Euclid Book One

1. On a given straight line to construct an equilateral triangle.

2. To place, at a given point [as an extremity] a straight line equal to a given straight line.

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

4. The SAS Prop.
If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, "they will be coincide" (paraphrased)..

5. Pons Assinorum
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. CONVERSE OF FIVE
If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

7. Given 2 straight lines constructed on a straight line [from its extremities] and meeting at a point, "you can't have two more straight lines equal to them (from the same extremities and on the same side) that meet at a different point."

8. The SSS Prop
If two triangles have two sides equal to two sides respectively, and the base equal to the base "their angles are equal, and they coincide."

9. To bisect a given rectilinear angle.

10. To bisect a given finite straight line.

11. To draw a straight line at right angles to a given straight line from a given point on it.

12. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.

13. If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.

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.

15. Vertical Angles Prop
If two straight lines cut one another, they make the vertical angles equal to one another.

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

17. In any given triangle, two angles taken together in any manner are less than two right angles.

18. In any triangle, the greater side SUBTENDS the greater angle.

19. CONVERSE OF 18
In any triangle, the greater angle IS SUBTENDED BY by the greater side.

20. In any triangle, two sides taken together in any manner area greater than the remaining one.

21. If on one side of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, those straight lines will be less than the remaining two sides of the triangle, but will contain a greater angle.

22. Out of three lines equal to three given straight lines, to construct a triangle. (In the given straight lines, any two taken together must be greater than the remaining one.)

23. CONSTRUCT AN EQUAL ANGLE
On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

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

25. CONVERSE OF TWENTY- FOUR
If two triangles have two sides equal to two sides respectively, but have the base greater than the base, they will also have one angle contained by the equal straight lines greater than the other.

26. The ASA, & AAS Prop
If two triangles have two angles equal to two angles respectively, and one side equal to one side (either one adjoining the equal angles, or subtending one of them) they will "coincide."

27. ALTERNATE ANGLES PROP
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to each other.

28. If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.

29. ULTIMATE PARALLEL GO-TO PROP
(when you know lines are parallel)
A straight line falling on parallel straight lines makes alternate angles equal, the ext. angle equal to int. and opp. angle, and int. angles on same side equal to two right angles.

30. Straight lines parallel to the same straight line are parallel to each other.

31. Through a given point to draw a straight lie parallel to a given straight line.

32. In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

33. Straight lines joining equal and parallel straight lines [at extremities] in the same direction, are themselves also equal and parallel.

34. In parallelogrammic areas, the opposite sides and angles are equal to one another and the diameter bisects the areas.

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

36. Parallels which are on equal bases and in the same parallels are equal to one another.

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

38. Triangles which are on equal bases and in the same parallels are equal to one another.

39. Equal triangles which are on the same base and on the same side are also in the same parallels.

40. Equal triangles which are on equal bases and on the same side are also in the same parallels.

41. If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.

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

43. In any parallelogram, the complements of the parallelograms about the diameter are equal to one another.

44. To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.

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

46. On a given straight line to describe a square.

47. PYTHAGORIAN THEORUM
In right-angled triangles, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

48. If in a triangle, the square of one of the sides be equal to the squares on the remaining two sides, the triangle is right.

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