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An Adventure in Language and Logic

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Introduction.  Geometry: The study of figures

Introduction to Logic.

Hypothesis and conclusion

Necessary and sufficient

Valid arguments

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BOOK I

Definitions

Postulates

Axioms or Common Notions

CONSTRUCTIONS

Proposition 1

On a given straight line to construct an equilateral triangle.

Proposition 2

From a given point to draw a straight line equal to a given straight line.

Proposition 3

Given two unequal straight lines, to cut off from the longer line
a straight line equal to the shorter line.

CONGRUENT TRIANGLES

Proposition 4 (Side-Angle-Side)

If two triangles have two sides equal to two sides respectively, and if the angles contained by those sides are also equal, then the remaining side will equal the remaining side, the triangles themselves will be equal areas, and the remaining angles will be equal, namely those that are opposite the equal sides.

THE ISOSCELES TRIANGLE

Proposition 5

In an isosceles triangle the angles at the base are equal.

Proposition 6

If two angles of a triangle are equal, then the sides opposite them will be equal.

CONGRUENT TRIANGLES 2

Proposition 8 (Side-Side-Side)

If two triangles have two sides equal to two sides respectively, and if the bases are also equal, then the angles will be equal that are contained by the two equal sides.

BISECTIONS

Proposition 9

To bisect a given angle.

Proposition 10

To bisect a given straight line.

PERPENDICULARS

Proposition 11

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

Proposition 12

To a given straight line that may be made as long as we please, and from a given point not on it, to draw a perpendicular line.

RIGHT ANGLES, VERTICAL ANGLES

Proposition 13

When a straight line that stands on another straight line makes two angles, either it makes two right angles, or it makes angles that together are equal to two right angles.

Proposition 14

If two straight lines are on opposite sides of a given straight line, and, meeting at one point of that line they make the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.

Proposition 15

When two straight lines intersect one another, the vertical angles are equal.

THE SIDES AND ANGLES OF A TRIANGLE

Proposition 16

If one side of a triangle is extended, then the exterior angle is greater than either of the opposite interior angles.

Proposition 17

Any two angles of a triangle are together less than two right angles.

Proposition 18

A greater side of a triangle is opposite a greater angle.

Proposition 19

A greater angle of a triangle is opposite a greater side.

Proposition 20

Any two sides of a triangle are together greater than the third side

Proposition 22

To construct a triangle whose sides are equal to three given straight lines: thus any two of them taken together must be greater than the third.

Proposition 23

On a given straight line and at a given point on it, to construct an angle equal to a given angle.

CONGRUENT TRIANGLES 3

Proposition 26 (Angle-Side-Angle)

If two triangles have two angles equal to two angles respectively, and one side equal to one side, which may be either the sides between the equal angles or the sides opposite one of them, then the remaining sides will equal the remaining sides (those that are opposite the equal angles), and the remaining angle will equal the remaining angle.

THE THEORY OF PARALLEL LINES

Proposition 27

If a straight line that meets two straight lines makes the alternate angles equal, then the two straight lines are parallel.

Proposition 28

If a straight line that meets two straight lines makes an exterior angle equal to the opposite interior angle on the same side, or if it makes the interior angles on the same side equal to two right angles, then the two straight lines are parallel.

Proposition 29

If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle equal to the opposite interior angle on the same side, and it makes the interior angles on the same side equal to two right angles.

Proposition 30

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

THE THREE ANGLES OF A TRIANGLE

Proposition 31

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

Proposition 32

If one side of a triangle is extended, then the exterior angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to two right angles.

PARALLELOGRAMS

Proposition 33

The straight lines which join the extremities on the same side of two equal and parallel straight lines, are themselves equal and parallel.

Proposition 34

In a parallelogram the opposite sides and angles are equal, and the diagonal bisects the area.

EQUALITY OF NON-CONGRUENT FIGURES

Proposition 35

Parallelograms on the same base and in the same parallels are equal.

Proposition 36

Parallelograms on equal bases and in the same parallels are equal.

Proposition 37

Triangles on the same base and in the same parallels are equal.

Proposition 38

Triangles on equal bases and in the same parallels are equal.

Proposition 39

Equal triangles that are on the same base and on the same side of it, are in the same parallels.

Proposition 41

If a parallelogram and a triangle are on the same base and in the same parallels, the parallelogram is double the triangle.

CONSTRUCTION OF A SQUARE

Proposition 46

On a given straight line to draw a square.

THE PYTHAGOREAN THEOREM

Proposition 47

In a right triangle the square drawn on the side opposite the right angle
is equal to the squares drawn on the sides that make the right angle.

Proposition 48

If the square drawn on one side of a triangle is equal to the squares drawn on the other two sides, then the angle contained by those two sides is a right angle.