Proof
Opposite faces share parallel sides (by XI.16) and equal angles (by
XI.10), so they are congruent parallelograms by I.33 / I.34.
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Depends on (4)
- I.33Proposition I.33The straight lines joining equal and parallel straight lines (at the extremities which are in the same directions) are…
- I.34Proposition I.34In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
- XI.10Proposition XI.10If two straight lines meeting one another be parallel to two straight lines meeting one another, not in the same plane,…
- XI.16Proposition XI.16If two parallel planes be cut by any plane, their common sections are parallel.
Required by (dependents) (5)
- XI.25Proposition XI.25If a parallelepipedal solid be cut by a plane parallel to opposite planes, then, as the base is to the base, so will…
- XI.27Proposition XI.27On a given straight line to construct a parallelepipedal solid similar and similarly situated to a given…
- XI.28Proposition XI.28If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid will be bisected…
- XI.29Proposition XI.29Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides…
- XI.38Proposition XI.38If the sides of the opposite planes of a cube be bisected, and planes be carried through the points of section, the…
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