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.
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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.ht
m
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?
C n
F
E
P
D
l
• Are points E,F and D Collinear?
• Name line m in three other ways.
m
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
C
B
A
D
Plane ABC
P
Plane P
Another Example
• Name 3 planes
H
G
E
F
C
D
A
B
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.
A
B
• 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.
Endpoint A
A
• This is AB, or “Ray AB”
B
More Rays
• Opposite Ray: These are two collinear rays
with the same endpoint.
• Opposite rays always form a line.
R
Q
S
• 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
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:
C
B
A
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.
H
G
E
F
C
D
A
B
Unit 1 Quiz 1
Name a Point
Name a Line
Name a Plane
Name 2 Lines that are
parallel
5. Name 2 Lines that are
skew
6. Name 3 Points that are E
Coplanar
7. Name 2 Points that are
Collinear
8. Opposite Rays are ______
9. (T/F) Opposite Rays have
the same endpoint
10. (T/F) Line Segments have
one end point
1.
2.
3.
4.
H
G
F
C
D
A
B
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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
How many points fit on the head of a pin?
An intersection of 2 lines is a _______________?
An intersection of 2 planes is a ______________?
For a 2 lines to be parallel, they must be _______________
and ______________.
For 2 lines to be skew, they must _______________ and
______________.
Coplanar means _______________________________
Collinear means _______________________________
Imagine 3 nonlinear points. Can you draw a plane through
them?
Imagine 4 nonlinear points. Do they have to be coplanar?
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
A
B
C
• If DT = 60, find the value of x. Then find DS
and ST.
2x-8
D
3x-12
S
T
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.
A
B
C
• If segment AB ≈ Segment BC, then . B is a
midpoint.
Examples
• B is a midpoint, what is X?
A
x
B
5
C
• B is a midpoint, what is X?
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
T
• B, TBQ, QBT, or 1.
• Note that the vertex B is always the middle
letter.
Q
Examples
• Name
1 in two different ways
– Angle ABC
– Angle CBA
• Name
A
C
1
2 in two different ways
B
2
– Angle EBC
– Angle CBE
E
D
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
• 2?
• 3?
• 4?
•
1?
2
3
1
A
•
ABC?
4
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.
2
Congruency symbol
for angles
1
Classifying Angles
• There are 4 basic angles:
• Acute angle
x
o
• 0 < x < 90
• Right angle
• X = 90
• Obtuse angle
xo
xo
• 90 < x < 180
• Straight Angle
• X = 180
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
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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
A
B
C
D
• 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
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
A
• Draw a second line from the endpoint of the ray
Through the intersecting arcs, thus recreating the arc
B
Point 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.
C
A
M
D
C
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.
You now have a
X
• Label the points of intersection
perpendicular bisector –you
• Draw a line between points
have made 4 right angles,
and cut the segment in half
X and Y
A
B
Y
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
B
and draw another arc so that it intersects the first arc
• Label the point of intersection Point X
A
• Draw the ray XY
X
• AX is the bisector of Angle A.
C
Assignment
• Worksheet 1-5 (Guided/Independent Practice)
Unit 1 Quiz 3 (2 Points each)
Identify the Construction
1. Congruent Segments ____ B)
2. Congruent Angles ____
A
3. Perpendicular Lines ____
C)
4. Bisected Angles ____
A
5. Solve this (no calculator):
– Hint: write it out, and look
for a pattern –answer is a fraction
1/2 x 2/3 x 3/4 x 4/5 x 5/6 x 6/7 x 7/8
A)
X
A
Y
B
B
A
A
D)
B
A
X
C
Distance Formula
• The Pythagorean Theorem States that:
• A2 + B2 = C 2
• 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
x 2 - 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 Point
2. A Line
3. A Plane
4. A Postulate/Axiom
5. Collinear
6. Coplanar
7. Skew
8. Parallel
Draw
9. The intersection of 2
planes and a line
10. 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 angle
7. Draw a straight angle
8. Draw an acute angle
9. Draw an obtuse angle
10. 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