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
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
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!
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
If element a relates to element b and element b relates to element c, then element a relates to element c. Transitive
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?
“Similarity” and equilateral triangles…does it work?
Definition?
See the book for more good examples and failures see pages 38 and 39
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 (Euclidean can be made to fit in here, too, with care)
III Riemannian
IV Algebraic
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.
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!
The SSA dilemma.
35° sideA sideB -- oops!
Not to mention losing control of the class if you say it wrong: SSA
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?
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?
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?
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.
For this triangle, . This equation comes from the Pythagorean Theorem.
There is a generalization of this theorem that applies to ALL triangles not just right 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.
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?
Check out the lengths of sides on this triangle. Use the Law of Cosines:
to find the side lengths from B to C. Use your calculator to do this. Round to two decimal places.
What is the cosine of 30 degrees? 45? KBH chart!
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.
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
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
H is parallel to every other line showing in the disc.
Since H intersects H on the circle, these two have a type of parallelism called “asymptotically parallel”.
H and H are “divergently parallel” to H .
So we have H and a point not on it: Point D and we have 3 lines parallel to H 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:
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!
Solve for x!
Note the two transversals…proportion problems
Simple closed curves and congruence of polygons. See Dear Dr. Math on page 55 and note that convexity shows up again!
What do you think the sum of the interior angles should be?
What’s wrong with this picture?
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”?
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?
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!
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:
Find the area…WHY can you do this? A19 says you can.
Else you couldn’t!
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!
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
Do the Pick’s Theorem in the ACTIVITIES handout right now.
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.
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 to degrees
Area of a sector, page 72 and 73 in the text.
We have so (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
How about naming the central angle and noting that the whole circle is 2. The fraction of the whole that represents theta is .
When we multiple this fraction times the area we get the area of a sector:
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.
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).
Let’s look at the volumes
V of the cylinder:
V of the sphere: