# Words

#### Basics

• point A: $A$
• line passing through points A and B: $\overleftrightarrow{AB}$
• ray starting at endpoint A and passing through point B: $\overrightarrow{AB}$
• $\overrightarrow{ST}$ and $\overrightarrow{SR}$ are opposite rays if $S$ lies between $R$ and $T$ on $\overleftrightarrow{RT}$
• segment starting at point A and ending at point B: $\overline{AB}$
• length of segment starting at point A and ending at point B: $AB$
• the distance between any two points equals the absolute value of the difference of their coordinates
• plane $M$ defined by any three points in $\{A, B, C, D\}$
• space is the set of all points
• colinear points lie on the same line
• coplanar points lie in the same plane
• the intersection of two figures is the set of points that are in both figures
• postulates or axioms are statements accepted without proof
• congruent ( $\cong$ ) objects have the same size and shape
• congruent segments have equal lengths
• the midpoint of a segment is the point that divides the segment into two congruent segments
• a bisector of a segment is a line, segment, ray, or plane that intersects the segment at its midpoint
• an angle ( $\angle$ ) is formed by two rays with the same endpoint
• the common endpoint is called the vertex of the angle
• the measure (degrees or radians between sides) of angle $\angle AOC$ with vertex $O$ and sides corresponding to rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is denoted $m \angle AOC$
• an acute angle has measure $\lt 90^\circ = \frac{\pi}{2} rad$
• a right ( rt. ) angle has measure $= 90^\circ = \frac{\pi}{2} rad$
• an obtuse angle has measure $\gt 90^\circ = \frac{\pi}{2} rad$
• a straight angle has measure $= 180^\circ = \pi rad$
• congruent angles have equal measures
• adjacent ( adj. ) angles are two angles in the same plane with a common vertex and a common side but no common interior points
• the bisector of an angle is the ray that divides the angle into two congruent angles

#### Axioms and Theorems

• A1 : The distance between any two points equals the absolute value of the difference between their coordinates
• A2 : If $B$ is between $A$ and $C$, then $AB + BC = AC$
• A3 : The difference between two angles that share a ray $\angle AOQ$ and $\angle AOP$ is the absolute value of the difference of their measures $m \angle AOQ - m \angle AOP$
• A4 : If point $B$ lies in the interior of  $\angle AOC$, then $m \angle AOB + m \angle BOC = m \angle AOC$ .  If $\angle AOC$ is a straight angle, then $m \angle AOB + \angle AOC = 180^\circ$
• A5 : A line is defined by two points, a plane is defined by three points, and a space contains four points not all in the same plane
• A6 : There is exactly one line through any two points
• A7 : There is at least one plane through any three points, and exactly one plane through three non-collinear points
• A8 : If two points are in a plane, the line that contains the points is in that plane
• A9 : If two planes intersect, their intersection is a line

• T1-1 : If two lines intersect, they intersect at exactly one point
• Pf. Given that two arbitrary (but distinct) lines $l1$ and $l2$ intersect at point $X$, assume that $l1$ and $l2$ also intersect at point $Y$.  This means that $l1$ passes through both points $X$ and $Y$, and $l2$ also passes through both points $X$ and $Y$.  Since there is exactly one line through any two points (by A6), $l1$ and $l2$ must be the same line.  This contradicts our given, so $l1$ and $l2$ must intersect at exactly one point ($X$).
• T1-2 : Through a line and a point not on the line there is exactly one plane
• Pf.  Given an arbitrary line $l$ and an arbitrary point $X$ not on line $l$, let line $l$ be defined by points $A$ and $B$ as $\overleftrightarrow{AB}$ (by A5).  Since $X$ is not on $\overleftrightarrow{AB}$, points $X, A$, and $B$ are not collinear.  Since there is exactly one plane through three non-collinear points (by A7), points $X, A$, and $B$ define exactly one plane (call this plane $M$).  Since points $A$ and $B$ define line $l$, point $X$ and line $l$ define exactly one plane ($M$).
• T1-3 : If two lines intersect, then exactly one plane contains the lines
• Pf. Given that two arbitrary (but distinct) lines $l1$ and $l2$ intersect at point $X$, there is some point $Y$ on $l2$ but not $l1$.  By T1-2, there is exactly one plane (call this plane $M$) that contains line $l1$ and point $Y$.  Since line $l1$ is in plane $M$, point $X$ on line $l1$ is also in plane $M$.  Since points $X$ and $Y$ are on line $l2$, and there is exactly one line through any two points (by A6), line $l2$ is also in plane $M$ (by A8).

# SVG

### Basic Shapes

(from Dashing D3, mostly so that I have a reference for the SVG). W3C doc.