## Euclid's Elements

### Book I Translation Project

#### Book I    Propositions

Proposition 1.
It is possible to construct an equilateral triangle on a given finite straight line.

Proposition 2.
It is possible to attach a line equal to a given straight line to a given point.

Proposition 3.
Given two unequal straight lines, it is possible to take from the greater a part equal to the smaller.

Proposition 4.
Letting two triangles have two sides of one equal to two sides of the other respectively and the angle contained by the sides of the one equal to the angle contained by the sides of the other, then also the base of the one will equal to the base of the other and the one triangle will be equal to the other and the remaining angles of the one will be equal to the remaining angles of the other respectively, being those which the equal sides subtend.

Proposition 5.
The angles at the base of an isosceles triangle are equal to one another, and the equal sides having been extended further, the angles under the base will be equal to one another.

Proposition 6.
If two angles of a triangle are equal to one another, the legs subtending the equal angles will also be equal to one another.

Proposition 7.
Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), 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.

Proposition 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.

Proposition 9.
It is possible to bisect a given rectilineal angle.

Proposition 10.
It is possible to bisect a given finite straight line.

Proposition 11.
From a given point on a given straight line, it is possible to draw a straight line making right angles (with it?).

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

Proposition 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.

Proposition 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.

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

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

Proposition 17.
In a triangle two angles taken together in any manner are less than two right angles.

Proposition 18.
In any triangle the greater side subtends the greater angle.

Proposition 19.
In any triangle the greater angle is subtended by the greater side.

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

Proposition 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.

Proposition 22.
It is possible to construct a triangle out of three straight lines equal to three given straight lines. Of course it is necessary that any two of the lines are greater than the remaining line [because any two sides of a triangle are greater than the remaining side].

Proposition 23.
It is possible to construct a rectilineal angle equal to a given rectilineal angle on a given straight line and at a given point on it.

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

Proposition 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 that the other.

Proposition 26.
If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, of that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle. [ASA and AAS]

Proposition 27.
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

Proposition 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.

Proposition 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 30.
Straight lines parallel to the same straight line are also parallel to one another.

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

Proposition 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.

Proposition 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

Proposition 47.
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.

Proposition 48.
If in a triangle the square on one of the sides 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.