Euclid's Elements Book I

1. Let ABC, DEF be two triangles having the two angles ABC, BCA equal to the two angles DEF, EFD respectively, and one side equal to one side. First let the equal sides be those adjoining the equal angles: BC equal to EF. The claim: the remaining sides and the remaining angle are also equal. (A, B, C, D can be dragged.)
2. If AB is unequal to DE, one of them is greater. Let AB be greater; let BG be made equal to DE, and let GC be joined. (G is drawn as this assumption would place it.)

   Proposition I.3Postulate 1

   Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

   Postulate 1: To draw a straight line from any point to any point.
3. Since GB is equal to DE and BC to EF, and the angle GBC is equal to the angle DEF, the triangle GBC is equal to the triangle DEF, and the angle GCB is equal to the angle DFE.

   Proposition I.4

   Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.
4. But the angle DFE is equal by hypothesis to the angle BCA; therefore the angle BCG is equal to the angle BCA, the less to the greater, which is impossible. Therefore AB is equal to DE.

   Common Notion 1

   Common Notion 1: Things which are equal to the same thing are also equal to one another.
5. So the two sides AB, BC are equal to the two sides DE, EF, and the angle ABC is equal to the angle DEF; therefore AC is equal to DF, and the angle BAC to the angle EDF.

   Proposition I.4

   Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.
6. Next let the sides subtending equal angles be equal: AB equal to DE. If BC is unequal to EF, let BC be greater; let BH be made equal to EF, and let AH be joined. (H too is drawn as the assumption would place it.)

   Proposition I.3Postulate 1

   Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

   Postulate 1: To draw a straight line from any point to any point.
7. Since AB is equal to DE and BH to EF, and they contain equal angles, the angle BHA is equal to the angle EFD, that is, to the angle BCA.

   Proposition I.4Common Notion 1

   Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.

   Common Notion 1: Things which are equal to the same thing are also equal to one another.
8. Then in the triangle AHC the exterior angle BHA is equal to the interior and opposite angle BCA, which is impossible. Therefore BC is equal to EF, and the triangles are equal in every respect.

   Proposition I.16Proposition I.4

   Proposition I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.

   Proposition I.4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.

Being what it was required to prove.