| 1 |
On any given finite
straight line to construct an equilateral triangle |
Illustration |
| 2 |
To place
at a given point (as an extremity) a straight line equal to a given straight
line. |
Illustration |
| 3 |
Given two
unequal straight lines, to cut off from the greater a straight line equal to
the less. |
Illustration |
| 4 |
If two
triangles have the same sides equal to two sides respectively, and have the
angles contained by the equal straight lines 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, namely
those which the equal sides subtend. |
|
| 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. |
Illustration |
| 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. |
|
| 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. |
|
| 8 |
If two
triangles have the two sides equal to two sides respectively, and also have the
base equal to the base, they will also have the angles equal whhich are
contained by the equal straight sides. |
|
| 9 |
To bisect
a given rectilinear angle. |
Illustration |
| 10 |
To bisect
a given finite straight line. |
Illustration |
| 11 |
To draw a
straight line at right angles to a given straight line from a given point on
it. |
Illustration |
| 12 |
To given
infinite straight line, from a given point which is not on it, to draw a
perpendicular straight line. |
Illustration |
| 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 a point on it, two straight lines not lying on the same
side make the adjacent angles equal to two right angles, the straight lines
will be in a straight line with one another. |
|
| 15 |
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 interior and opposite angles. |
|
| 17 |
In any
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 |
In any
triangle the greater angle is subtended by the greater side. |
|
| 20 |
In any
triangle two sides taken together in any manner are greater than the remaining
one. |
|
| 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. |
|
| 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 any manner should be greater than the remaining one. |
Illustration |
| 23 |
On a given
straight line and at a point on it to construct a rectilineal angle equal to a
given rectilineal angle. |
Illustration |
| 24 |
If two
triangles have the two sides equal to two sides respectively, but have one of
the angles contained by the equal straight lines greater than the other, they
will also have the base greater than the base. |
|
| 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 one of the angles contained by the
equal straight lines greater than the other. |
|
| 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, or 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. |
|
| 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. |
|
| 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 |
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. |
|
| 30 |
Straight
lines parallel to the same straight line are also parallel to one
another. |
|
| 31 |
Through a
given point to draw a straight line parallel to a given straight
line. |
Illustration |
| 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 |
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. |
Illustration |
| 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. |
Illustration |
| 36 |
Parallelograms which are on equal bases and in the same parallels are
equal to one another. |
Illustration |
| 37 |
Triangles
which are on the same base and in the same parallels are equal to one
another. |
Illustration |
| 38 |
Triangles
which are on equal bases and in the same parallels are equal to one
another. |
Illustration |
| 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. |
Illustration |
| 41 |
If a
parallelogram has the same base with a triangle and be in the same parallels,
the parallelogram is double of the triangle. |
Illustration |
| 42 |
To
construct, in a given rectilineal angle, a parallelogram equal to a given
triangle. |
Illustration |
| 43 |
In any
parallelogram the complements of the parallelogram about the diameter are equal
to one another. |
Illustration |
| 44 |
To a given
straight line to apply, in a given rectilineal angle, a parallelogram equal to
the given triangle. |
Illustration |
| 45 |
To
construct, in a given rectilineal angle, a parallelogram equal to a given
rectilinear figure. |
|
| 46 |
On a given
straight line to describe a square. |
Illustration |
| 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. |
Illustration |
| 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. |
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