Proposition 4 of 48 • Theorem

If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively.

If two triangles have two sides the same length and the angle between those sides is also equal (SAS - Side-Angle-Side), then the triangles are identical in every way.

Before You Read

Draw two triangles that look different but have this in common: two sides are the same length, and the angle between those sides is the same. Do you think the rest of the triangle has to match too? Can you make two triangles with the same side-angle-side but different third sides?

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What Euclid Is Doing

Setup: We have two triangles, ABC and DEF. We know that AB = DE, AC = DF, and ∠BAC = ∠EDF. We want to prove the triangles are congruent (identical in all measurements).

Approach: Euclid uses 'superposition'—imagine picking up triangle ABC and placing it on DEF. Since AB = DE, point B lands on E. Since ∠A = ∠D and AC = DF, point C must land on F. Therefore the triangles match exactly.

Conclusion: The base BC = EF, and all angles match: ∠ABC = ∠DEF, ∠BCA = ∠EFD. The triangles are congruent.

Key Moves

Mentally place triangle ABC on triangle DEF, aligning A with D
Since AB = DE, point B coincides with E
Since ∠BAC = ∠EDF, ray AC falls along ray DF
Since AC = DF, point C coincides with F
Therefore BC coincides with EF (two points determine a line)
By Common Notion 4, coincident things are equal

Try It Yourself

Using a protractor and ruler, draw a triangle with sides 5 cm and 7 cm meeting at a 40° angle. Now draw another triangle with the exact same measurements. Compare everything: the third side, the other two angles. Are they identical? Try to make them different—you can't.

Why this proposition is different

Euclid's original proof uses a method called superposition.

In this proposition, Euclid imagines placing one triangle on top of another to compare them. This idea is called superposition.

Euclid's argument is historically important, but later mathematicians pointed out that this method is not fully explained by Euclid's original rules. Today, mathematicians usually justify this result in a different, more precise way.

So on this page, we show Euclid's original idea, but we do not use the normal drag-and-drop proof exercise.

Euclid's original proof idea

This is Euclid's historical argument using superposition, not a standard drag-and-drop proof on this site.

Mentally place triangle ABC on triangle DEF, aligning A with D
Since AB = DE, point B coincides with E
Since ∠BAC = ∠EDF, ray AC falls along ray DF
Since AC = DF, point C coincides with F
Therefore BC coincides with EF (two points determine a line)
By Common Notion 4, coincident things are equal

Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 4.

✓Lesson Plan (prop-04-lesson-plan.pdf)
✓Student Worksheet (prop-04-worksheet.pdf)
✓Answer Key (prop-04-answer-key.pdf)

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Why It Matters

This is the first of the triangle congruence theorems (SAS). It's used constantly throughout the Elements to prove triangles are equal. When you know two sides and the included angle match, everything else must match too.

Modern connection: SAS congruence is fundamental in engineering and construction. If you build two triangular trusses with the same measurements, you're guaranteed they're identical. It's why triangles are so stable—three measurements completely determine the shape.

Historical note: The 'superposition' proof is controversial. Euclid essentially assumes figures can be moved without changing them—but he never states this as a postulate. Modern treatments often take SAS as an axiom instead of proving it.

Discussion Questions

Why does Euclid need to 'move' one triangle onto another? Is this really a proof?
What if we knew two sides and a NON-included angle? Would that be enough?
Why is the angle between the two sides (the 'included' angle) so important?

Euclid's Original Proof

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 the angle BAC equal to the angle EDF.

I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

For, if the triangle ABC be applied to the triangle DEF, and if the point A be placed on the point D and the straight line AB on DE, then the point B will also coincide with E, because AB is equal to DE.

Again, AB coinciding with DE, the straight line AC will also coincide with DF, because the angle BAC is equal to the angle EDF; hence the point C will also coincide with the point F.

But B also coincided with E; hence the base BC will coincide with the base EF and will be equal to it. [C.N. 4]

Thus the whole triangle ABC will coincide with the whole triangle DEF, and will be equal to it. And the remaining angles will also coincide with the remaining angles and will be equal to them.

Therefore etc. (Being) what it was required to prove.

What's Next

SAS gives us a powerful tool for proving triangles are identical. Next up is Proposition 5—the famous 'Pons Asinorum' (Bridge of Donkeys). It proves that if a triangle has two equal sides, the angles opposite those sides must be equal too. This was historically the first real test of a geometry student's ability.