, and FG = DE. By SSS congruence (Proposition 8), triangle AFG is congruent to triangle CDE. Therefore ∠FAG = ∠DCE. The angle has been copied to line AB at point A. ✓

Key Moves

  1. Take arbitrary points D, E on the two rays of the given angle ∠DCE
  2. Join DE to form triangle CDE (Postulate 1)
  3. On line AB at point A, construct triangle AFG with AF = CD, AG = CE, FG = DE (Proposition 22)
  4. By SSS congruence (Proposition 8): triangle AFG ≅ triangle CDE
  5. Therefore ∠FAG = ∠DCE — the angle has been copied ✓

Try It Yourself

Draw an acute angle DCE on your paper and a separate line AB with a marked point A. Try to construct the same angle at A without using a protractor. Euclid's insight is to 'solidify' the angle as a triangle: pick two points on the angle's rays, copy those three side lengths to a new triangle based at A, and let SSS congruence do the rest.

Proof Challenge

Available Justifications

1.

Take points D and E on the two rays of the given angle, forming triangle CDE

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2.

On line AB at point A, construct triangle AFG with AF = CD, AG = CE, FG = DE

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3.

Triangles CDE and AFG have three pairs of equal sides

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4.

By SSS, ∠FAG = ∠DCE — the angle has been copied

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 23.

  • ✓Lesson Plan (prop-23-lesson-plan.pdf)
  • ✓Student Worksheet (prop-23-worksheet.pdf)
  • ✓Answer Key (prop-23-answer-key.pdf)
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Why It Matters

Angle copying is one of the most fundamental compass-and-straightedge operations. Without it, we could not transfer geometric relationships from one location to another. This construction is used immediately in Proposition 23 itself becomes a workhorse tool for constructing parallel lines (Prop 31) and many later constructions.

Modern connection: Angle copying is built into every CAD (computer-aided design) system and geometry software. When an architect says 'replicate this roof pitch on the other wing,' they are performing Prop 23. In robotics, copying angles is essential for programming joint rotations that replicate observed movements.

Historical note: This construction elegantly reduces an angle problem to a triangle problem. Euclid could have tried to copy the angle directly with arcs, but instead he materializes the angle as a rigid triangle and rebuilds it elsewhere. This 'encode as triangle, reconstruct from sides' strategy is a hallmark of Greek geometric thinking.

Discussion Questions

  • Why does Euclid form a triangle from the angle rather than trying to copy the angle directly with arcs? What advantage does the triangle approach give?
  • The choice of points D and E on the rays is arbitrary. Does the size of triangle CDE affect the result? Why or why not?
  • This construction depends on Proposition 22 (triangle from three segments) and Proposition 8 (SSS). Could you copy an angle without SSS congruence?
Euclid's Original Proof
On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

Let AB be the given straight line, A the point on it, and the angle DCE the given rectilineal angle;

thus it is required to construct on the given straight line AB, and at the point A on it, a rectilineal angle equal to the given rectilineal angle DCE.

On the straight lines CD, CE respectively let the points D, E be taken at random;

let DE be joined, [Post. 1]

and out of three straight lines which are equal to the three straight lines CD, DE, CE let the triangle AFG be constructed in such a way that CD is equal to AF, CE to AG, and DE to FG. [I.22]

Then, since the two sides DC, CE are equal to the two sides FA, AG respectively, and the base DE is equal to the base FG, the angle DCE is equal to the angle FAG. [I.8]

Therefore on the given straight line AB, and at the point A on it, the rectilineal angle FAG has been constructed equal to the given rectilineal angle DCE.

(Being) what it was required to do.

What's Next

With angle-copying in hand, Euclid now has the key tool he needs for a more dramatic result: comparing triangles with two equal pairs of sides but different included angles. Proposition 24—the hinge theorem—asks what happens to the opposite side when you open or close that angle like a hinge, and it uses Proposition 23 at a critical moment in the proof.