Unit 1 Introduction/Constructions

•This unit covers the course introduction and class expectations. •It lays the basic groundwork for the entire year with definitions and commonly used terms and symbols.•This unit also covers how to manually construct various geometric figures using a compass and a straight edge.

Standards• SPI’s taught in Unit 1:• SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. • SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. • SPI 3108.4.1 Differentiate between Euclidean and non-Euclidean geometries. • CLE (Course Level Expectations) found in Unit 1:• CLE 3108.1.4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to

solve problems, to model mathematical ideas, and to communicate solution strategies. • CLE 3108.1.6 Employ reading and writing to recognize the major themes of mathematical processes, the historical development of mathematics, and

the connections between mathematics and the real world. • CLE3108.2.3 Establish an ability to estimate, select appropriate units, evaluate accuracy of calculations and approximate error in measurement in

geometric settings. • CLE 3108.4.4 Develop geometric intuition and visualization through performing geometric constructions with straightedge/compass and with

technology. • CFU (Checks for Understanding) applied to Unit 1:• 3108.1.3 Comprehend the concept of length on the number line. • 3108.1.4 Recognize that a definition depends on undefined terms and on previous definitions. • 3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized

vocabulary (e.g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polydrons, measurement tools, compasses, PentaBlocks, pentominoes, cubes, tangrams).

• 3108.1.12 Connect the study of geometry to the historical development of geometry. • 3108.1.14 Identify and explain the necessity of postulates, theorems, and corollaries in a mathematical system. • 3108.2.6 Analyze precision, accuracy, and approximate error in measurement situations.• 3108.4.1 Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true and discuss unique properties

of each. • 3108.4.6 Describe the intersection of lines (in the plane and in space), a line and a plane, or of two planes. • 3108.4.7 Identify perpendicular planes, parallel planes, a line parallel to a plane, skew lines, and a line perpendicular to a plane. • 3108.4.21 Use properties of and theorems about parallel lines, perpendicular lines, and angles to prove basic theorems in Euclidean geometry (e.g.,

two lines parallel to a third line are parallel to each other, the perpendicular bisectors of line segments are the set of all points equidistant from the endpoints, and two lines are parallel when the alternate interior angles they make with a transversal are congruent).

• 3108.4.22 Perform basic geometric constructions using a straight edge and a compass, paper folding, graphing calculator programs, and computer software packages (i.e., bisect and trisect segments, congruent angles, congruent segments, a line parallel to a given line through a point not on the line, angle bisector, and perpendicular bisector).

Euclidean Geometry

• Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria (300 BC).

• Euclid's text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content.

• The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms.

• Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.

Beyond Euclidean Geometry

Or…A boy’s dream to figure out the

weirdness in the world…

Explaining those other things

• Think of Nature, or other shapes that aren’t normally found/defined in Euclidean Geometry

• In other words, weird shapes which have unusual 3 dimensional properties –or even 2 dimensional properties- that aren’t easily calculated with our standard rules

The Geometry of Graphs

• In the early 1700s the city of Konigsberg Germany was connected by 7 bridges.

• People wondered if it was possible to walk through the city and only cross each bridge only once

• After trying several times, you might think no But this really isn’t a proof –without trying each and every possible combination

A simple look at the City

Leonhard Euler

• A famous Swiss mathematician at the time• They took the problem to him and asked him if

there was a mathematical model he might be able to devise to solve the problem of the bridges

• He invented a whole new kind of geometry called “Graph Theory.”

• Graph theory is now used to design city streets, analyze traffic patterns, and determine the most efficient public transportation routes –i.e. buses

He Couldn’t do it either, but…• Euler recognized that in order to succeed, a traveler in the middle of the

journey must enter a land mass via one bridge and leave by another, thus that land mass must have an even number of connecting bridges. Further, if the traveler at the start of the journey leaves one land mass, then a single bridge will suffice and upon completing the journey the traveler may again only require a single bridge to reach the ending point of the journey. The starting and ending points then, are allowed to have an odd number of bridges. But if the starting and ending point are to be the same land mass, then it and all other land masses must have an even number of connecting bridges.

• Alas, all the land masses of Konigsberg have an odd number of connecting bridges and the journey that would take a traveler across all the bridges, one and only one time during the journey, proves to be impossible!

New Terminology

• Vertex: This is a point• Edge: This is a line segment or curve that starts and ends

at a vertex• Graph: Formed by vertexes and edges• Odd Vertex: A vertex with an odd number of attached

edges• Even Vertex: A vertex with an even number of attached

edges• Traversable: A graph is traversable if it can be traced

without lifting the pencil from the paper and without tracing an edge more than once

Rules of Traversability

1. A graph with all even vertices is traversable. You can start at any vertex and end where you began

2. A graph with two odd vertices is traversable. You must start at either of the odd vertices and finish at the other

3. A graph with more than two odd vertices is not traversable

Example

• Let’s try a simple one:• Is this graph traversable?• If it is, describe the route• Solution

– Determine the number of even / odd vertices

– It has 2 odd, and one even vertice

– According to the 2nd rule, it is traversable

Can you go through this building, and only go through each door only once?

Topology

• A branch of modern geometry which looks at shapes in a new way

• In Euclidean Geometry, shapes are rigid and unchanging

• In Topology, shapes can be twisted, stretched, bent and shrunk

• A topologist does not know the difference between a coffee cup and a doughnut

Topology Classification

• In Topology, objects are classified according to the number of holes in them

• This is called their genus– Since Coffee Cups and Donuts both have one hole, they

are considered the same • The genus gives the largest number of complete

cuts that can be made in the object without cutting the object into two parts

• Objects with the same genus are topologically equivalent

Hyperbolic Geometry

• Developed by Russian mathematician Nikolay Lobachevsky (1792-1856) and Hungarian mathematician Janos Bolyai (1802-1860)

• Based on the assumption that given a point not on a line, there are an infinite number of lines that can be drawn through the point parallel to the given line

Elliptic Geometry

• Proposed by German Mathematician Bernhard Riemann (1826-1866)

• Assumes that there are no parallel lines• Based on a sphere• Used by Albert Einstein when he created his theory of the

universe• One aspect of this theory is that if you begin a journey in

space, and go in the same direction, eventually you’ll come back to where you started

• This is where we get the idea that “space is curved”

Fractal Geometry• An attempt to replicate or describe nature• A close look at nature reveals patterns,

repeated over and over, in smaller and smaller detail

• Self Similarity: A pattern that repeats, as well as adding new and unexpected patterns to the whole

• Fractal: Comes from the Latin word Fractus, meaning “broken up” or “fragmented.

Fractal Geometry

• Iteration: The process of repeating a rule (rules are used to create patterns) to create a self similar pattern

• Computers can easily create fractals because you establish rules, and they can repeat those rules thousands or millions of times

• http://www.coolmath.com/fractals/gallery.htm

Fractal

Fractal

Fractal

Fractal

Points

• Point –this is a location. A point has NO SIZE. It is represented by a small dot, and named by a capital letter. A geometric figure consists of a set of points.

• Space –this is defined as the set of all points in existence.

. A . B

Lines

• Line –this can be thought of as a series of points that extends in two opposite directions forever.

• You can name a line by choosing any two points on the line, such as AB –we read this as “Line AB” . A . B

• Another way to name a line is with a single lowercase letter, such as “Line t”

• Points that lie on the SAME line are Collinear Points

Example

• Are points E, F, and C collinear? – If so, what line do the lie on?

• Are points E,F and D Collinear?• Name line m in three other ways.

E

P

F C

D

nm

l What do you think arrowheads are used to show when drawing a line, or naming a line such as EF?

Planes

• Plane –A plane is a flat surface that has NO thickness. A plane extends forever in the directions of all of it’s lines.– How many lines do you think a plane may contain?– How many points do you think a plane may

contain?• You can name a plane by a single Capital letter, or by

at least three of its noncollinear points. • Points and lines that are within the same plane are

called coplanar.

Example

Plane P

P

Plane ABC

D

CBA

Another Example

A B

CD

E F

GH• Name 3 planes

Postulates and Axioms

• A postulate or axiom is an accepted statement of fact

• It is something we hold to be true, and we do not need to prove it -it has either been proven already, or the proof is self evident

Postulate 1-1,1-2,1-3, 1-4

• 1-1 Through any two points there is EXACTLY ONE line

• 1-2 If two lines intersect, then they intersect in EXACTLY ONE point

• 1-3 If two planes intersect, then they intersect in EXACTLY ONE line

• 1-4 Through any three non-collinear points there is EXACTLY ONE plane

Example

• Imagine you have a cube –a dice for example. Just using edges, sides, and corners, answer this:– How many lines are there? –remember, what does

it take to make a line?– How many planes are there?– While there are infinite numbers of points, we also

know that the intersection of two lines creates a point. How many intersections are there? –i.e. how many points can you name?

Segment and Ray

• Segment: A part of a line. It consists of two endpoints (which we have to label) and all of the points in between.

• We would write this as AB –or segment AB• Ray: A ray is the part of a line consisting of one

endpoint and all of the points of the line on one side of the endpoint.

• This is AB, or “Ray AB”

A B

A B

Endpoint A

More Rays

• Opposite Ray: These are two collinear rays with the same endpoint.

• Opposite rays always form a line.

• Name the two opposite rays presented here:– Ray QR, and Ray QS. To be opposite, we must

imagine them going away from each other, so we must use the center point as our starting point

R Q S

Example

• Question: Ray LP and Ray PL form a line. Are they opposite rays? Why or why not?

• Name the segments and rays formed by this figure:

A

B

C

A Closer Look at Lines

• Parallel Lines: These are coplanar, and do not intersect.

• Are all lines that do not intersect Parallel?• No. What if they are not in the same plane,

and do not intersect?• Skew: Lines that do not intersect, but are NOT

coplanar.

Example• Name a pair of parallel lines. Then name another pair.• Name one pair of skew lines, then name another pair.

A B

CD

E F

GH

Unit 1 Quiz 11. Name a Point2. Name a Line3. Name a Plane4. Name 2 Lines that are

parallel5. Name 2 Lines that are

skew6. Name 3 Points that are

Coplanar7. Name 2 Points that are

Collinear

8. Opposite Rays are ______9. (T/F) Opposite Rays have

the same endpoint10. (T/F) Line Segments have

one end point

A B

CD

E F

GH

Assignment

• Page 16/17 8-22, 27-32, 40-45 (guided practice)

• Worksheet 1-2 and 1-3 (independent practice)

Unit 1 Quiz 2

1. How many points fit on the head of a pin?2. An intersection of 2 lines is a _______________?3. An intersection of 2 planes is a ______________?4. For a 2 lines to be parallel, they must be _______________

and ______________.5. For 2 lines to be skew, they must _______________ and

______________.6. Coplanar means _______________________________7. Collinear means _______________________________8. Imagine 3 nonlinear points. Can you draw a plane through

them?9. Imagine 4 nonlinear points. Do they have to be coplanar?10. What is a postulate?

Perpendicular Lines

• Perpendicular Lines: 2 lines that intersect to form Right Angles. The symbol means “is perpendicular to.” In the diagram below, line AB line CD, and line CD line AB.

A

D

C B

This symbol means“Right Angle”There are actually 4 symbols here, since there are 4 right angles

The Ruler Postulate

• The points of a line can be thought of as points on a ruler. We can therefore measure the distance between two points.

• The distance between any two points is the absolute value of the difference of the corresponding numbers.

• This is common sense. If you have a nail at 4 inches on a board, and another nail at 7 inches, how far apart are they?

Congruent

• Two items (in math) are considered congruent if they have the same dimensions (size) and shape. This is a loose definition, and we narrow it for various concepts.

• ≈ -in other math disciplines this means “almost equal to” here, it means the same length

• Congruent segments: Segments which have the same length. They do not have to be on the same line, just have the same measure.

Example

• We use “hashmarks” to indicate congruency in Geometry. We will do this all year, to show segments are congruent, angles are congruent, triangles are congruent and so on.

A B C D• When we look at this picture, we can conclude that

segment AB is congruent to segment CD

Segment Addition Postulate

• If three points, A,B, and C are collinear and B is between A and C, then

• AB +BC = AC

• If DT = 60, find the value of x. Then find DS and ST.

A B C

D S T

2x-8 3x-12

Midpoint

• Midpoint: the point of a segment that divides the segment into two congruent (equal length) segments. A midpoint, or any line, ray, or other segment through a midpoint is said to bisect the segment.

• If segment AB ≈ Segment BC, then . B is a midpoint.

A B C

Examples

• B is a midpoint, what is X?

• B is a midpoint, what is X?

A x B 5 C

A x B 5x-4 C

Assignment

• Page 24 8-25 (Guided Practice)

Keep this assignment –we will add to it.

Angles • An Angle ( ) is formed by two rays with the

same endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle.

• The sides of the angle shown here are BT and BQ. The vertex is B.

• You can name this angle 4 ways: B• 1• B, TBQ, QBT, or 1. • Note that the vertex B is always the middle

letter.

T Q

Examples

• Name 1 in two different ways – Angle ABC– Angle CBA

• Name 2 in two different ways– Angle EBC– Angle CBE

1A

B

C

D

E

2

Measurement (m)

• We often measure angles in degrees. • To indicate the size or degree measure of an

angle, we write a lower case “m” in FRONT of the angle symbol.

• Here the degree measure is 80. We would show this by writing m A = 80.

80o

A

Protractor Postulate

• You can add or subtract the measure of angles. And all angles on one side of a line will add up to 180 degrees

• The measure of a straight line is 180 degrees. We also call this a straight angle.

Example

• What is the m of 1?• 2?• 3?• 4? •

• ABC?

2

1 4

3

A B C

Congruent Angles

• Angles with the same measure are congruent angles. In other words, if m 1 = m 2, then 1≈ 2

• We consider these statements interchangeable: they mean the same thing.

1

2Congruency symbol for angles

Classifying Angles

• There are 4 basic angles:• Acute angle

• 0 < x < 90

• Right angle• X = 90

• Obtuse angle• 90 < x < 180

• Straight Angle• X = 180

xo

xo

xo

xo

BellRinger

• OK, time to evaluate the class and the teacher.• Write what you like, and dislike about the class

(and me) so far.• I expect to see a paragraph. Not two

sentences. At least three.• 10 minutes tops…

Assignment

• Page 31 6-14,18-21 (Guided Practice)(add to previous assignment)

• Worksheet 1-4 (independent Practice)

Simon Says• Point• Line• Plane• Parallel Lines• Skew Lines• Intersecting Lines• Intersecting Planes• Intersection of a Line and a Plane• How many Lines are there between 2 points• How many points are there on the head of a pin?• Perpendicular Lines• Congruent• Obtuse Angle• Acute Angle• Right Angle• When 2 lines intersect, how many points do they intersect on?

Basic Constructions

• A construction requires a compass and a straight edge to draw geometric figures. You can use a ruler, or the side of your protractor for the straight edge.

• You will NOT get credit for hand drawn figures. • You will NOT get credit for figures you

measured with your ruler, instead of using the compass properly.

Constructing Congruent Segments• Given Segment AB• Construct Segment CD so that CD ≈ AB

• Draw a Ray• Measure AB with your compass• Put the point of your compass on the endpoint of your ray,

and make a mark on the ray with the pencil end of the compass

• You now have a new segment the same length as the first one

A B

C D

Constructing Congruent Angles• Given angle A, construct angle B so that angle b ≈ angle A• Draw a ray• Draw an arc on angle A, then draw the SAME arc on ray B.• Place the point of the compass on the intersection of the arc and the side

of the angle, and measure across the angle to the other side and mark it lightly.

• Mark this second arc on the first arc of the ray• Draw a second line from the endpoint of the ray Through the intersecting arcs, thus recreating the arc

A

BPoint of Compass

Point of Compass

Perpendicular Bisector

• A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

A M C

C

D

Constructing a Perpendicular Bisector

• Given segment AB• Construct segment XY so that XY AB at the midpoint M of AB (which

we will determine) • Put the point of the compass at point A, and draw a long arc –make sure

the arc is pas the half way point of the line.• Using the same compass setting, put the compass point on point B and

draw another long arc.• Label the points of intersection• Draw a line between points

X and Y

A B

X

Y

You now have a perpendicular bisector –you have made 4 right angles, and cut the segment in half

Angle Bisector

• An angle bisector is a ray that divides an angle into two congruent coplanar angles. Its endpoint is the angle vertex. Within the ray, a segment with the same endpoint is also an angle bisector – in other words a segment or a ray can bisect an angle but they both start with a ray.

• You can also say that a ray or segment bisects the angle.

Construct the Angle Bisector• Construct the bisector of an angle• Given angle A, construct ray AX, the bisector of angle A.• Put the compass point on vertex A, and draw an arc that intersects the

sides of the angle. Label the points of intersection B and C.• Put the compass point on Point B and draw an arc• Without changing the compass, put the point on Point C

and draw another arc so that it intersects the first arc• Label the point of intersection Point X• Draw the ray XY• AX is the bisector of Angle A.

B

C

X

A

Assignment

• Worksheet 1-5 (Guided/Independent Practice)

Unit 1 Quiz 3 (2 Points each)

Identify the Construction1. Congruent Segments ____2. Congruent Angles ____3. Perpendicular Lines ____4. Bisected Angles ____5. Solve this (no calculator):

– Hint: write it out, and lookfor a pattern –answer is a fraction

B

C

XA

D)

A B

X

Y

A)

A A

C)

A B

A B

B)

1/2 x 2/3 x 3/4 x 4/5 x 5/6 x 6/7 x 7/8

Distance Formula• The Pythagorean Theorem States that:• A2 + B2 = C2 • If we wanted to find C, we would square root both sides, to

get:• √(A2 + B2) = C• On a graph in the coordinate plane, the value of “A” is the

distance between “x” values –which we calculate by subtracting the smaller “x” value from the larger “x” value –or x2 - x1

• Therefore A2 is the same as (x2 - x1)2

• We do the same for Y values, and get (y2 - y1)2 • Finally, we can calculate diagonal distance on a graph by

stating that: C = √(x2 - x1)2 + (y2 - y1)2 • Or we can use the calculator….

Unit 1 Quiz 3

In your own words define: 1. A Point2. A Line3. A Plane4. A Postulate/Axiom5. Collinear 6. Coplanar7. Skew8. Parallel

Draw9. The intersection of 2

planes and a line10. The intersection of 2

planes

Unit 1 Quiz 3(5 points each)

• Write the Distance Formula• State where the Distance Formula comes from

(What equation did I derive it from in class?)

Unit 1 Quiz 2• Calculate the distance between these points (leave answer in

decimal form):1. A (2,5) B (4,9)2. C (1,4) D (-3,5)3. E (0,0) F (-3,-3)4. G (5,5) H (-1,8)5. J (2,11) K (2, 19)6. Draw a right angle7. Draw a straight angle8. Draw an acute angle9. Draw an obtuse angle10. Angle ABC is 135 degrees. What kind of angle is it?

Unit 1 Final Exam Extra Credit

• (5 points)• On the answer sheet provided, CONSTRUCT a

135 degree angle, using the techniques taught in class. Show all work, and highlight the finished angle (I have highlighters if you need to borrow one).

Unit 1 Final Exam Extra Credit (2 points per sentence)

1. Write a sentence explaining the difference between coplanar and collinear

2. Write a sentence explaining three ways to tell if a point is a midpoint on a line segment

3. Write a sentence explaining what perpendicular means

4. Write a sentence explaining what parallel means

5. Write a sentence explaining what a postulate is