Book I: Plane Geometry
48 propositions covering triangles, parallels, and area
Constructions & Basic Properties
Fundamental constructions and triangle properties
Proposition 1
Construct an equilateral triangle on a given line
Proposition 2
Copy a line segment to a given point
Proposition 3
Cut a segment equal to a shorter line from a longer line
Proposition 4
Side-Angle-Side (SAS) Congruence
Proposition 5
Base angles of an isosceles triangle are equal
Proposition 6
Converse of Proposition 5
Proposition 7
Uniqueness of triangle construction
Proposition 8
Side-Side-Side (SSS) Congruence
Proposition 9
Bisect an angle
Proposition 10
Bisect a line segment
Proposition 11
Draw a perpendicular from a point on a line
Proposition 12
Draw a perpendicular to a line from an external point
Proposition 13
Adjacent angles on a line are supplementary
Proposition 14
Converse of Proposition 13
Proposition 15
Vertical angles are equal
Triangle Congruence
Congruence theorems and angle properties
Proposition 16
Exterior angle is greater than either remote interior angle
Proposition 17
Two angles of a triangle are less than two right angles
Proposition 18
Greater side opposite greater angle
Proposition 19
Greater angle opposite greater side
Proposition 20
Triangle inequality
Proposition 21
Interior triangle has smaller sides but larger angle
Proposition 22
Construct triangle from three given lengths
Proposition 23
Copy an angle
Proposition 24
SAS inequality (hinge theorem)
Proposition 25
Converse of Proposition 24
Proposition 26
ASA and AAS Congruence
Parallel Lines
Theory of parallel lines and angles
Proposition 27
Alternate angles imply parallel lines
Proposition 28
Corresponding or co-interior angles imply parallel
Proposition 29
Converse: parallel lines give equal angles
Proposition 30
Transitivity of parallel lines
Proposition 31
Construct a parallel line through a point
Proposition 32
Angle sum of a triangle is two right angles
Proposition 33
Equal and parallel lines form a parallelogram
Proposition 34
Properties of parallelograms
Area & Pythagorean Theorem
Area relations culminating in the Pythagorean theorem
Proposition 35
Parallelograms on the same base and between same parallels are equal
Proposition 36
Parallelograms on equal bases and between same parallels are equal
Proposition 37
Triangles on the same base and between same parallels are equal
Proposition 38
Triangles on equal bases and between same parallels are equal
Proposition 39
Equal triangles on the same base are between same parallels
Proposition 40
Equal triangles on equal bases are between same parallels
Proposition 41
Parallelogram is double the triangle
Proposition 42
Construct a parallelogram equal to a given triangle
Proposition 43
Complements of parallelograms are equal
Proposition 44
Apply a parallelogram equal to a triangle to a line
Proposition 45
Construct a parallelogram equal to any rectilinear figure
Proposition 46
Construct a square on a given line
Proposition 47
The Pythagorean Theorem
Proposition 48
Converse of the Pythagorean Theorem