Euclid - The Elements - Book I   Notes

Propositions

Proposition
Statement
Proposition
Statement
1
 On a finite straight line, to construct an equilateral triangle.
25
 If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.
2
 To place at a given point a straight line equal to a given straight line.
26
 If two triangles have the two angles equal to the two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining angle equal to the remaining angle.
3
 Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
27
 If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.
4
 If two triangles have the two sides equal to two sides respectively, and have the angles contained by the sides 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.
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.
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.
29
 A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angles equal to the interior and office angle and the interior angles on the same side equal to two right angles.
6
 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.
30
 Straight lines parallel to the same straight line are also parallel to one another.
7
 Given two straight lines constructed on a straight line and meeting in a point, there cannot be constructed on the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively namely each to that which has the same extremity with it.
31
 Through a given point to draw a straight line parallel to a given straight line.
8
 If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.
32
 In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.
9
 To bisect a given rectilinear angle.
33
 Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.
10
 To bisect a given finite straight line.
34
 In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.
11
 To draw a straight line at right angles to a give straight line from a given
35
 Parallelograms which are on equal bases and in the same parallels are equal to one another.
12
 To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.
36
 Parallelograms which are on equal bases and in the same parallels equal one another.
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.
37
 Triangles which are on the same base and in the same parallels equal one another.
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.
38
 Triangles which are on equal bases and in the same parallels equal one another.
15
 If two straight lines cut one another, they make the vertical angles equal to one another.
39
 Equal triangles which are on the same base and on the same side are also in the same parallels.
16
 In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
40
 Equal triangles which are on equal bases and on the same side are also in the same parallels.
17
 In any triangle two angles taken together in any manner are less than two right angles.
41
 If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.
18
 In any triangle the greater side subtends the greater angle.
42
 To construct a parallelogram equal to a given triangle in a given rectilinear angle.
19
 In any triangle the greater angle is subtended by the greater side.
43
 In any parallelogram the complements of the parallelograms about the diameter equal one another.
20
 In any triangle two sides taken together in any manner are greater than the remaining one.
44
 To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.
21
 If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.
45
 To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.
22
 Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in an manner should be greater than the remaining one.
46
 To describe a square on a given straight line.
23
 On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
47
 (Pythagorean theorem)In right angled triangles, the square on the side subtending the right angle is equal to the squares of the sides containing the right angle.
24
 If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal stright lines greater than the other, they will also have the base greater than the base.
48
 If in a triangle the square on one of the side be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.