Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.24 · Theorem

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

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. 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 angle at A be greater than the angle at D. The claim: the base BC is also greater than the base EF. (A, B, C, D can be dragged.)
  2. On the straight line DE, at the point D, let the angle EDG be constructed equal to the angle BAC, and let DG be made equal to either of the two straight lines AC, DF.
    Proposition I.23Proposition I.3
    Proposition I.23: On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  3. Let EG, FG be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  4. Since AB is equal to DE and AC to DG, and the angle BAC is equal to the angle EDG, the base BC is equal to the base EG.
    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.
  5. Again, since DF is equal to DG, the angle DGF is equal to the angle DFG.
    Proposition I.5
    Proposition I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
  6. Therefore the angle DFG is greater than the angle EGF, and the angle EFG is much greater than the angle EGF.
    Common Notion 5
    Common Notion 5: The whole is greater than the part.
  7. In the triangle EFG the angle EFG is greater than the angle EGF, and the greater angle is subtended by the greater side; therefore EG is greater than EF. But EG is equal to BC; therefore BC is also greater than EF.
    Proposition I.19
    Proposition I.19: In any triangle the greater angle is subtended by the greater side.