College Geometry Test 2

Front 1

Exterior angle

Back 1

An angle that is a supplement to an angle in a trianlge.

Front 2

Remote Interior angles

Back 2

With respect to the angle in a triangle for which an exterior angle is the supplement, the other to angles are referred to as this.

Front 3

Exterior Angle Theorem

Back 3

In any Hilbert Plane, an exterior angle of a triangle is greater than either remote interior angle.

Front 4

The Hypotenuse leg Criterion

Back 4

Two right triangles are congruent if the hypotenuse and a leg of one are congruent, respectively, to a leg and hypotenuse of the other.

Front 5

Prop 4.3 (Midpoints

Back 5

Every line segment has a unique midpoint.

Front 6

Angle Bisector

Back 6

Given angle BAC and suppose ray AG is interior to BAC and BAG is congruent to GAC then AG bisects BAC.

Front 7

Perpendicular Bisector of line AB

Back 7

a line l such that line AB is perpendicular to l at the midpoint of line AB.

Front 8

Prop 4.4 (Existence of Bisectors

Back 8

a) Every angle has aunique Bisector

b) Every segment has a unique perpendicular bisector.

Front 9

Prop 4.5

Back 9

In triangle ABC, the greateer angle lies opposite the greater side and the greater side lies oposite the greater angle. hence segment BA > BC iff angle C > A.

Front 10

Archimedes' Axiom

Back 10

If CD is any segment, A any point, and R any ray emanating from point A, then for every point B not equal to A on r, there is a natural number n such that when CD is laid off n times along r starting at A a point E is reached along r so that n*CD=AE and either B=E or B is between A and E.

Front 11

Thm 4.4 Angle Measurement Thm

Back 11

0) m angle A is a real number such that 0<mA<180.

1)mA = 90 iff A is a right angle

2)mA = mB iff A is congruent to B

3)If ray AC is interior to angle DAB, then mDAB=mDAC+mCAB.

4)For every real number x with 0<x<180, there is an angle A such that mA=x.

5) If angle B is supplementary to A, then A+B=180

6)angle A>B iff mA>mB.

7)Absolute Value of AB is a positive real number and OI =1. (unit segment)

8)AB congruent to CD iff (av)AB = (av)CD

9)A*B*C iff (av)AC=(av)AB+BC

10)AC<AC iff (av)AB<AC

11)For every positive real number x, there is a line segment AB such that (av)AB =x

Front 12

Thm 16.1 The triangle inequality

Back 12

Given triangle ABC, the sum of the lengths of any two sides is greater than the length of the 3rd side.

Front 13

Euclid's 5th postulate

Back 13

if 2 lines are intersected by a transversal such that the sum of the degree measuress of the two interior angles on one side of the transversal is less than 180, then the two lines meet on that side of the transversal.

Front 14

Bi-right

Back 14

Given a quad ABCD suppose that adjacent angles A and B are both right anngles. then it is this.

Front 15

Base

Back 15

segment joining vertex A to vertex B is called this (bottom one)

Front 16

Summit

Back 16

Side opposite the base is this

Front 17

Sides

Back 17

Other 2 sides of quad

Front 18

Summit Angles

Back 18

Angles C and D will be referred to as this (top angles of quad)

Front 19

Isosceles bi-right quad

Back 19

If the sides of a bi-right quad are congruent it is this

Front 20

Saccheri Quad

Back 20

a bi-right quad with congruent sides

Front 21

Lambert Quad

Back 21

A quad with at least 3 right angles. The remaining angle is referred to as the 4th angle.

Front 22

Equidistant locus from l to P

Back 22

let l be a line and let PQ be a segment that is perpendicular to l at point Q. This is the set of all points P' on the same side of l as P such that if Q' is the foot of the perpendicular from P' to l, then PQ congruent to P'Q'.

Front 23

Clavius' Axiom

Back 23

For any line l and any point P not on l, the equidistant locus to l through P is the set of all points on a line through P.

Front 24

Similar

Back 24

2 Triangles are this iff there is a 1-1 correspondence between their vertices so that corresponding angles are congruent.

Front 25

Wallis'

Back 25

Given any triangle, ABC, and any line segment, DE, there is a triangle DEF with DE as one of its sides that is similar to ABC.

Front 26

Clairaut's Axiom

Back 26

Rectangles exist.

Front 27

Proclus' Thm

Back 27

The euclidean parallel postulate holds in a Hilbert plane if and only if the plane is semi euclidean and Aristotle's unboundedness Axiom holds. EPP holds in an archimedian semi Euclidean plane.

Front 28

Euclid's Proposition 1

Back 28

Given any segment there exists an equilateral triangle with that segment as one of its sides.

Front 29

Circle circle continuity principle

Back 29

If Circle Y has 1 point inside and 1 point outside the circle Y' the 2 circles meet in 2 points

Front 30

Line Circle continuity principle

Back 30

If a line passes through a point inside a circle, then the line intersects the circle in 2 points.

Front 31

Segment Circle continuity principle

Back 31

If an endpoint of a segment is inside a circle and the other endpoint outside the circle then the segment intersects the Circle at some point in between.

Front 32

Aristotle's angle unboundedness Axiom

Back 32

Given any side of an acute angle and any segment AB there exists a point Y on the given side of the angle such that X is the foot of the perpendicular across from Y and XY>AB.

Front 33

Hilbert's euclidean Axiom on parallelism

Back 33

For every line L and every Point P not incident with L there is at most one line M through p and parallel to l.

Front 34

The alternate interior angle theorem

Back 34

In any Hilbert plane if 2 lines are cut by a transversal so that a pair of alternate interior angles are congruent then the 2 lines are parallel.

Front 35

SAA

Back 35

In triangle ABC and triangle DEF if AC is congruent to DF and angle A is congruent to angle D and angle B is congruent to angle E then the triangles are congruent.

Front 36

Converse to alternate interior angle theorem

Back 36

If two lines are parallel and T is the transverse to both then the corresponding alternate interior angles are congruent.

Front 37

Uniformity theorem

Back 37

In any Hilbert plane if the summit angles of a saccheri quadrilateral are acute (respectively obtuse or right) then all Summit angles of saccheri quadrilaterals are acute (respectively obtuse or right).

Front 38

Semi euclidean

Back 38

Hilbert plane is semi euclidean if all saccheri quadrilaterals and Lambert quadrilaterals are rectangles. If the fourth angle of any Lambert quad is acute (respectively obtuse) we say it satisfies the acute angle hypothesis (respectively obtuse).

Front 39

Saccheri angle theorem

Back 39

In any Hilbert plane

A) if there exists a triangle with an angle sum < 180 then all triangle angle sums are <180. Equivalently the fourth angle of a Lambert quad and Summit angles of saccheri quads are acute.

B) if there exists a triangle with an angle sum= 180 then all triangles angle sum =180. Plane is semi euclidean (rectangles exist)

C) if there exists a triangle with an angle sum>180 then all triangles angle sums >180. Quads angles obtuse

Front 40

The non obtuse angle theorem

Back 40

A Hilbert plane in which Aristotle's Axiom is valid either is semi euclidean or satisfies the acute angle hypothesis.

Front 41

Saccheri-legendre theorem

Back 41

In an archimedian Hilbert plane all triangles angle sums are less than or equal to 180.

Front 42

Convex

Back 42

Quad ABCD is convex iff it has a pair of opposite sides such that one side is completely contained in the half plane of the other side.

Front 43

Defect of a triangle

Back 43

The difference of a triangles angles sums and 180

Front 44

Legendres theorem

Back 44

Taking Archimedes Axiom as valid there exists an acute angle A and if a point D is in the interior of the angle, there exists a line through D that intersects both sides of A then the angle sum of any triangle must be 180

Front 45

Absolute geometry (AKA neutral geometry)

Back 45

Geometry implied by all the axioms besides parallelism axioms.

Front 46

Euclidean plane

Back 46

A Hilbert plane in which Hilbert's euclidean Axiom of parallelism is valid as well as circle circle continuity principle

Front 47

Interior angles

Back 47

The angles between two lines formed by a transversal

Front 48

Alternate interior angles

Back 48

The angles between two lines that are on opposite sides of the transversal