*point**line*- A set of points.*distance*- The distance between any two points`A`and`B`is a non-negative real number`d`(`A`,`B`) such that`d`(`A`,`B`) =`d`(`B`,`A`)*angle*- An angle is formed by three ordered points`A`,`O`,`B`(`A`≠`O`,`B`≠`O`: ∠`AOB`such that`m`∠`AOB`is a real number (mod 2π)

**Between**- If
`A`,`B`and`C`are distinct points, we say that`B`is*between*points`A`and`C`(A*B*C) if and only if .`d`(`A`,`B`) +`d`(`B`,`C`) =`d`(`A`,`C`) **Line Segment**- The points
`A`and`C`together with all points`B`between`A`and`C`form the*line segment*AC. **Half-Line; Endpoint**- The
*half-line*`m'`with*endpoint*`O`is defined by two points`O`,`A`in line`m`(`A`≠`O`) as the set of all points`A'`of`m`such that`O`is not between`A`and`A'`. **Parallel**- If two distinct lines have no points in common they are
*parallel*. A line is always regarded as parallel to itself. **Straight Angle; Right Angle; Perpendicular**- Two half-lines
`m`,`n`through`O`are said to form a*straight angle*ifm∠ Two half-lines`mOn`=π.`m`,`n`through`O`are said to form a*right angle*ifm∠ in which case we say that`mOn`= ±π/2,`m`is*perpendicular*to`n`. **Triangle; Vertices; Degenerate Triangle**- If
`A`,`B`,`C`are three distinct points the three segments`AB`,`BC`,`CA`are said to form a*triangle*with sides`AB`,`BC`,`CA`and*vertices*`A`,`B`,`C`. If`A`,`B`and`C`are collinear then triangle`ABC`is said to be*degenerate*. **Similar; Congruent**- Any two geometric figures are
*similar*if there exists a one-to-one correspondence between the points of the two figures such that all corresponding distances are in proportion and corresponding angles have equal measures (except, perhaps, for their sign). Any two geometric figures are*congruent*if they are similar with a constant of proportionality,`k`=1.

**Postulate I.***Postulate of Line Measure*. The points`A`,`B`, ..., of any line can be put into 1:1 correspondence with the real numbers`x`so that| for all points`x`-_{b}`x`| =_{a}`d`(`A`,`B`)`A`and`B`.**Postulate II.***Point-line Postulate*. One and only one line,`l`, contains any two distinct points`P`and`Q`.**Postulate III.***Postulate of Angle Measure*. The half-lines (or rays)`l`,`m`,`n`, ..., through any point`O`can be put into 1:1 correpondence with the real numbers`a`(mod 2π) so that if`A`and`B`are points (other than`O`) of`l`and`m`, respectively, the difference of the numbers associated with lines`a`-_{m}`a`(mod 2π)_{l}`l`and`m`is m∠`AOB`.**Postulate IV.***Postulate of Similarity*. If in two triangles`ABC`and`A'B'C'`and for some constant`k`>0, ,`d`(`A'`,`B'`) =`kd`(`A`,`B`) , and`d`(`A'`,`C'`) =`kd`(`A`,`C`)m∠ `B'A'C'`= m±∠`BAC`, then also ,`d`(`B'`,`C'`) =`kd`(`B`,`C`)m∠ and`C'B'A'`= ±m∠`CBA`,m∠ `A'C'B'`= ±m∠`ACB`.

- A Set of Postulates for Plane Geometry (Based on Scale and
Protractors), G. D. Birkhoff,
*Annals of Mathematics*, 33, 1932 *Roads to Geometry*, E. Wallace & S. West, Prentice-Hall (Sect 2.5 & App. C)

R. I. Campbell, campbell@math.umbc.edu 10 Feb, 2002