Euclid's Elements Book I

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

Proposition I.35 · Theorem

Parallelograms which are on the same base and in the same parallels are equal to one another.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABCD, EBCF be parallelograms on the same base BC and in the same parallels AF, BC. The claim: the parallelogram ABCD is equal to the parallelogram EBCF. (Try dragging A, B, C or E.)
  2. Since ABCD is a parallelogram, AD is equal to BC. For the same reason EF is also equal to BC; so that AD is also equal to EF.
    Proposition I.34Common Notion 1
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
  3. Let DE be added to each; therefore the whole AE is equal to the whole DF.
    Common Notion 2
    Common Notion 2: If equals be added to equals, the wholes are equal.
  4. But AB is also equal to DC, and the exterior angle FDC is equal to the interior angle EAB.
    Proposition I.34Proposition I.29
    Proposition I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
    Proposition I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
  5. Therefore the two sides EA, AB are equal to the two sides FD, DC, and they contain equal angles; so the triangle EAB is equal to the triangle FDC.
    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. Let BE, DC meet at G, and let the triangle DGE be subtracted from each; then the trapezium ABGD which remains is equal to the trapezium EGCF which remains.
    Common Notion 3
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.
  7. Let the triangle GBC be added to each; therefore the whole parallelogram ABCD is equal to the whole parallelogram EBCF.
    Common Notion 2
    Common Notion 2: If equals be added to equals, the wholes are equal.