College Geometry

Front 1

How do we define a plane?

Back 1

It is the universal set of all points under consideration.

Front 2

How do we define a line?

Back 2

It is a subset of a plane.

Front 3

What are our 3 undefined terms?

Back 3

1) a point (an element)
2) a line (subset)
3) a plane (universal set)

Front 4

What is an axiom?

Back 4

A statement we are going to accept as true w/o proof; it is fundamental--no one would question it.

Front 5

What is a theorem?

Back 5

It is a statement that can be proven through axioms.

Front 6

What are definitions?

Back 6

They define new words in terms of the undefined words.

Front 7

Axiom 1 - The point-line incidence axiom

Back 7

Given any two different poitns, there is exactly one line which contains them. OR A unique straight line may be drawn from any point to any other point.

Front 8

Axiom 2 - The ruler axiom

Back 8

States that there is a function f: L-->R such that f is
1) 1 to 1
2) Onto R
3) definition of function
4) AB = |f(A) - f(B)|

Front 9

Definition 2.1 - Betweeness

Back 9

A, B, C are points on L
f = function
B is btw A & c iff f(B) is btw. f(A) & f(C). We write A-B-C.

Front 10

Definition 2.2 - Congruence of line segments: AB is congruent to CD means AB = CD

Back 10

Given points A & B.
There is a line containing A & B (Axiom 1).
Define AB = { x = L | f(A) ≤ f(x) ≤ f(B)}
AB ≅ CD (distance of two is same.)

Front 11

Definition: Ray AB

Back 11

ray AB = AB U {x ∈ L | ~( x-A-B )}

Front 12

Definition 2.5 (Angle)

Back 12

Let A, B, and C be points which are not on the same line. Then ray AB U ray AC is called the angle BAC and denoted by angle BAC or angle CAB. We may use the shorthand notation angle A to denote angle BAC.

Front 13

Axiom 3 (Pasch's Separation Axiom for a Line)

Back 13

Given a line L in the plane, the points in the plane which are not on L form two sets, H1 and H2, called half-planes. L is called the boundary of the half-planes. The half-planes are called the sides of L. The half-planes satisfy the following two conditions:
1) if A & B are opints on the same half-plane then segment AB is wholly in that half-plane
2) if A & B are points not on the same half-plane, then line segment AB intersects L.

Front 14

Definition 2.6 ( Interior of an Angle)

Back 14

Let A, B, and C be three noncolinear points. Let Hb(AC) be the half-plane determined by AC containing B and let Hc (AB) be the half-plane determined by AB containing C. The inside or interior of angle BAC is defined to be Hb(AC intersect Hc (AB).

Front 15

Axiom 4 (The angle measurement axiom)

Back 15

To every angle, there corresponds a real number between 0 and 180 degrees called the measure or size of the angle. We denote the measure of angle BAC by m<BAC.

Front 16

Definition 2.7 (Congruent angles)

Back 16

If m<BAC = m<PQR then we say the angles <BAC and <PQR are congruent.

Front 17

Axiom 5 (The Angle Construction Axiom)

Back 17

Let AB lie entirely in the boundary of some half-plane H. For every r such that 0 deg. < r < 180 deg, there is exactly 1 ray AC where C is in H and m<CAB = r.

Front 18

Axiom 6 (The Angle Addition Axiom)

Back 18

If D is a point in the interior of <BAC, then m<BAC = m<BAD + m<DAC.

Front 19

Definition 2.8 (Supplementary Angles)

Back 19

If A-B-C and D is any point not on line AC then the angles <ABD and <DBC are said to be supplementary (180) angles.

Front 20

Definition 2.9 Vertical Angles

Back 20

Suppose two liens AA' and BB' intersect at point X. Then the angles <AXB and A'XB' are said to be vertical angles.

Front 21

Axiom 7 (The supplementary angles axiom)

Back 21

If two angles are supplementary, then their measures add to 180 degrees.

Front 22

Definition 2.10 (Triangle)

Back 22

Let A, B and C be 3 noncolinear pts. We define the triangle ABC to be AB U BC U CA.

Front 23

Definition 2.11 (Interior of a Triangle)

Back 23

WE define the interior of traingle ABC to be Ha(BC) inter. Hb(AC) inter. Hc(AB)

Front 24

Definition 2.12 (Congruent Triangles)

Back 24

Two triangles are congruent if it is possible to find a 1-to-1 correspondence from the vertices of one triangle to the vertices of the other such that:
1) corresponding angles are congruent
2) sides with corresponding endpoints are congruent.

Front 25

Axiom 8 (The Side-Angle-Side congruence axiom)

Back 25

If

1) 1 angle in a triangle is congruent to an angle in a 2nd triangle

2) one side forming the angle in the 1st triangle is congruent to one side forming the congruent angle in the 2nd triangle

3) The remaining side forming the angle in the first triangle is congruent to the remaining side forming the congruent angle in the 2nd triangle

then the two triangles are congruent.

Front 26

2.4 The Famous Parallel Axiom
Axiom 9 (Playfair's parallel Axiom)

Back 26

Given a point P off a line L, there is at most 1 linein the plane through P not meeting L.

Front 27

Theorem 2.13: Two lines intersect in at most

Back 27

one point.

Front 28

Theorem 2.14 (measurement and betweenness)

Back 28

1. AB = BA
2. If A-B-C, then AC = AB + BC
3. If A, B, and C are 3 different points on a line, then exactly 1 of them is between the other two.

Front 29

Theorem 2.15 (Extendibility)

Back 29

If A and B are any two points, then the segment AB can be extended by any positive distance on either side of segment AB.

Front 30

Theorem 2.16 (Midpoints)

Back 30

Every segment has a midpoint.

Front 31

Theorem 2.17.

Back 31

Consider a ray AB on line L. AB = AB U {X subset L | A-B-X}

Front 32

Theorem 2.18

Back 32

If a ray AB has an endpoint A on line L and B is not on L. Then all points of the ray AB, except A, lie on teh same side of L as B.

Front 33

Theorem 2.19

Back 33

Consider triangle ABC. Extend AC past C to a point D. Let M be any point on BC other than B and C and extend AM to a point E past M. THen E is in the interior of <BCD.

Front 34

Theorem 2.2

Back 34

Vertical angles are congruent.