# 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 **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**).