COLLEGE GEOMETRY - MANUAL to accompany COLLEGE GEOMETRY A DISCOVER...

T est Bank with SOLUTIONS

MANUAL

to accompany

COLLEGE GEOMETRY

A DISCOVERY APPROACH

SECOND EDITION

DAVID C. KAY

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Addison Wesley 1 / 4

CONTENTS

Unit One Tests Banks for Chapters 2-7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Unit Two Moments for Discovery Solutions ........................ 121 Unit Three Solutions to Problems ................................. 143 2 / 4

UNIT l Test Banks For Chapters 2-7 The purpose of this unit is to provide problems for making up tests. Since instructors will no doubt custom design their tests, the material in this unit was made to be flexible, yet organized for easy usage.For convenience, we refer to each section of questions, usually consisting of IO problems, as a "test".However, none of these units is intended as a test by itself since only one section of the text is covered by them, and a reasonable one-hour test in geometry should probably not consist of more than six to eight nontrivial problems. Moreover, the problems on these tests have not been scrutinized as to level of difficulty or average working time. Rather, th is material is intended to provide a choice of problems different from those in the text for possible use on an examination created by the instructor. The optional sections are also covered here (with the exception of Chapter I, first two sections of Chapter 2, and the first section of Chapter 6).Since using this material will no doubt require the copying of isolated problems throughout several sections, this manual has been designed for ease in making photocopies, and assembly. Problems are separated by ample space for cut-and-paste procedure, and figures are included for those problems that require a diagram. Future plans are to make this material accessible for downloading through the website for this book. For information on this, contact the Addison Wesley Longman website (www.aw.com ) ,or your local representative.Answers appear at the end of each test to aid in the process of choosing problems for a particular class, and also for use in making an answer key. The accuracy of these answers have, as yet, not been carefully checked. Future revisions of this material wil I eventually take care of such matters. 3 / 4

• UNIT 1 • Test Banks

Test for Section 2.3: Incidence Axioms for Geometry (8 Problems}

• Considec the model, Point,: S-{I, 2, 3, 4, 5}; L;nes: {I, 2, 3, 4}, {I, 5}, {2, 5), {3, 5}, {4, 5};

and Plones: {I, 4, 5), {2, 4, 5), {3, 4, 5),

(a) Decide whether Axiom I-2 ("A plane is detennined by three noncollinear points") is valid.(b) Decide whether Axiom 1-3 ('"A line passing through two points of a plane lie in that plane") is valid.

• Considerthemodel: Points: S={l,2,3,4}; Lines:{l,2},{l,3},{l,4},{2,3},{2,4},{3,4};

and Planes: { I, 2, 3}, {I, 2, 4}, {2, 3, 4}.

(a) Decide whether Axiom I-2 ("A plane is determined by three noncollinear points"} is valid.(b) Decide whether Axiom 1-3 ("A line passing through two points ofa plane lie in that plane") is valid.

• A m(X}el for Axioms 1-1-1-2 has points S"" {l, 2, 3, 4), and one of the lines is J = { I, 2, 3 }.

(a) What sets must the remaining lines be?(b) What sets must the planes be?

• A model for Axioms I-l-l-5 has points S = {I, 2, 3, 4} and lines {I, 2}, { l, 3}, { l, 4}, and {2, 3}.

(a) What sets must the remaining lines be?(b) What sets must the planes be?

• For now, a quadrilateral is a "four-sided" figure havin11: consecutive vertices A, B, C, and D such

-~----that no three vertices lie on a line, with '•sides" AB, BC, CD, and DA. If diagonals AC and BD intersect at some point E, explain why the four vertices must all lie in the same plane.

• A line i meets two distinct planes at points R and S, and has a third point T * R, * S, in common

with one of them. Show that the two planes intersect in a line. Be sure to consider all possibilities.

• Two planes intersect along line J Another line m, distinct from J, intersects the two planes at two

distinct points E and F, respectively. Show that if line m belongs to the second plane, the planes coincide.

• In a certain finite model for Axioms 1-1-1-3 and 1-5, a computer is used to show that each pair of

planes meet in two or more points. Show that Axiom 1-4 ("the intersection oftwo distinct planes is a line") is valid. (Your argument should show that no curve or other set of points could be the intersection of two planes.)

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