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

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