euclidean geometry

goes through

a point lies on a line

intersect

two lines that cross at a point

collinear

three or more points that lie on the same line

line segment

set consisting of points A and B and all points C such that A-C-B

ray

set consisting of line segment (AB) and all points C such that A-B-C

triangle

consists of 3 distinct, noncollinear points A,B,C and line segments AB, BC, AC

angle

3 distinct, noncollinear points A,B,C such that A is the vertex and rays AC and AB are sides

convex

for any points A,B on a surface S, the line segment AB is a subset of S

point D is interior to angle ABC

D lies on the C-side of AB and on the A-side of BC

point E is exterior to angle ABC

if it is not on the angle or interior

ray BD lies between rays BA and BC

A, B, C are not collinear and D is interior to angle ABC

A and C are on opposite sides of B

A-B-C

A and C are on the same side of B

B-C-A or B-A-C

congruent triangles

corresponding parts are congruent

equivalence relation

reflexive (x=x), symmetric (if x=y then y=x), transitive (if x=y, y=z, then x=z)

midpoint of AC given A-B-C

two segments AB and BC are equal

angle bisector of angle ABC

two angles ABD and DBC are equal

circle centered at C passing through A

set of points X such that line segment CA=CX

radius

segment from the center to any point on the circle

equilateral triangle

all line segments are equal

supplementary

angles on a straight (collinear) line add to 2 right angles, or 180 degrees

right

supplementary angles are congruent

perpendicular

2 lines meet to form right angles

interior angles

angles on the inside of parallel lines

exterior angles

angles on the outside of parallel lines

subetends/subtended by

line subtends angle (angle subtended by line)

parallel lines

lines that do not intersect

transversal

a line that intersects two or more lines

adjacent interior angles

angle pairs between two parallel lines on the same side of the transversal

alternate interior angles

angle pairs between two parallel lines on opposite sides of the transversal

polygon

n noncollinear points with n line segments connecting them such that no segments intersect anywhere but the vertices

isosceles triangle

triangle with at least two congruent sides

diameter

line segment with endpoints on the circle which passes through the center

chord

line segment with endpoints on the circle

minor/major arc

component of a circle divided by a chord

semi circle

when major and minor chords are the same length

central angle

an angle whose vertex is the center of the circle and whose sides are radii

inscribed angle

an angle whose vertex is on the circle and whose sides are chords of the circle

Thm: Line Separation

a point on a line divides the points on the line (except for the splitting point) into 2 nonempty, disjoint sets

Thm: Pasch

if a line intersects a line segment of a triangle, then it must also intersect a different side or the opposite vertex

Thm: Crossbar

if a ray is between two other rays, then the ray intersects the line segment connecting the two other rays at a point between their endpoints.

Thm: Thales

an angle inscribed in a semi circle is a right angle

Thm: Inscribed Angle

the measure of the central angle is twice the measure of the inscribed angle

Prop I.14: Vertical Angle Thm

if adjacent angles add to 180, then the sides that are not share form a straight line

Prop I.20: Triangle Inequality

in any triangle, the sum of the lengths of 2 sides is greater than the other side length

Prop I.8: SSS

if 2 triangles have 3 pairs of congruent sides, then the triangles are congruent

Prop I.26: ASA and AAS

(ASA) if the sides joining the 2 pairs of equal angles are congruent, or (AAS) the sides subtending one of the equal angles are congruent, the triangles are congruent

Prop. I.30: Parallel Transitivity

lines parallel to the same line are parallel to each other

Post. 1: Uniqueness

given any two points, there is a unique line passing through the points

Post. 5: Parallel Conditions

if 2 lines are cut by a transeversal and the sum of adjacent interior angles is less than 180, then the lines meet on that side of the transversal and the lines aren’t parallel

Post.: Playfair's

given a line and a point not on the line, there is a unique line that can be drawn through the point which is parallel to the given line

Post. 6: Incidence

every line has at least 2 points, there is at least 1 plane, and every plan contains at least 3 noncollinear points

Post. 7: Betweeness

betweeness is symmetric/collinear, points in between, points on the outside, order matters

Post. 8: Plane Separation

a line splits the plane into 2 disjoint convex sets

Post. 9: Congruence

congruence is additive, equality in area is additive, congruent triangles have equal area

Post. 11: Circular Continuity

a line segment starting in the circle and ending out of it will intersect once, but a line in the circle and going out of it will intersect twice

Post. 12: SAS

2 triangles have 2 sides equal to 2 respective sides, and the angle between these sides are also equal, then the triangles are congruent