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point

line

straight line

surface

that which has no part

breathless length ... --the extremities of a line are points

a line which lies evenly with the points on itself

that which has length and breadth only ... --the extremities of…

point

that which has no part

line

breathless length ... --the extremities of a line are points

Point

Line

Straight line

Surface

That which has no part. The extremities of a line.

Breadthless length. The extremities of a surface.

A line which lies evenly with the points on itself.

That which has length and breadth only.

Point

That which has no part. The extremities of a line.

Line

Breadthless length. The extremities of a surface.

point

line

straight line

surface

that which has no part

breathless length ... --the extremities of a line are points

a line which lies evenly with the points on itself

that which has length and breadth only ... --the extremities of…

point

that which has no part

line

breathless length ... --the extremities of a line are points

#1: AB and CD are equally distant from…

Point D is center of the circle

Straight line joined from G to F and p…

A line joined from F to G passes throu…

#1: Circle ABCD, AB and CD are equal straight lines... #2: Circl…

Circle ABC with point D inside it, and DA, DB, and DC fall up…

Circles ABC, ADE touching internally at A, and F is the cente…

Circles ABC, ADE touching at A externally, F the center of AB…

#1: AB and CD are equally distant from…

#1: Circle ABCD, AB and CD are equal straight lines... #2: Circl…

Point D is center of the circle

Circle ABC with point D inside it, and DA, DB, and DC fall up…

Prop 4

Prop 5

Prop 6

Prop 8

SAS

If a triangle is isosceles then the base angles are congruent.

If two angles in a triangle are congruent then the two opposi…

If three corresponding sides of one triangle are congruent to…

Prop 4

SAS

Prop 5

If a triangle is isosceles then the base angles are congruent.

first postulate

second postulate

third postulate

fourth postulate

To draw a straight line from any point to any point.

To produce a finite straight line continuously in a straight…

To describe a circle with any center and distance.

That all right angles are equal to one another.

first postulate

To draw a straight line from any point to any point.

second postulate

To produce a finite straight line continuously in a straight…

Definition 1

Definition 2

Definition 4

Definition 10

A point is that which has no part.

A line is breadthless length.

A straight line is a line which lies evenly with the points o…

When a straight line set up on a straight line makes the adja…

Definition 1

A point is that which has no part.

Definition 2

A line is breadthless length.

I.1

I.2

I.3

I.4

Construct an equilateral triangle on a given line.

Construct a line equal to a given line at a given point.

Cut a line equal to a given line from a given longer line.

If two triangles have two sets of sides and the angle between…

I.1

Construct an equilateral triangle on a given line.

I.2

Construct a line equal to a given line at a given point.

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Midpoint

Slope

Distance

Parallel lines

The middle point of a line segment

Thus describes the direction and steepness of a line.

This describes the length of a segment or how far apart an ob…

Two lines that never meet.

Midpoint

The middle point of a line segment

Slope

Thus describes the direction and steepness of a line.

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

point

line

line segment

ray

A location; it has no dimension. Represented by a dot.

A straight path that continues forever in both directions; ha…

Part of a line; has two endpoints.

Part of a line; has one endpoint and the other end goes on fo…

point

A location; it has no dimension. Represented by a dot.

line

A straight path that continues forever in both directions; ha…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

1. a point

2. a line

3. ends (extremities) of a line

4. a straight line

A point is that which has no parts.

A line is length without width.... [We normally today use the te…

The ends of a line are points.

A straight line is a line that lies symmetrically with the po…

1. a point

A point is that which has no parts.

2. a line

A line is length without width.... [We normally today use the te…

Book 1

Book 2

Book 3

Book 4

-triangles, parallelograms, parallel lines... -1.47 Pythagorean…

-Divided lines and areas they give rise to... -straight lines di…

-Circles ... -diameter, tangent, angles in circles, segments of…

-Regular polygons and their relationship with circles... -inscri…

Book 1

-triangles, parallelograms, parallel lines... -1.47 Pythagorean…

Book 2

-Divided lines and areas they give rise to... -straight lines di…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Composition of Transformations

Congruent segments

Degree of Rotation

Dilation

combination of two or more transformations

line segments whose lengths are equal

number of degrees a figure is rotated about the point (center…

a similarity transformation in which a figure is enlarged or…

Composition of Transformations

combination of two or more transformations

Congruent segments

line segments whose lengths are equal

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

father of Geometry

segment addition postulate

Postulate 1:1

Point

Euclid

If A, B, C, are collinear and B is between A and C, then AB+…

through any 2 points there is exactly one line

A location in the coordinate plane. Symbol- A

father of Geometry

Euclid

segment addition postulate

If A, B, C, are collinear and B is between A and C, then AB+…

"The Knowledge of which geometry aims…

"There is no royal road to geometry"

"... being perpetually charmed by his…

"Which was to be proved."

Plato, The Republic

Euclid

Plutarch, The life of Archimedes

Euclid, The elements

"The Knowledge of which geometry aims…

Plato, The Republic

"There is no royal road to geometry"

Euclid

Undefined Terms

Angle

Acute Angle

Euclid

point, line, plane

the union of two rays with a common endpoint

An angle whose measure is less than 90º

The father of Geometry, wrote "The Elements"

Undefined Terms

point, line, plane

Angle

the union of two rays with a common endpoint

D1. Point

D2. Line

D3. Extremities of lines

D4. Straight line

A point is that which has no part.

A line is a breadthless length.

The extremities of a line are points.

A straight line is a line which lies evenly with the points o…

D1. Point

A point is that which has no part.

D2. Line

A line is a breadthless length.

Ways to prove lines parallel (7 of them)

corresponding angles

Alternative Interior angles

Alternative Exterior Angles

converse corresponding angle post... converse alt interior angle…

1 & 5,... 3 & 7,... 2 & 6, ... 4 & 8

3 & 6... 4 & 5

1 & 8... 2 & 7

Ways to prove lines parallel (7 of them)

converse corresponding angle post... converse alt interior angle…

corresponding angles

1 & 5,... 3 & 7,... 2 & 6, ... 4 & 8

Who is the "father of geometry"

What did Euclid write?

Point

Line

Euclid of Alexandria

The Elements

Indication of an exact location. A point has location and not…

A straight, unbroken set of points that goes on forever. It h…

Who is the "father of geometry"

Euclid of Alexandria

What did Euclid write?

The Elements

point

line

line segment

ray

A location; it has no dimension. Represented by a dot.

A straight path that continues forever in both directions; ha…

Part of a line; has two endpoints.

Part of a line; has one endpoint and the other end goes on fo…

point

A location; it has no dimension. Represented by a dot.

line

A straight path that continues forever in both directions; ha…

Euclid

point

line

line segment

Greek mathematician who organized known geometry in the 4th c…

shown by a dot, it is a location in space

extends infinitely

part of a line with a beginning and ending point

Euclid

Greek mathematician who organized known geometry in the 4th c…

point

shown by a dot, it is a location in space

Proposition 1

Proposition 2

Propostion 3

Proposition 4

On a given finite line to construct an equilateral triangle

To place a given point as an extremity a strsight line equal…

Given 2 unequal straight lines, to cut off from the greater a…

If 2 triangles have 2 sides equal to 2 sides amd have angles…

Proposition 1

On a given finite line to construct an equilateral triangle

Proposition 2

To place a given point as an extremity a strsight line equal…

line segment

collinear

noncollinear

concurrent

a part of a line bounded by 2 endpoints

points that are on the same line

points that are not on the same line

lines that contain the same point

line segment

a part of a line bounded by 2 endpoints

collinear

points that are on the same line

Postulate 1

Postulate 2

Postulate 3

Postulate 4

a straight line can be drawn from any point to any point

a finite straight line can be produced continuously in a stra…

a circle may be described with any center and distance

all right angles are equal to one another

Postulate 1

a straight line can be drawn from any point to any point

Postulate 2

a finite straight line can be produced continuously in a stra…

What is geometry about?

What does the word geometry mean?

Inductive reasoning/induction

In deductive reasoning, what is the "i…

Shapes, such as triangles, circles, squares, and rectangles.…

"earth measurement."

We look at several examples of something and find that a fact…

assumption (premise)

What is geometry about?

Shapes, such as triangles, circles, squares, and rectangles.…

What does the word geometry mean?

"earth measurement."

Inscribed Circle

Circumscribed Circle

Concurrent Lines

Incenter

A circle that is inside a triangle and is tangent to its sides.

A circle that goes around a triangle and passes through its v…

Lines that meet at a common point.

The center of the inscribed circle of a triangle.

Inscribed Circle

A circle that is inside a triangle and is tangent to its sides.

Circumscribed Circle

A circle that goes around a triangle and passes through its v…

Euclid

point

line

width;thickness

___________________ wrote the first book on Geometry

A _____________________ is a fixed position in space.

The shortest distance between two points is called a ________…

breadth

Euclid

___________________ wrote the first book on Geometry

point

A _____________________ is a fixed position in space.

antipodal points

great circle

arc

radius

the endpoints of a diameter of a sphere

circle whose center is also the center of a sphere

a part of the circumference of a circle or other curve

A line segment with one endpoint at the center of a circle an…

antipodal points

the endpoints of a diameter of a sphere

great circle

circle whose center is also the center of a sphere

Geometry

Thales of Miletus

Pythagoras

Euclid

Means "to measure the earth."

Used geometry to measure the height of the pyramids and the d…

Ancient Greek philosopher and mathematician. Proved Pythagore…

Ancient Greek philosopher and mzthematician. Father of Geometry

Geometry

Means "to measure the earth."

Thales of Miletus

Used geometry to measure the height of the pyramids and the d…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…

Post. 1

Post. 2

Post. 3

Post. 4

A straight line segment can be drawn joining any two points.

Any straight line segment can be extended indefinitely, to fo…

Given any straight line segment, a circle can be drawn having…

All right angles are congruent.

Post. 1

A straight line segment can be drawn joining any two points.

Post. 2

Any straight line segment can be extended indefinitely, to fo…

Geometry

Euclidean Geometry

Undefined terms

Point

word composed of two Greek terms, "geos" meaning earth and "m…

identifies a geometry which Euclid formalized and which was a…

terms in Euclidean geometry which mathematicians consider onl…

one of the undefined terms; it has only position and has no l…

Geometry

word composed of two Greek terms, "geos" meaning earth and "m…

Euclidean Geometry

identifies a geometry which Euclid formalized and which was a…