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.
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