Euclid's WorkshopBook I
← Back to Propositions
Proposition 25 of 48 • Theorem

If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other.

If two triangles have two pairs of equal sides but one has a longer base, then the triangle with the longer base also has the larger included angle. (The converse of the hinge theorem.)

Before You Read

Proposition 24 showed that a wider angle between two fixed sides produces a longer base. Now flip the question: if you know one base is longer than the other, can you conclude that its angle must be wider? The answer is yes, but proving it requires a different strategy—ruling out the alternatives one by one rather than constructing anything new.

Browse All Foundations

All Foundations

Definitions (23)
Postulates (5)
Common Notions (5)
Browse All Propositions

All Propositions

Basic Constructions (5)
Triangle Fundamentals (5)
Perpendiculars & Angles (5)
Exterior Angles & Inequalities (5)
Interior Triangles & Angle Copying (5)
Parallel Lines (5)
Parallel Constructions & Parallelograms (5)
Area Theorems (5)
Area Applications (4)
Grand Finale (4)
A B C D E F longer shorter

What Euclid Is Doing

Setup: We have two triangles ABC and DEF where AB = DE and AC = DF, but BC > EF. We want to prove that ∠BAC > ∠EDF.

Approach: Euclid uses proof by exhaustion, the same technique as Proposition 19. There are exactly three possibilities for how ∠BAC compares to ∠EDF. He eliminates two using earlier propositions, leaving only the desired conclusion.

Conclusion: There are three cases: ∠BAC = ∠EDF, ∠BAC < ∠EDF, or ∠BAC > ∠EDF. Case 1: If ∠BAC = ∠EDF, then by SAS (Proposition 4), the triangles are congruent, so BC = EF. But we are given BC > EF — contradiction. Case 2: If ∠BAC < ∠EDF, then by the hinge theorem (Proposition 24), BC < EF. But we are given BC > EF — contradiction. Case 3: ∠BAC > ∠EDF. This is the only remaining possibility, so it must hold. Therefore ∠BAC > ∠EDF. ✓

Key Moves

  1. Given AB = DE, AC = DF, and BC > EF. We want to show ∠BAC > ∠EDF.
  2. Case 1: Suppose ∠BAC = ∠EDF. Then by SAS (Proposition 4), BC = EF. Contradiction with BC > EF.
  3. Case 2: Suppose ∠BAC < ∠EDF. Then by the hinge theorem (Proposition 24), BC < EF. Contradiction with BC > EF.
  4. Case 3: ∠BAC > ∠EDF is the only remaining possibility.
  5. Therefore ∠BAC > ∠EDF ✓

Try It Yourself

Draw two triangles with the same two side lengths (say 5 cm and 8 cm) but make the bases clearly different—one about 4 cm and one about 9 cm. Measure the included angles. Which triangle has the larger angle at the vertex between the two equal sides? Try to construct a counterexample where the longer base has a smaller angle—Proposition 25 says you cannot.

Proof Challenge

Available Justifications

1.

Assume ∠BAC = ∠EDF

Drag justification
2.

Then BC = EF by SAS — contradicts BC > EF

Drag justification
3.

Assume ∠BAC < ∠EDF

Drag justification
4.

Then BC < EF by Prop 24 — contradicts BC > EF. So ∠BAC > ∠EDF.

Drag justification
0 of 4 steps completed

Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 25.

  • ✓Lesson Plan (prop-25-lesson-plan.pdf)
  • ✓Student Worksheet (prop-25-worksheet.pdf)
  • ✓Answer Key (prop-25-answer-key.pdf)
Browse Curriculum Bundles

Why It Matters

This completes the hinge theorem pair: Prop 24 says a wider angle implies a longer base, and Prop 25 says a longer base implies a wider angle. Together they establish a perfect two-way correspondence between the included angle and the opposite side (when the other two sides are fixed). This mirrors the Prop 18/19 pair for single triangles.

Modern connection: In mechanical engineering, the converse hinge theorem lets you infer the angular opening of a mechanism from a length measurement. If you measure that a scissor lift has extended further, you know the hinge angle has increased. This principle is used in structural monitoring, where sensors measure distances and software infers angular changes.

Historical note: The proof by exhaustion technique used here is identical in structure to Proposition 19 (and ultimately derives from the logical principle of trichotomy). Euclid recognized this as a powerful proof pattern: to prove A > B, show that A = B leads to contradiction and A < B leads to contradiction.

Discussion Questions

  • This proof has the same structure as Proposition 19. What is that proof technique called, and why does it work?
  • Could you prove this proposition directly, without eliminating cases? What would a direct proof look like?
  • Propositions 24 and 25 together say: with two sides fixed, the angle and opposite side increase together. How does the law of cosines express this same relationship algebraically?
Euclid's Original Proof
If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.

Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF; and let the base BC be greater than the base EF;

I say that the angle BAC is also greater than the angle EDF.

For, if not, it is either equal to it or less.

Now the angle BAC is not equal to the angle EDF; for then the base BC would also have been equal to the base EF, [I.4]

but it is not;

therefore the angle BAC is not equal to the angle EDF.

Neither again is the angle BAC less than the angle EDF; for then the base BC would also have been less than the base EF, [I.24]

but it is not;

therefore the angle BAC is not less than the angle EDF.

But it was proved that it is not equal either;

therefore the angle BAC is greater than the angle EDF.

Therefore etc.

Q.E.D.

What's Next

Props 24 and 25 complete the hinge theorem pair, giving a perfect two-way relationship between the included angle and the opposite side. With all the triangle comparison results now in place, Euclid turns to a new topic: the congruence theorem AAS/ASA in Proposition 26, which also closes out all the triangle congruence cases before the theory of parallel lines begins.