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Ultimate Geometry (Terms, Postulates, Theorems, etc.)

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point
a location (undefined term)
plane
a flat surface made up of points; has no depth and extends indefinitely in all directions (undefined term)
line
made up of points and has no thickness or width (undefined term)
collinear points
points on the same line
axioms/postulates
statements that are accepted without proofs
theorems
statements that can be proven
coplanar
points that lie on the same plane
space
a boundless, three-dimensional set of all points; can contain lines and planes
congruent line segments
two line segments with equal lengths
ray
a type of line that has one endpoint
opposite rays
two rays are opposite to each other iff they have the same endpoint and they are collinear
angle
two rays that share an endpoint
angle bisector
rays, line segments, or lines that divide and angle into two equal parts
complimentary angles
two angles whose measures add up to 90°
supplementary angles
two angles whose measures add up to 180°
90-x
compliment of an angle
180-x
supplement of an angle
adjacent angles
two angles that share a side, share a vertex, and have no common interior point
linear pairs
a pair of adjacent angles whose non-common sides are opposite rays
regular polygon
a polygon where all the sides and angles are congruent
triangle
3 sides
quadrilateral
4 sides
pentagon
5 sides
hexagon
6 sides
heptagon
7 sides
octagon
8 sides
nonagon
9 sides
decagon
10 sides
hendecagon
11 sides
dodecagon
12 sides
conditional statement
a statement that can be written in if-then form
converse
the statement formed by exchanging the hypothesis and conclusion of a conditional statement
inverse
the statement formed by negating both the hypothesis and conclusion of a conditional statement
contrapositive
the statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement
symmetric property
if a=b then b=a
skew lines
if two lines are not on the same plane, they cannot intersect
transversal
a line that intersects two (or more) lines at two different points
corresponding angles (postulate)
if two parallel lines are cut by a transversal, then the corresponding angles are congruent
converse of corresponding angles (postulate)
if corresponding angles are congruent, then the lines are parallel
scalene
type of triangle where all sides are different lengths
isosceles
type of triangle where at least two sides have equal lengths
equilateral
type of triangle where all sides are equal in length
acute
type of triangle where all angles are acute
obtuse
type of triangle that contains exactly one obtuse angle
right
type of triangle that contains exactly one right angle
angle sum theorem
given a triangle, the sum of the measures of the interior angles is 180°
alternate interior angle (theorem)
if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent
consecutive interior angle (theorem)
if two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary
alternate exterior angle (theorem)
if two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent
third angle theorem
if two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent
exterior angle theorem
the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
sss
if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent
sas
if two sides and the included angle of one triangle are congruent to two sides and the included angle another triangle, then the triangles are congruent
asa
if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent
aas
if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent
HL
if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent
isosceles triangle theorem
if two sides of a triangle are congruent, then the angles opposite those sides are congruent
diagonal
a line segment that connects two non-adjacent vertices
CPCTC
corresponding parts of congruent triangles are congruent
perpendicular bisector (of a line segment)
a line that cuts through another line segment at its midpoint and bisects it at a 90° angle
circumcenter
point of concurrency for perpendicular bisectors; equidistant from the vertices
centroid
point of concurrency for medians; 1/3 of the whole
incenter
point of concurrency for angle bisectors; equidistant from each side
orthocenter
point of concurrency for altitudes
right triangle
in what type of triangle is the orthocenter a vertex?
equilateral triangle
in what type of triangle is the orthocenter and circumcenter the same?
never
in what type of triangle is the centroid outside of the triangle?
triangle inequality theorem
in a triangle, the sum of the measures of two sides is greater than the measure of the third side
hinge theorem
if two sides of one triangle are congruent to two sides of another triangle and the included angle in the first triangle is greater than the included angle in the second triangle, then the remaining side of the first triangle is greater than the remaining side of the second triangle
converse of hinge theorem
if two sides of one triangle are congruent to two sides of another triangle and the third side of the first triangle is greater than the third side of the second triangle, then the angle opposite the third side in the first triangle is greater than the angles opposite the third side in the second triangle