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- StudyBlue
- Michigan
- University of Michigan - Dearborn
- Mathematics
- Mathematics 331
- Jabbusch
- final review

Natasha G.

Interpretation

Particular way of giving meaning to undefined terms

Model

Interpretation is model when axioms hold true in that interpretation

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Two lines m,l are parallel if...

There is not a point P such that P is on m and P is on l

A point B is **between** A,C if.....

A,B,C are collinear and AB+BC=AC

Two line segments AB and CD are **congruent** if....

They have the same length

Two angles are **congruent** if....

their angle measures are the same

An angle is a **right angle** if...

the measure of the angle = 90

A function f is a **coordinate function** if...

f:*l*->**R** is both 1-1 and onto such that if points P,Q on *l* correspond to x,y in **R**, then PQ=abs. value (f(x)-f(y))

The ray AD is **between rays** AB and AC if....

A,B,C,D noncollinear points such that D is in the interior of angle CAB

Two lines m,n are **perpendicular** if....

points A,B are on m and points B,C are on n such that the measure of angle ABC =90

Angles that form a linear pair with interior angles

Draw picture

A **quadrilateral** is **convex** if...(draw picture)

Each vertex of the quadrilateral is contained in the interior angle formed by the other 3 vertices

A metric is a function D:PxP->R ST

1. D(P,Q)=D(Q,P)

2. D(P,Q) >=0

3. D(P,Q)=0 iff P=Q

Midpoint, M, of two points (draw picture)

Let A,B be two distinct points. M is called the midpoint of AB if A*M*B and AM=MB

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Theorem of existence and uniqueness of midpoints

If A,B are two distinct points, then there exists a unique point M ST M is the midpoint of AB

Angles BAC and DAE form a vertical pair if...(draw picture)

rays AB and AE are opposite and rays AC and AD are opposite, or if rays AB and AD are opposite and rays AC and AE are opposite

Vertical Angles theorem

vertical angles are congruent

Two angles BAD and DAC form a linear pair if... (draw picture)

AB and AC are opposite rays

Linear Pair Theorem

If two angles form a linear pair, then the sum of their angles is 180

Two Triangles are congruent if...

there exists a correspondence between vertices of the first triangle and vertices of the second triangle ST corresponding sides and angles are congruent

Which scenarios guarantee triangle congruence in NG?

SAS, AAA, AAS, ASA

A triangle is called isosceles if....

it has a pair of congruent sides

Isosceles triangle theorem

Base angles of an isosceles triangle are congruent

Exterior Angles Theorem

The measure of an exterior angle is strictly greater than the measure of either remote interior angle

Theorem: existence and uniqueness of perpendiculars

For every line l and every point p, there exists a line m ST P lives on m and m is perpendicular to l

In neutral geometry, do 3 angles determine a triangle?

no

Scalene inequality

In any triangle, the greater side lies opposite the greater angle, and the greater angle lies opposite the greater side

Triangle inequality

If A,B,C are 3 noncollinear points, then AC<AB+BC

Alt. Int. Angle Theorem

If l, l' are lines cut by transversal t in such a way that alt. int. angles are congruent, then l and l' are parallel

Theorem: existence of parallels

If l is a line and P is an external point, then there is a line m ST P lies on m and m is parallel to l

T/F: elliptic parallel postulate is true in any model for NG

false

If l,m,n are 3 lines ST n is parallel to l and m is parallel to l, then

either m=n or m is parallel to n

Pythag theorem

If ABC is a right triangle with a right angle at C, then the square of the hypotenuse is equal to the square of the sum of the sides.

Saccheri-Legendre theorem

If ABC is any triangle, then the sum of the angles is less than or equal to 180

\Statements equivalent to Euclidean parallel postulate

CTAIAT

Euclid's 5th postulate

Angle sum postulate

Clairaut's Axiom

Transitivity of parallelism

Euclid's 5th postulate

Angle sum postulate

Clairaut's Axiom

Transitivity of parallelism

Clairaut's Axiom (NG)

There exists a rectangle

Saccheri Quad (draw picture)

quad ST base angles are right angles and shorter sides are congruent

Lambert Quad

quad in which 3 of the angles are right angles

review properties of Lambert and Saccheri quads

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