Chapter 2 Handouts

Chapter 2 – Section 1 need the balls and rubber bands! Calculator w/ trig functions

Chapter 2 – Section 2 usual supplies

Chapter 2 – Section 3 usual supplies

Chapter 2 – Section 4 usual supplies

Chapter 2 – Section 5 usual supplies

Chapter 2 – Section 6 4 blank sheets of copying paper, tape, scissors

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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 (1, 0) and from (2, 3) to (2, 4).

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.

Equivalence Relations – 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

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Now, let’s look at examples of what is and is NOT an ER

Less than and the Natural numbers

Less than or equal to and the Integers

Similarity and equilateral triangles

Definition?

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See the book for more good examples pages 38 and 39

Also note, Euclid “proved” the SAS Axiom while we know now that it can’t be proved…it has to be listed as an Axiom:

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.

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Naming Rules: page 37 in the text!

Let’s discuss this question!

Is triangle ABC congruent to triangle C''B''A''?

B''

A''

C''B

C

A

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The SSA dilemma.

side Bside B

side A

45°45°35°

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 and put them in the context of the Big Three Geometries (EG, SG, and HG)

Part A Euclidean geometry in the plane

Measure one of the exterior angles at B. Measure the remote interior angles. Thm true?

C

A B

Part B 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?

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With respect to the isosceles triangle theorem, let’s look at perpendicular bisectors!

Check out the distances from the points to the endpoints with your protractor!

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Page 43 Scalene Triangle Inequality Theorems

Scalene means?

C

A B

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In Class assignment: Scalene Polygons Theorem

Is the scalene inequality for triangles true of polygons in general? If you have a polygon with more than 3 sides, is the shortest side always across from the shortest angle?

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Generalization of the Pythagorean Theorem

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 opposite B divided by the length of the hypotenuse. All cosine values range from −1 to 1. In particular cos(90°) = 0.

c

a

b

C

A B

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.

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How do you find cosines? Which ones are the “know by heart” ones?

Check out the lengths of sides on this triangle. Assume that a = 5 units. Use the Law of

Cosines: to find the side lengths for b and c

c

b a

4530°

C

A B

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In class assignment: The Law of Cosines:

the Law of Cosines:

Find all angle measures and all side lengths…use exact numbers not approximates

8 cm

45°60°

C

A B

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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.

Review the Big Three.

A small exercise:

C

A B

Snip out the triangle. Cut off the angles. Arrange the angles so that the vertices are coincident on the line below.

__________________________________________________________

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Let’s check out all the theorems on pages 49 – 51. Let’s especially check the sum of the interior angles of a triangle in Spherical Geometry!

And Hyperbolic Geometry:

m1+m2 = 75.80m1+m2+m3 = 111.44 mLBM = 144.4

mABH = 35.6mAHB = 40.5mBAH = 35.3

Disk Controls F

A

B

H

L

M

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Exterior Angles and Euclidean Exterior Angle Thm…..let’s review this with an eye toward Spherical and Hyperbolic…page 42 in the text…

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Problems on page 53

x + 6

x

5

12

Solve for x!

Note the two transversals…proportion problems

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Simple closed curves and congruence of polygons. See Dear Dr. Math on page 55

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Chapter 2, Section 3

Discuss all the words on page 57 and then write out the set diagram, showing quadrilaterals, proper trapezoids, parallelograms, rectangles, squares, rhombi.

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:

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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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Chapter 2 Section 4

Review Area formulas – pages 61 – 64.

Page 62 – where have you seen that before…a Connection!

Page 62 – Heron’s formula….

What’s the SEMI talk?

Check it out with a 3 – 4 – 5 right triangle!

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Page 66 Pick’s Theorem

Put a 6x6 array of points below. Make a rectangle with 2 sides length 3 and the top & bottom length 4. 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 interior

Now add some array points to your 6x6…Let’s put in a convex, irregular pentagon and find the area using Pick’s theorem.

Check your work by decomposing the item and using the old formulas.

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Chapter 2, Section 5

Let’s look at all the vocabulary pages 69 and 70

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Inclass assignment: Circle Vocabulary

Fill in the names for the following items

Note that there may be more than one!

G'

C

DB

GI

J

H

F

E

Tangent line Radius

Secant line

Chord

Central Angle Inscribed Angle

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Together let’s check out the Inscribed angle theorem:

Create two circles with central angles and inscribed angles that SHARE the same endpoints on the circles. Measure both angles. Compare the central angle to the inscribed angle, what did you find?

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Area of a sector p 72

Radian Measure review – conversion factors! P 70

We have so (s is arc length)

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 .

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.

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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.

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:

V of the sphere:

V of the cone:

What is the ratio of these very special volumes?

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Inclass assignment: A “circle net”

How would you make a net of a circle? Something you’d NEVER ask your kids to do all by themselves…

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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

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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