Euclid's Propositions: Definitions and Theorems from Geometry

I.1

To make an equilateral triangle on a segment

I.2/3

To copy a given segment at a point or from a longer segment

I.4

SAS congruence; given two triangles, triangle ABC and triangle DEF, if AB=DE and AC=DF

I.5

Pons Asinorum; in an isosceles triangle the angles at the base will be equal and if the equal sides be produced the angles under the base will be equal

I.6

converse of I.5, in a triangle in which we have equal sides then the sides opposite the equal angles will be equal

I.8

SSS congruence; given triangle ABC and triangle DEF if AB=DE, AC=DF, BC=EF then angle A=angle D, angle B=angle E, and angle C=angle F

I.9

Bisect the given angle

I.10

bisect a segment

I.11/12

Make a perpendicular angle to the line through the point

I.13

If a line stands on another line, then the adjacent angles are either both right or the sum of the two angles is 180 degrees, aka supplementary angles add up to two right angles

I.14

Converse of I.13; If adjacent angles add to two right angles, ten the sides not in common lie in a straight line

I.15

Vertical angles are equal

I.16

exterior angle theorem; given triangle ABC if we extend AB by BD, the exterior angle formed CBD will be greater than each of the remote or non-adjacent interior angles (angle A and C)

I.17

Converse of postulate 5; in a triangle the sum of any two of its interior angles will be strictly less than two right angles

I.18/19

In any triangle, the greater angle or side will be opposite the greater side or angle

I.20

the triangle inequality; in any triangle, the sum of any two sides will be greater than the third side

I.22

converse of triangle inequality; given 3 segments AB, CD, EF for which the sum of any two is greater than the third, then there is a triangle whose sides equal the three segments

I.23

copy angles; given angle A and line BC to make an angle at B with BC as a side to angle A