| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 15 |
If two straight lines cut one another, they make the vertical angles
equal to one another. |
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| 16 |
In any triangle, if one of the sides be produced, the exterior angle
is greater than either of the interior and opposite angles. |
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| 17 |
In any triangle two angles taken together in any manner are less than
two right angles. |
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| 18 |
In any triangle the greater side subtends the greater
angle. |
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| 19 |
In any triangle the greater angle is subtended by the greater
side. |
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| 20 |
In any triangle two sides taken together in any manner are greater
than the remaining one. |
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| 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. |
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| 30 |
Straight lines parallel to the same straight line are also parallel to
one another. |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| 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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