Euclid's Elements Book I

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

Proposition I.45 · Problem

To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABCD be the given rectilineal figure and E the given rectilineal angle. It is required to construct, in the given angle E, a parallelogram equal to ABCD. (Every point can be dragged.)
  2. Let DB be joined, dividing ABCD into the two triangles ABD, DBC.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  3. Let the parallelogram FKHG be constructed equal to the triangle ABD, in the angle HKF equal to the angle E.
    Proposition I.42
    Proposition I.42: To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.
  4. To the straight line GH let the parallelogram GHML be applied equal to the triangle DBC, in the angle GHM equal to the angle E.
    Proposition I.44
    Proposition I.44: To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.
  5. Since each of the angles HKF, GHM is equal to the angle E, let the angle KHG be added to each; then the angles KHG, GHM are equal to the angles FKH, KHG, that is, to two right angles; therefore KH is in a straight line with HM. Similarly FG is in a straight line with GL.
    Common Notion 1Common Notion 2Proposition I.29Proposition I.14
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
    Common Notion 2: If equals be added to equals, the wholes are equal.
    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.
    Proposition I.14: If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.
  6. Therefore KFLM is a parallelogram; it is equal to the two triangles ABD, DBC, that is, to ABCD, and it has the angle FKM equal to the angle E.
    Proposition I.30Proposition I.33Common Notion 2
    Proposition I.30: Straight lines parallel to the same straight line are also parallel to one another.
    Proposition I.33: The straight lines joining equal and parallel straight lines at the extremities which are in the same directions respectively are themselves also equal and parallel.
    Common Notion 2: If equals be added to equals, the wholes are equal.