Propositions
Book I contains 48 propositions that build on the foundations. Each one uses only what has already been proved — definitions, postulates, common notions, and earlier propositions. Pick any proposition to dive in.
Basic Constructions
Equilateral triangles, copying segments, cutting to length, and the first congruence proof (SAS).
- Proposition 1CTo construct an equilateral triangle on a given finite straight line.
- Proposition 2CTo place a straight line equal to a given straight line with one end at a given point.
- Proposition 3CTo cut off from the greater of two given unequal straight lines a straight line equal to the less.
- Proposition 4TIf two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively.
- Proposition 5TIn isosceles triangles the angles at the base equal one another.
Triangle Fundamentals
Isosceles triangles, uniqueness of constructions, SSS congruence, and angle/segment bisection.
- Proposition 6TIf in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
- Proposition 7TGiven two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of 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.
- Proposition 8TIf two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.
- Proposition 9CTo bisect a given rectilinear angle.
- Proposition 10CTo bisect a given finite straight line.
Perpendiculars & Angles
Right angles from perpendiculars, supplementary angle sums, and vertical angles.
- Proposition 11CTo draw a straight line at right angles to a given straight line from a given point on it.
- Proposition 12CTo draw a straight line perpendicular to a given infinite straight line from a given point not on it.
- Proposition 13TIf a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.
- Proposition 14TIf with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.
- Proposition 15TIf two straight lines cut one another, then they make the vertical angles equal to one another.
Exterior Angles & Inequalities
The exterior angle theorem, triangle side-angle relationships, and the triangle inequality.
- Proposition 16TIn any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.
- Proposition 17TIn any triangle the sum of any two angles is less than two right angles.
- Proposition 18TIn any triangle the angle opposite the greater side is greater.
- Proposition 19TIn any triangle the side opposite the greater angle is greater.
- Proposition 20TIn any triangle the sum of any two sides is greater than the remaining one.
Interior Triangles & Angle Copying
Triangles within triangles, constructing triangles from sides, copying angles, and the hinge theorem.
- Proposition 21TIf from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides.
- Proposition 22CTo construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one.
- Proposition 23CTo construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it.
- Proposition 24TIf two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base.
- Proposition 25TIf two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other.
Parallel Lines
AAS/ASA congruence, alternate and corresponding angles with parallels, Proposition 29 (the parallel postulate in action), and transitivity of parallelism.
- Proposition 26TIf two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle.
- Proposition 27TIf 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.
- Proposition 28TIf a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another.
- Proposition 29TA 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 sum of the interior angles on the same side equal to two right angles.
- Proposition 30TStraight lines parallel to the same straight line are also parallel to one another.
Parallel Constructions & Parallelograms
Drawing parallels, the angle sum theorem (Prop 32), forming parallelograms, their properties, and equal areas on the same base.
- Proposition 31CTo draw a straight line through a given point parallel to a given straight line.
- Proposition 32TIn 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.
- Proposition 33TStraight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.
- Proposition 34TIn parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.
- Proposition 35TParallelograms which are on the same base and in the same parallels equal one another.
Area Theorems
Equal bases between parallels, triangle area equality, and their converses.
- Proposition 36TParallelograms which are on equal bases and in the same parallels equal one another.
- Proposition 37TTriangles which are on the same base and in the same parallels equal one another.
- Proposition 38TTriangles which are on equal bases and in the same parallels equal one another.
- Proposition 39TEqual triangles which are on the same base and on the same side are also in the same parallels.
- Proposition 40TEqual triangles which are on equal bases and on the same side are also in the same parallels.
Area Applications
Parallelogram-triangle doubling, area construction, complements, and application of areas.
- Proposition 41TIf a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.
- Proposition 42CTo construct a parallelogram equal to a given triangle in a given rectilinear angle.
- Proposition 43TIn any parallelogram the complements of the parallelograms about the diameter equal one another.
- Proposition 44CTo a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.
Grand Finale
Rectilinear area, square construction, the Pythagorean Theorem (Prop 47), and its converse.
- Proposition 45CTo construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.
- Proposition 46CTo describe a square on a given straight line.
- Proposition 47TIn right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
- Proposition 48TIf in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.