Geometry
Chapter 2 Answers
The Foundations of Geometry
Advertisers use different methods to convince the public to buy a product or to believe their message. Below are several ads that attempt to influence the way the readers think. You will use precise language to write statements about what the ads are trying to convey and then use what you learn in this Superlesson to rewrite these statements in several different forms.
Part A, Conditional Statements
1. A Clio is the advertising industry's award for excellence. Browse through some of the advertisements that have won a Clio.
a. Look at the advertisement for the Valleyfair Amusement Park. Write an "if-then" statement that describes what the ad is trying to convey.
[Sample: If you ride this roller coaster, then your stomach will be left behind.b. Look at the advertisement for Detangler. Write an "if-then" statement that describes what the ads are trying to convey.
[If you use detangler, then you will be able to comb out your hair.]
c. Look at the advertisement for Mints. Write an "if-then" statement that describes what the ad is trying to convey.
[If you eat mints, then your breath won't smell bad.]
Part B, Related Conditional Statements
2. Pick one of the conditional statements you wrote in 1 and write:
a. the converse of your statement.
(Students' answers may vary. "If you live in Florida, then you live in Miami.")b. the inverse of your statement.
(Students' answers may vary. "If you do not live in Miami, then you do not live in Florida.")c. the contrapositive of your statement.
(Students' answers may vary. "If you do not live in Florida, then you do not live in Miami.")
3. If your original statement was true, which of the statements that
you wrote in question 2 is (are) true?
[The contrapositive statement]
Part C, The Rules of Logic
3. Look at the Anti-smoking advertisement.
a.Write two conditional statements implied by the anti-smoking ad in which the conclusion of one statement is used as the hypothesis of the other.
[Sample: If you smoke, you will get out of breath easily. If you get out of breath easily, you will have trouble climbing the stairs.]b. Apply the chain rule to write a third conditional statement using the two statements you wrote in a.
[Sample: If you smoke you will have trouble climbing stairs.]c. If the two original statements in a are true, is the statement you wrote in b necessarily true? Explain.
[Yes.]
When we refer to the Bill of Rights, we are usually talking about the first ten amendments to the United States Constitution. However, many organizations have their own bills of rights. Below, you will examine the bills of rights of the United States and of the real estate industry. Analyze the significance of assumptions and undefined terms within these documents.
Part A, Undefined Terms and Definitions
1. Look at Amendment VIII of the Bill of Rights of the United States.
a. Although the words used in the amendment are not defined within the amendment, most have simple definitions that most English-speaking people will understand. Which words and phrases in the amendment are unclear in their meanings?
[Answers may vary. "Excessive" and "cruel and unusual" can be interpreted in a variety of ways. Whichever words or phrases students choose, they should be able to explain why they feel the words or phrases are unclear.]b. How could these unclear meanings lead to legal arguments?
[Answers will vary. Two people might interpret the word "excessive" differently. For one person, excessive bail might be more than $5,000. For another, it might be more than $500,000.]c. What assumptions are made in this amendment?
[Bail will exist. Fines will be imposed. Punishments will be given.]d. How could a lawyer logically argue that Amendment VIIIof the Bill of Rights was violated if her client was sentenced to 100 hours of community service for vandalizing a street sign? Do you think she would get very far with this argument? Explain.
[Answers will vary. A lawyer could argue that 100 hours of community service is too severe a punishment for the infraction. She may or may not succeed, depending on the size of the sign that was vandalized, how much damage was done, if this had happened before, and other factors.]e. The last line of Amendment V reads "...nor shall private property be taken for public use, without just compensation." Rewrite this as an "if-then" statement.
[If private property is taken for public use, then just compensation will be provided.]f. Can this sentence be rewritten as a biconditional? If so, rewrite the sentence, or else explain why it cannot be done.
[Yes, because the converse--"If just compensation is provided, then private property may be taken for public use."--is true. The biconditional statement is "Private property may be taken for public use if and only if just compensation will be provided."]
Part B, Postulates
2. The mathematical symbol "=" is roughly equivalent to the word "is." For example, the phrases "He is nice." and "Nice is he." both convey the same meaning (One just sounds a little odd!). This property is called the symmetric property. Because of it, you can use properties of equality to rewrite sentences that contain the word "is." (Note: Other forms of the verb "to be," such as "am," "are," and "shall be," are also symmetric.)
a. Read Article I of the United States Constitution. Use the symmetric property to rewrite the first phrase of the sentence.
[In a Congress of the United States shall be vested all legislative powers herein granted.]b. Find another sentence in the Constitution that you can rewrite using the symmetric property.
[Answers will vary. Article II, Section 1 (1) ("The executive power shall be vested in a President of the United States of America") can be rewritten "In a President of the United States of America shall be vested the executive power."]c. Write a sentence that use the reflexive property.
[Answers may vary. A rose is a rose.]d. Pick another property and write a pair of sentences that illustrate it.
[ Answers will vary. Transitive Property: If Meg is the sister of Fran and Fran is the sister of Carrie, then Meg is the sister of Carrie. Caution students that the transitive property is not always valid with language. In the example, "If Sam is the father of Bill and Bill is the father of Ted, then Sam is the father of Ted," the conclusion is clearly invalid.]
Part C, Working in a Deductive System
3. Like Euclid, the framers of the Declaration of Independence began with self-evident truths from which they built a system of laws. Euclid's geometry was based on postulates and undefined terms; the Declaration of Independence and, in turn, the United States government, are based on assumptions about human nature that are presumed to be universal.
a. Read the Declaration of Independence. What were the framers' self-evident truths?
[That all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.]b. What are some laws based on these truths?
[Answers will vary. The assumption that people have a right to life is the basis of laws prohibiting murder. The assumption that people have a right to liberty is the basis of laws prohibiting kidnapping.]c. What were the self-evident truths on which Euclid built his system of geometry?
[Euclid's self-evident truths were his postulates and undefined terms. Modern geometry systems use some of the same postulates that Euclid used, including his famous "Parallel Postulate," which students will encounter in Section 2-3 ("Through a given point P, not on a line l, exactly one line may be drawn parallel to line l.")]d. The next document you will examine is the Real Estate Bill of Rights. Read the "To Account" section describing the agent's duties. Which words and phrases are undefined?
[Answers will vary. Examples: "promptly," "report," and "request"]e. What assumptions are made in this section?
[Answers will vary. It is assumed that the agent acts as the intermediary for money and property received.]f. Describe a situation where the "To Account" section of the Real Estate Bill of Rights could be interpreted differently by the agent and the client.
[Answers will vary. One possibility: An agent reports something verbally when the client expects a report in writing.]
There are many different techniques artists use to give a realistic two-dimensional image of a three-dimensional object or scene. Two techniques are perspective drawing and orthographic drawing.
Part A, Perspective Drawing
1. Perspective drawings help the observer to visualize depth in a two-dimensional picture. In a one-point perspective drawing, lines intersect at a vanishing point.. Read the discussion of Perspective drawing at the site.
a. At the end of this Web page, copy or print out the picture of the hallway with a single window.
b. Draw in two more windows further down the hallway using what you have learned about one-point perspective.
Part B, Orthographic and Isometric Drawing
You've looked at how perspective drawing gives depth to a picture. Now, you will see how you can use orthographic drawings to give accurate pictures of objects from different views.
2. Look at a picture of the Leaning
Tower of Pisa. Sketch three views of the Tower: a front view (in
which you can see the tilt in the building), a side view (in which you cannot
see the tilt), and a top view.
Part C, Making Connections
3. Explain when an artist would produce a perspective drawing and when
she/he would produce an orthographic drawing.
[Example: An artist would use a perspective drawing
if she/he wanted to produce a picture of an entire scene but would use an
orthographic drawing to give an accurate sketch of a single object.]