Math 3305
Chapter 2 Notes
Beem, F’14
Chapter 2 – Section 1
Ball and rubber bands, protractor and ruler
Calculator with trig functions
Chapter 2 – Section 2
Ball, rubber bands, protractor
Chapter 2 – Section 3
Big vocabulary chapter – if you make flash cards, bring the blanks!
Chapter 2 – Section 4
begun
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Second class meeting on Chapter 2
Chapter 2 – Section 4 finished
Chapter 2 – Section 5
Chapter 2 – Section 6
4 blank sheets of copying paper, tape, scissors
New material on Euclidean, Spherical, and Hyperbolic Geometries
Compare and contrast these three important geometries
2
Chapter 2, Section 1
Let’s look closely at Euclid’s common notion and see why it’s been discarded:
4.
Things that coincide with each other are equal to each other.
Take the line segment, including the endpoints from (0, 0) to (0, 1) and from (2, 3)
to (2, 4).
Equality means the same points with two different names. CONGRUENCE means
having the same PROPERTY (in this case: length) and perhaps different point
sets.
These can be made to “coincide” by rigid motions in the plane – let’s do that.
BUT the problem is that the points that make up the two segments are NOT the
same points. Euclid didn’t have set theory!
3
We use the word “congruent” to refer to properties of the point set like length but
“equal” is reserved for the concept of the same points in each set.
What needs to be true for two triangles to be congruent?
page 37 in the text
Congruence is an Equivalence Relation in Math. This actually says a whole lot!
Let’s unpack it.
An Equivalence Relation (~) on a set A with elements, a, b, c… is one that has the
following 3 properties:
Reflextive,
Symmetric, and
Transitive.
Element a relates to itself
Reflexive
a~a
If element a relates to element b, then element b relates to element a
Symmetric a ~ b  b ~ a
If element a relates to element b and element b relates to element c, then element a
relates to element c.
Transitive (a ~ b  b ~ c)  a ~ c
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Now, let’s look at examples of what is and is NOT an ER. It is always important
to show why we have the words. If every relation were an equivalence relation,
there would be no need to have this vocabulary.
Congruence is an Equivalence Relation! Let’s pick the set of all triangles and use
3 congruent ones: T1, T2, and T3. Be sure to check each of the 6 items on
congruence and then the 3 items on ER. Now the general comparative character
“~” will be replaced with the symbol for congruence “  ”.
“Less than” and the Natural numbers…does it work? [ ~ becomes <]
“Less than or equal to” and the Integers…does it work?
5
“Similarity” and equilateral triangles…does it work?
Definition?
See the book for more good examples and failures see pages 38 and 39
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Now why do we care:
We are working with axiomatic systems for our many geometries. It turns out that
some properties of geometries need to be equivalence relations. For example,
here’s an outline of Geometries currently in use with some mathematicians
Geometry:
I.
Mobius
Euclidean
Elliptical (Spherical is one of these)
Hyperbolic
II
Projective
Finite
Infinite
III
Riemannian
IV
Algebraic
(Euclidean can be made to fit in here, too, with care)
The Distance function in Mobius Geometries needs to be an equivalence relation
for the geometries to meet the definition of Mobius Geometries. This is one reason
for the classification.
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Note, Euclid “proved” the SAS Axiom while we know now that it can’t be
proved…it has to be listed as an Axiom, a statement that is accepted and not
proved:
A15. The SAS Postulate:
page 39 in text
Given an one-to-one correspondence between two triangles (or between
a triangle and itself). If two sides and the included angle of the first
triangle are congruent to the corresponding parts of the second triangle,
then the correspondence is a congruence.
Please take the time to read through the theorems for ASA, SSS, AAS on your
own.
Naming Rules: page 37 in the text! Let me read them to you
Let’s discuss this question!
Is triangle ABC congruent to triangle C''B''A''?
A''
C
C''
B
A
B''
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The SSA dilemma.
side A
side B
side B
35°
45°
45°
35° sideA sideB -- oops!
Not to mention losing control of the class if you say it wrong: SSA
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Now, in the text: pages 41 and 42
Isosceles Triangle Theorem and the Exterior Angle Theorem…let’s read those.
Is everybody familiar with these?
Measure one of the exterior angles at B. Measure the remote interior angles and
add the measures.
Does this illustrate the Theorem?
C
A
B
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With respect to the Isosceles Triangle Theorem, let’s look at the form and the proof
and then let’s look at perpendicular bisectors!
Page 41. Read the theorem. Do you see the “iff”? Now look at the proof.
Part 1
Two sides congruent  Angles opp congruent
Part 2
Angles opp congruent  Two sides congruent
Definition: page 45: Perpendicular Bisector Check out the distances from the
points to the endpoints with your protractor in the activities handout.
Page 43 Scalene Triangle Inequality Theorems
What does scalene means?
C
A
B
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The set of all triangles organized by side lengths:
An “arbitrary triangle” is what kind of triangle?
Ambiguity on isosceles and equilateral…
Scalene, Isosceles, Equilateral
Where do right triangles fit into this set diagram?
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Generalization of the Pythagorean Theorem
Who knows a little trig?
Based on right triangles
Why do we care about it?
Similarity allows small models
Let’s review the trig function “cosine”
(adjacent/hypotenuse!)
Given a right triangle, with A the right angle, the cosine of angle B is the length of
the side adjacent B divided by the length of the hypotenuse. All cosine values
range from −1 to 1. In particular cos(90°) = 0.
C
a
b
A
c
B
For this triangle, a 2  b2  c 2 . This equation comes from the Pythagorean Theorem.
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There is a generalization of this theorem that applies to ALL triangles not just right
a 2  b 2  c 2  2bc cos A
triangles. It is called the Law of Cosines:
It has one more term than the Pythagorean Theorem equation and this term adjusts
the length of the side opposite A for exactly how “not a right triangle” you have
your hands on.
Use the Law of Cosines to find the distance from A to B.
B
C
A
m AC = 4.00 cm
CB = 4.00 cm
mACB = 120.00°
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How do you find cosines? Using a calculator
37°
59°
120°
Which ones are the “know by heart” ones? What is the easy way to put them down
on scratch paper for a test?
15
Check out the lengths of sides on this triangle. Use the Law of Cosines:
a 2  b 2  c 2  2bc cos A
to find the side lengths from B to C. Use your calculator to do this. Round to two
decimal places.
BE = 7.50 cm
EC = 5.30 cm
mEBC = 30.00°
mECB = 45.00°
E
B
C
What is the cosine of 30 degrees? 45? KBH chart!
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Another example do this in groups!
Given the obtuse triangle with side lengths 3 and 5 around an angle measuring 135
degrees.
Find the measure of the side across from the angle. Use your calculator and round
to two decimal places.
Do the Law of Cosines assignment now.
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Chapter 2 – Section 2
Consequences of the Parallel Axiom.
A16 The Parallel Postulate:
Through a given external point there is at most one line parallel to a given
line.
We use this axiom to prove that the sum of the interior angles of a triangle is 180
degrees in Euclidean Geometry. Page 51. Let’s find this in the proof.
Do the cut out exercise in the Assigments Booklet now
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Spherical Geometry – do you remember that there are no parallel lines in this
geometry?
What models a line in SG? Try to follow the EG axiom…how many times do the
lines intersect? VERY non-Euclidean, no?
Parallel Lines in Hyperbolic Geometry
Points, Lines
F
Disk Controls
H
B
G
A
D
E
H
AB
is
paral
lel to
every
other
line
show
ing
in the
disc.
Since H AB intersects H DF on the circle, these two have a type of parallelism
called “asymptotically parallel”.
H DH and H DE are “divergently parallel” to H AB .
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So we have H AB and a point not on it: Point D and we have 3 lines parallel to H
AB through D right there on the sketch. This illustrates our choice of parallel
axiom. And we now have two types of parallelism: asymptotic and divergent.
Let’s sketch this together:
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Page 53 – Exterior Angle Equality Theorem
We actually discussed this in Section 1
Problems on page 53
An important property of parallel lines in EG is that they cut transversals into
proportional ratios. We saw something like this with calculating square roots!
12
5
x
x-6
Solve for x!
Note the two transversals…proportion problems
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22
Simple closed curves and congruence of polygons. See Dear Dr. Math on page 55
and note that convexity shows up again!
mABC = 48.79°
mBCD = 32.54°
mCDA = 110.63°
mDAB = 29.29°
mABC + mBCD + mCDA + mDAB = 221.25°
B
D
A
C
What do you think the sum of the interior angles should be?
What’s wrong with this picture?
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Chapter 2, Section 3
Discuss all the words on page 57 and then look at the set diagram, showing
quadrilaterals, proper trapezoids, parallelograms, rectangles, squares, rhombi.
See the caution about trapezoids!
Is this set diagram clearer than the paragraph?
Check out the theorems! Note particularly the “iff” one on page 58. Let’s write
that out completely:
Thm 2.3.1 A quadrilateral is a parallelogram iff opposite sides are of equal
length.
IFF
“of equal length”…could we have said “congruent”?
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Thm 2.3.2 The diagonals of a parallelogram bisect each other.
Which other types of quadrilaterals is this true for and why?
Thm. 2.3.3 A quadrilateral is a parallelogram iff each pair of consecutive angles
are supplementary.
Break down the proof into the two parts…what are they?
Corollary 2.3.4
All four angles of a rectangle are right angles.
What is a “corollary” and why is this one?
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Chapter 2, Section 4
Area and Perimeter confusions
Put one triangle in between. Using the same base, draw a second
triangle with a new vertex. Measure the perimeter of each. Calculate
the area of each.
Same area but not congruent!
A18. If two triangles are congruent, then the triangular regions have the
same area.
NOT iff!
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Area axioms:
A17.
To every polygonal region there corresponds a unique positive number called its area.
A18.
If two triangles are congruent, then the triangular regions have the same area.
A19.
Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at
most in a finite number of segments and points, then the area of R is the sum of the areas of
R1 and R2.
A20.
The area of a rectangle is the product of the length of its base and the length of its altitude.
Now A19 is the one all kinds of problems are based on:
90°
1 cm
Find the area…WHY can you do this? A19 says you can.
Else you couldn’t!
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Review Area formulas together – pages 61 – 64. KBH
Page 65 – Heron’s formula….can be very convenient. Discuss semi-perimeter
Let’s illustrate it with two triangles:
3 – 4 – 5 right triangle
appointed groups
Isosceles right triangle with side length 1. The rest of the groups!
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Page 66 Pick’s Theorem
Make a rectangle with 2 sides length 3 and the top & bottom length 4 in
the array on the next page. Calculate the area the old way. Calculate the
area using Pick’s Theorem.
I = the number of interior lattice points
B = the number of lattice points on the boundary
area  I  ( B / 2)  1
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Do the Pick’s Theorem in the ACTIVITIES handout right now.
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Chapter 2, Section 5
Let’s look at all the vocabulary pages 69 and 70
Definition of a circle
Tangent
Secant and chord
Central angle and Inscribed angle
Devise a way to keep these definitions straight right now out loud
Do the circle vocabulary exercise in the ACTIVITIES handout now
Note that there may be more than one example of a word.
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Together let’s check out the Inscribed angle theorem:
p. 71 text
Do the illustration of this theorem in the ACTIVITIES Handout now.
Area of a sector
p 72
Radian Measure review – conversion factors!
P 70
Multiplication by 1….multiplicative identity….
Convert 45 degrees to rads
Convert
5
6
to degrees
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Area of a sector, page 72 and 73 in the text.
We have  
s
r
so s  r (s is arc length). The measure of an angle in rads is the
ratio of the arclength divided by the radius…why? Where did pi come from?
If you have a central angle the area of the central angle will be a fraction of the
area of the circle.
Working in radian measure
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How about naming the central angle  and noting that the whole circle is 2  . The
fraction of the whole that represents theta is

.
2
When we multiple this fraction times the area we get the area of a sector:
  r2
A r 

2
2
2
note that we’re working in RADIAN measure here
Problem: A circle with radius 6 has a given central angle of 15°. Find the area
of the sector created by this angle.
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Chapter 2, Section 6
Page 76
Nets! Why do nets?
Page 78…let’s do the exercise on the bottom of the page out loud
together.
Now go to the ACTIVITIES handout and cook up a circle net!
Going 3D here gives us more to work with: Both surface area and
VOLUME!
Lateral surface area page 81
Now spheres: page 82
Now let’s compare cylinders, spheres, and cones. Suppose you have 3
solids.
A right circular cylinder with a base radius of r and a height of 2r.
A sphere of radius r (the same r as above).
A right circular cone with base radius r and height 2r (the same r).
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Let’s look at the volumes
V of the cylinder:
Vcl   r 2 h  2 r 3
V of the sphere:
4
Vs   r 3
3
V of the cone:
1
2
Vc   r 2 h   r 3
3
3
What is the ratio of these very special volumes?
36
Classroom discussion: Two easy to make right circular cylinders:
Given one right circular cylinder with height 8.5, and a second one with
height 11…make these! Which has the greater volume? What does it
look like?
Note that they both have the SAME side (lateral) surface area.
What is the volume of each? Use
V   r 2h
How will we find the radius?
Let’s do the math and compare volumes!
This highlights common perception problems with most everybody!
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Now for The Big Three:
Each of these geometries has its own set of axioms!
Each of these geometries has a basic point set. What are these point
sets?
Each of these geometries has objects that model lines. What are these
lines in each geometry?
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Spherical geometry on the ball
Take your sphere and 3 rubber bands. Put one on the equator and 2 through the
North Pole perpendicular to the equator.
Measure the angles – interior and exterior. What do you find?
What is the sum of the interior angles? Let’s write these on the board
How about that exterior angle and the sum of the remote interior angles?
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Now another look at Hyperbolic Geometry:
We are using the Poincaré Disc model. Let’s take a minute to look at these lines
and then we can check out exterior angles.
Unit DISC, orthogonal circles are lines!
Triangles and Exterior Angles
F
Disk Controls
L
mBAH = 35.3
B
mAHB = 40.5
mABH = 35.6
A
M
mLBM = 144.4
m1+m2+m3 = 111.44
m1+m2 = 75.80
H
Yes, we have triangles. No, the sum of the interior angles is not equal to 180; it is
LESS THAN 180. The difference between 180 and the sum of the interior angles
of a given Hyperbolic triangle is called the DEFECT of the triangle. In Spherical
geometry the difference between the sum of the interior angles of a spherical
triangle and 180 is called the EXCESS of the triangle.
In fact, the measure of an exterior angle is greater than the sum of the remote
interior angles. Do you see this in the diagram?
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Note that the defect of the triangle is 58.6. (The sum of the angles is 121.4) We
use the lower case Greek letter delta for defect (  ).
Right triangles
Euclidean Geometry
Yes we have them. How many right angles are there in a right triangle? What is
the sum of the interior angles of a right triangle in EG?
Spherical Geometry
Make a right triangle with THREE right angles. What is the sum of the interior
angles of this triangle. Can we have triangles like this in EG?
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Hyperbolic Geometry:
Disk Controls
mCDB = 90.0
C
mDCB = 17.6
mCBD = 18.1
D
B
DC = 1.81
DB = 1.79
A
CB = 2.96
Distance2+Distance2 = 6.48
Here’s a right triangle – the measures of the angles are shown. I built it using an H
AB and the Hperpendicular bisector of that line (again: Hyperbolic tools on the
left menu in GSP).
 = 54.3
Does the Pythagorean Theorem hold in Hyperbolic Geometry?
Demonstrably not – see the calculations in the sketch.
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A review break:
What are some similarities and differences between the 3 Geometries we’ve looked
at so far?
Each geometry has its own set of axioms! Are they the SAME axioms?
Do they each have triangles?
What’s different about triangles in each?
Do they each have lines?
What about parallel lines?
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What are some differences?
Biangles! Where are these? What are these?
Pythagorean Theorem in each?
What about circles?
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