The Elements

A geometric diagram at the top left showing a triangle with internal lines and points labeled with Greek letters, demonstrating an inequality proof related to Proposition 24 or 25.

of elz original: "ΕΛΖ"; and around the triangle dez, [set] el original: "ΕΛ" equal to ba, and ez original: "ΕΖ" equal to ag, and the angle bag equal to the angle lez; therefore [the base bc] is equal to the base lz; since, therefore, bc is greater than ez, lz is therefore also greater than ez; and the angle lez is greater than the angle elz; but lez is equal to bag; therefore bag is greater than elz; and elz is greater than edz; therefore, by much, bag is greater than edz; which was to be demonstrated.

25

If two triangles have two sides equal to two sides, each to each, but have the base greater than the base, they will also have the angle contained by the equal straight lines greater than the [other] angle.
Let there be two triangles abg and dez, having two sides ab and ag equal to two sides de and dz, each to each; ab to de, and ag to dz; and let the base bg be greater than the base ez; I say that the angle bag is also greater than the angle edz; for if not, it is either equal to it or less. But the angle bag is not equal to edz; for if it were equal, the base bg would also be equal to the base ez; but it is not; therefore, the angle bag is not equal...

Two small congruent triangles side-by-side.

This block of text in a smaller, denser hand appears to be a scholion or commentary providing further clarification on the logic of Proposition 25.
Since, therefore, bag is equal to edz, and the two sides ab and ag are equal to the two sides de and dz, each to each, therefore the base bg is equal to the base ez (according to the 4th theorem). But it is also hypothesized to be greater; therefore, it is not equal. If the angle bag is less than the angle edz, then the base bg is also less than ez (according to the 24th theorem). But it is also hypothesized to be greater; therefore, it is not less. Since, therefore, it is neither equal nor less, the angle bag is therefore greater than the angle edz.

Two triangles labeled Α, Β, Γ and Δ, Ε, Ζ illustrating the primary elements of the proposition.

elz; therefore, edz is also not equal to elz; and it is greater; therefore, edz is neither greater nor less than elz; edz is therefore equal to elz; which was to be demonstrated; but this is absurd. Therefore bg is greater than ez; for the triangles whose angles are equal, the sides opposite the equal angles are equal to each other; and to the equal angles, the side subtending them must be equal to each other; which was what had to be demonstrated.

26

If two triangles have two angles equal to two angles, each to each, and one side equal to one side, either the side by the equal angles or the one subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides, each to each, and the remaining angle equal to the remaining angle. Let there be two triangles abg and dez, having two angles abg and bag equal to two angles dez and edz, each to each; abg to dez, and bag to edz; and let them also have one side equal to one side, first the one by the equal angles, ab to de; I say that they will also have the remaining sides equal to the remaining sides...

Two congruent triangles placed near each other, showing the setup for Proposition 26.
A large geometric diagram featuring a right-angled triangle ΑΒΓ with an internal line segment and another triangle ΔΕΖ. A circular library stamp is visible to the right of the diagram.