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AESTRACT

Gecmetry. Mathematics Curriculum Guide.Gary City Public School System, Ind.6641p.

ERRS Price ME-$0.65 EC-$3.29*Curriculum Guides, *Geometry, *Mathematics,*Secondary Education, *Seccndary Schcol Mathematics

GRADES OR AGES: Seccndary. SUEJECT MATTER: Geometry.ORGANIZATION AND FRYSICAL AFFEARANCE: The subject content ct theguide is arranced in fcur cclumns--major areas, significant outcomes,observations and suggestions, references and films. The guide ismimeographed and spiral bcund with a sctt cover. CEJECTIVES ANrACTIVITIES: General objectives are listed in the intrcductorymaterial, with mcre sFecific cbjectives in the significant outcomescolumns. Activities are not listed in detail. INSTRUCTIONALMATERIALS: Texts, films, ana tilmstriFs are listed for the majorareas, and there is a brief bibliography. STUDENT ASSESSMENT: Amultiple choice test, with answers, is included to provide a means ofevaluation. (`:EM)

MATHEMATICS CURRICULUM GUIDE

BOARD OF

SCHOOL TRUSTEES

GEOMETRYU.S. DEPARTMENT OF HEALTH,

EDUCATION & WELFAREOFFICE OF EDUCATION

THIS DOCUMENT HAS BEEN REPRO.DUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIG.1NATING IT, POINTS OF VIEW OR OPIN-IONS STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF EDU-CATION POSITION OR POLICY.

Mr. Donald Belec, President

'Mr. Andrew D. White, Vice-President

Mr. Theodore Nering, Jr., Secretary

Dr, Montague M. Oliver, Member

Dr. James F. Wygant, Member

Dr. Clarence E. Swingley, Acting Superintendent of SchoolsDr. Heron J. Battle, Assistant Superintendent-InstructionDr. Don T. Torreson, General Supervisor-Secondary EducationDr. William H. Watson, General Supervisor-Curriculum DevelopmentMr. Paul Bohney, Mathematics Consultant

SCHOOL CITY OF'GARYGary, Indina

1968

1

Participants in the Preparation of this Guide

Mathematics Department Representative Committee

1966-67

Ernest BennettPaul BohneyDoris BooseLillian BradleyRobert CroweJoseph DolakGeorge DunleavyWilma FiewellingCharles HubbardHarold JonesWayne MorganJohn PittsMaurice SmolinskiThomas Decker *

Geometry Curriculum Committee

Maurice SmolinskeJames YearyPaul Bohney

Supervisory Personnel

1967-68

Ernest BennettPaul BohneyDoris BooseJoseph DolakGeorge DunleavyCharles HubbardElwood LumpkinWayne MbrganJohn PittsMaurice SmolinskiThomas DeckerAnn Van Beek

Paul Bohney, Mathematics ConsultantWilliam H. Watson, General Supervisor-Curriculum DevelopmentDon T. Torreson, General Supervisor-Secondary Education

Secretarial Personnel

Sharon Marley

Procedure

the Department Representative Committee, working in their sessions, reviewthe mathematics program and recommend specific improvements needed tostrengthen instruction. Some of the recommendations require the serviceof special committees.

This supplement was proposed to up-date the geometry curriculum by theDepartment Representative Committee. The writing of the first draft tookplace during the summer of 1967. The first draft was submitted to theentire mathematics faculty for suggestions for incorporation in this edition.All materials were reviewed and edited by the Mathematics Consultant.

2

FOREWORD

Learning mathematics becomes more important to our society as tech-

nology advances. More people are needed to fill entry positions which

require skill and knowledge of mathematics. Their skills and knowledge

must be developed increasingly at higher levels and schools must help

students strive toward this attainment. Well-prepared, energetic and

enthusiastic mathematics teachers are primary contributors to these

goals.

This mathematics guide represents another step in the on-going process

of developing a strong mathematics curriculum in the Gary Public Schools.

The guide outlines major areas of study in geometry, defines signifi-

cant outcomes in behavioral terms, provides teaching suggestions,

identifies learning materials, and includes evaluation items which

can.aid in determining students' achievement.

Appreciation is expressed to each individual, to the committee and

supervisory personnel who have contributed to the development and the

strengthening of the mathematics curriculum. We hope that the mathe-

matics teachers will use this guide diligently, evaluate the results

carefully, and share in identifying further revisions that will help

keep the program moving forward in accord with the many worthwhile

changes occurring in the field of mathematics.

CLARENCE E. SWINGLEY HARON J. BATTLEActing Superintendent of Schools Assistant Superintendent-Instruction

3iii

SCHOOL CITY OF GARYGary, Indiana

General Objectives of a Secondary Course in Geometry

In developing objectives for the course in geometry, the curriculum committee

reviewed the experiences currently being given to students in the elementary

grades and in the junior high school through an intuitive, experimental, and

informal study of geometry. It was the view of the committee that these early

experiences with geometry are not only of an enrichment nature, but also

essential for an introduction to formal demonstration which to a great extent

characteri,:es a secondary course in geometry. Experiences with geometric

ideas of points, lines, planes, sets, intersecting and parallel lines and

planes, skew lines, rays, angles, as well as plane and space figures along

with some geometric constructions and notions about congruency, similarity,

and measurement are vital prerequisites for a meaningful and worthwhile study

of formal geometry in the secondary school. Secondary teachers can lend

encouragement to elementary teachers in underlining the importance of this

aspect of elementary school mathematics.

The general objectives of a secondary course in

curriculum committee are as follows:

1. To develop certain basic geometric concepts.

2. To develop the ability to do deductive appli

3. To develop a command of precise and concise

4. To encourage and increase ability to make c

5. To apply and extend algebraic skills and un

6. To further knowledge of coordinate geometry

7. To develop spatial perception through studygeometric elements.

8. To stimulate systematic and logical thoughtwritten work.

iv

4

geometry set forth by the

cations.

language.

onjectures.

derstandings.

of relationships among

through orderly and neatly

I

1

GENERAL TEACHING SUGGESTIONS

Most films available for teaching geometry can best be used as a summary

of ideas that have been previously. presented.

It is recognized that single concept film loops are currently being develo

and their value will have to be determined from actual use by students.

The overhead projector is a valuable aid in presenting most topics of

geometry. Not only its dramatic appeal, but also its conservation of

time in presenting ideas makes this so. Every effort should be made to

have overhead projectors available on a permanent basis in each geometry

classroom.

Time can be well spent using various geometric models during classroom

presentations. The student will also benefit from constructing models from

cardboard and other appropriate materials.

While references other than the currently adopted textbook have been include

in this guide, their major purpose is to give teachers suggestions on alter-

nate opproathes and methods. They may also be used as sources for other

exercises or test items. When necessary, care must be taken to rewrite

si.aTrzuterts and symbols which agree, with those in the currently adopted

Finally, it should be noted, in the spirit of this guide, that success

of students in studying geometry. can be related to adequate daily teacher

preparation, including clear ideas of outcomes desired, interesting and

enthusiastic presentations, and total involvement of students in learning

situations.

5

1.

GEOMETRY

MAJCR AREAS

Basic Concepts of Geometry

A, Sets1. Representing2. Relationships

B. Real Numbers and the Number Line1. Rational and irrational numbers2. 1-1 correspondence and order

relation3. Distance between points

I. 1.

2.

3.

4.

C. Undefined Terms 5.1. Point2. Line3. Plane 6.

D. Definitions1. Space

a. sets of pointsb. line, segments

c. rays

7.

2. Anglesa. measurementb. classification

8.

c. special pairsa. bisector

9.

10.

11.

12.

13.

14.

6 1.-

SIGNIFICANT OUTCCMES

Specify sets by roster and rulemethods.

State whether a set is finite orinfinite, and if finite, statewhether null or not.

Identify subsets and union orintersection of given sets.

Use Venn diagrams to interpretset relations.

State whether numbers are ra-tional or irrational.

Portray a one-to-one correspon-dence between two sets.

Write ordering symbols whichproduce true statements aboutcoordinates on a line.

Compute distance between pointson a number line.

Use the definitionof absolutevalue to determine the solutionof open sentences.

State whether or not conditionsgiven are valid examples of points,lines, planes and relationshipssuch as "contains", "on", "passesthrough", "intersection".

Draw figures which portray rela-tionships of points, lines andplanes.

Identify a term with its den-nition.

Use a protractor to measure anddraw angles.

Compute with denominate measuresof angles.

I

I

GEOMETRY

OBSERVATIONS AND SUGGESTIONS REFERENCES AND FILMS

I. This is a review of ideas which havebeen introduced intuitively in earliergrades. The teacher's major emphasisin this area should be to develop thepreciseness of definitions necessaryto enable students to think aboutgeometry from a more abstract view-point.

The teacher might emphasize thefollowing qualities cf a good defini-tion:

1. The term to be defined shouldbe placed in its nearest class.

2. The necessary distinguishingproperties should be given.

3. The definition should be rever-sible.

4. The definition should involvepreviously defined terms.

It is necessary at this beginningpoint that students experience success.Their apprehension about extremedifficulty in the study of geometrycan thereby be removed.

A student cannot ignore any facet ofthe information presented in thistopic because it is all needed tocomprehend geometry.

I. 4 Chapter 1*1 Chapter 2,32 Chapter 13 Chapter 1,36 Chapter 27 Chapter 1,6,78 Chapter 2,3,411 Chapter 1,2,3

Films:

Angles. (KB) B&W, 10 min.

*The first reference listed is the currentlyadopted text.

-2-

7

GEOMETRY

MAJOR AREAS

II. Inductive and Deductive Reasoning

A. Intuition - Induction1. Triangles

a. definitionsb. intuitive and inductive

experiences2. Polygons

a. definitions and nomencla-ture

b. intuitive and inductiveexperiences

3. Quadrilaterz..ls

a. definitionb. intuitive and inductive

experiences4. Circle and Spheres

a. definitionsb. intuitive and inductive

experiencesc. relationships with other

figures

B. Deduction and Proof1. General experiences

a. comparison with inductionb. if - then statements

2. Characteristics of definitionsand postulates

3. Postulatesa. real number propertiesb. equality propertiesc. properties of inequalitiesd. points, lines, and planes

4. Proof of theorems and exercises

-3-

O

SIGNIFICANT OUTCOMES

II 1. State whether a specific con-clusion has been reached byintuition, inductive or de-ductive reasoning.

2. From a set of facts state induc-tive or deductive conclusions.

3. Name and measure parts of a tri-angle and state relationships aboutthe sum of the measures of theangles and about the sum of thelengths of two sides compared tothe length of the third side.

4. Classify triangles by angle andside measurements.

5. Derive conclusions about polygonsby induction and apply these con-clusions to other polygons and testthe results for accuracy.

6. Classify quadrilaterals accordingto given conditions.

7. Name lines and segments as theyrelate to a circle.

8. Draw figures to illustrate andderive conclusions from concen-tric circles, tangent circles,common tangent, circumscribedand inscribed figures.

9. State hypothesis and conclusionsof a given statement.

10. State whether or not a givendefinition is acceptable.

11. State which properties of the realnumbers are illustrated in givenequalities and inequalities.

12. Determine validity of statementsby use of postulates and theorems.

13. Write two column deductive proofsincluding those which require theuse of the property of between-ness and other properties of realnumbers.

(.760MErRY

OBSERVATIONS AND SUGGESTIONS

II. This is the students' first introduc-tion to formal. proof. Section. B ofthis major area must be introducedwith much student-teac:her verbalintaraction. The unieretanding of4hat -.:onstitutea a proof must be wellfounded La each ..tite:kt. Wdde varia-tion in stude.nts' rate of c;Amprehen-sion. of this idea is preseKt in eery

Curre:It mathematical literature shovea preference. for "postulates" ratherthan "axioms.'

The reasoning diseuseed inthis major area is not mathematicalinduction based on the wall-orderingprinciple of the positive tategers.

By exmmiAing concepts consideredthis ms.:;or areal, it was determinad thastunt7. must eit7;:er ha from theirpast .7zperienes or mu,tlearn a mini-m= of ninety-nine voeabulary v,ords.

00casional matning rolv7es of termswith defl.7.itin and opening quissesmay in711te a):-e:.:.s in which special

houlf.1 te.

Emphasize that the student shouldnot rely on the figure for reasonsin a proof. Some optical illusionsmight convince students of this..

REFEIZENCES AND FILMS

TI, 4 C apter 2, :3

1 CFapter I, 42 Ch pter 1, 13

3 Chtpter 2, 46 Ch'pter 6

7 Chapter 3

8 Cha7ter 1, 2

11 Chapter 3, 5

-4-9

Films:

Triangl(s: _Types and Uses. (Cor)

B&W, 11 min.Poluons. (KB) B&W, 10 min.9tradrila- erals. (KB) B&W, 10 min.

The Circle. (KB) B&W, 10 min.

Filmstrips:

MG-542-5 Proof (SVE)

r.

GEOMETRY

MAJOR AREAS SIGNIFICANt OUTCOMES

Angles

A. Angle-MeasurementI. Postulates2. Betweenness of rays3. Angle-addition theorem4. Uniqueness of angle measure

and bisector

B. Theorems about Angles1. Straight angle and right

angle. equalities2. Conditions pertaining to

perpendicular lines3. Supplementary and complemen-

tary angles4. Vertical angles

C. Requirements for Formal Demonstra-tion of Theorem1. Statement2. Figure3. Given4. To prove5. Analysis6. Proof

III. 1. Define an anglen

m10

2. State the difference betweenequality of angles and equalityof measures of angles.

3. Recognise instances where theangle-addition theorem justi-fies conclusions.

4. Write proofs requiring the useof the angle-addition theorem.

5. State the relationship betweenperpendicular lines and rightangles.

6. From given statements and anassociated drawing, recogniseright angles, perpendicularlines, adjacent angles, ver-tical angles, opposite rays,complemeiitary and supplementaryangles by indicating the truthor talsity oE statements or byresponse. to oral questiono.

7. Express the measure. of an anglewhen the measure of its comple-ment nr supplement is given asaz algebraic expression.

8. Express the measure of comple-mentary or supplementary angleswhen the ratio of their mea-sures is given.

9. State in order eix essentialparts of a demonstration of athee: rem.

10. Gnven a statement, draw a figure,list given conditions based uponthe figure drwn, and list con-elusions to be proved in termsof the. figure and the givenstatement.

GEOMETRY


III. the teacher should note with care the III. 4 Chapter 4limitations placed upon theMeasureof an angle in the text being used.The text by Jurgensen1 et. al.) placesthese limits greater than zero andless than or equal to 180, while thetext by Noise and Downs specifiesthe open interval between zero and 180

Direct students to draw an angle asfollows: Have students locate a ver-tex and choose two points on the halfcircle having the vertex as center.As the student draws the rays fromthe vertex to the chosen points on thehalf-circle emphasize that the raysrepresent sets of points and that theangle formed consists of the union ofthese points. This exercise ca^ beextended to include the concept ofangle measure.

Th.e studet might profit from measur-ing am angle using two or more pro-tre...;tors of different radii.

The rationale for these postulates ofmeasured angles rests on the defini-.Lion that angles consist of two distinct rays.

StufLets c.an profit from the knowledgethat in studying trigonometry, angleswill be redefined to include anglemeasure of all real numbers.

Stude can profit from many exerciseWh:Ife7a require the first four essentialparts e a demonstration of a theo-rem; namely, statememt, figure, givenand to prove.

To further acquaint students withreasoning patterns, a short study ofelements of logic is desirable. Theappendix of (4) may be used

-6-

1 Chapter 3, 1

2 Chapter 1

3 Chapter 1

6 Chapter 4, 5

7 Chapter 2, 48 Chapter 411 Chapter 1, 5, 9

Filmstrips:

MG-542-3 Angles (SVE)

GEOMETRY

MAJOR AREAS SIGNIFICANT OUTCOMES

IV. Parallel Lines and Planes

A. Basic Properties

B. Angles Formed by Transversals1. Definitions

a. transversalsb. corresponding anglesc. alternate interior anglesd. alternate exterior angles

2. Equal pairs3. Perpendicular transversals

C. Indirect Proof1. Negative of a statement2. Arriving at a contradiction

D. Proving Lines Parallel1. Parallel postulate2. Induction or conjecture of

the converse of postulates andtheorems.

3. Use of auxiliary lines

E. Applications to Angle Measure inTrianglesI. Sum of the measures of angles2. Corollaries

IV. 1. Give examples of skew, parallel,and intersecting lines and ofparallel and intersecting planesfrom the classroom and from afigure.

-7-

:12

2. Identify alternate interior,alternate exterior, andcorresponding angles.

3. Compute the measure of anglesformed by parallel lines cut bytransversals when given themeasure of one angle.

4. Prove angle relationships.

5. State the negative of statements.

6. State the converse of statements.

7. Determine the validity of astatement and its converse.

8. Write an indirect proof inargumentative form.

9. State conditions for lines beingparallel.

10. Prove lines parallel.

11. Compute the measures of anglesof triangles using theoremseither about the sum of theinterior angles of a triangleor about the equality of amexterior angle of a triangleto the sum of its two remoteinterior angles.

GEOMETRY

.1


IV. While alternate interior anglesgenerally are studied in detail, thematter of alternate exterior anglesin this major area is often slighted.An excellent opportunity is availableto show students that additionaltheorems are possible beyond thoselisted in some textbooks. For exampleconsider:

"If two parallel lines are cut bya transversal, alternate exteriorangles are equal", and the converseof this theorem.

Students can prove these theorems andhave than available for use throughoutthe remainder of the course andthereby develop a better idea of aconverse.

Students should be cautioned regard-ing the improper writing of "ABI!CD"where AB and CD represent measure ofline segments.

The teacher might prefer to approachindirect proof through the conceptof a contrapositive.

Special attention is given to thefollowing postulate: "If two parallellines are cut by a transversal,corresponding angles are equal." Itis recommended that it be inter-preted to emphasize the condition ofparallelism. The same should be donein theorems and converses related tothis postulate. For example, thismight be reworded by the teacher:

When two lines are cut by atransversal, if they are parallel,corresponding angles are equal."

Possible symbols:

Segment >IT

Measure of Segment > IABI

Line AB

-8-

IV. 4 Chapter 51 Chapter L, 5, 6, 92 Chapter 3, 4, 133 Chapter 56 Chapter 3, 9, 107 Chapter 4, 7, 108 Chapter 69 Chapter 9

11 Chapter 5, 7, 17

43

Films:

Properties of Triangles. (KB)

B&W, 10 min.Parallel Lines. (IV) B&W, 10 min.

Filmstrips:

1148 - Sum of the Measures of Anglesof a Triangle (FOM)

1113 - parallelograms and their Pro-

221112.1 (FOM)1139 - The Parallel Postulate (FOM)1140 - Angle Sums for Polymps (FOM)

GEOMETRY


V. Congruent Triangles and the Propertiesof Quadrilaterals

A. Triangle Congruency1. Definitions2. Conditions

a. corresponding partsb. postulates, theorems, and

corollaries3. Proofs

a. non - overlapping trianglesb. overlapping trianglesc. equality of corresponding

partsd. limitations upon auxiliary

lines

e. special trianglea

V.

B. Properties of Quadrilaterals1. Distance definitions

a. frcm a point to a line orplane

b. between parallel lines orplanes

c. from a point to a figure2. Special quadrilaterals

a. angles and segment relation._b. conditions for proof that

quadrilaterals are special

-9-

1. Pair corresponding parts of twotriangles.

2. Write and interpret congruencyof parts.

3. Select postulates, theorems, andcorollaries necessary to provecongruency of triangles.

4. Write proofs of trianglegruency.

con-

5. Redraw overlapping triangles sothat they appear as separatetriangles.

6. List all corresponding parts oftriangles proved congruent.

7. Write proofs of segment or anglerelationships which requiretriangle congruency.

8. State whether or not conditionsfor an auxiliary line determinethe line.

9. Write proofs requiring the useof auxiliary lines.

10. Prove and use properties ofspecial quadrilaterals.

GEOMETRY

OBSERVATIONS AM) SUGGESTIONS REFERENCES AND FILMS

V. While congruency is a special case ofsimilarity, it is normally treatedbefore similarity.. It is suggestedthat the teacher follow this practice.Similarity ie a more difficult ideafor the student to apply and con-gmeney gieee him seme previous andeasier experiences which aid him indeveloping concepts for applicationto two or more figures.

It is recommended that the designa-tion of congruency of triangles bewritten such that corresponding ver-tices are arranged in order of theireorrespondenee. This enables studentsto select corresponding parts fromthe congruency statement as well asfrom the figure. T17A3 would helpstudents weoid errors in the selectionof the eorreopoederIcee.

While many teachers may be reminded ofthe iZes of seperpoeition in the dis-cussien ef eeagruency, it is hopedthat thc postulation of eongruency willbe maeh mo -egfut. The full.

ceagrueacy must extend tosolid figure:3 ec well as pteae figures--a ee,ea point where euperimposi-tier. wektid be .impoeeible. The otedelesheuld fully ..don:. taed that conegrueeeey emt.diea the fact that any orall parts of e cee-to-oae correspon-dence between twe figures have exactlythe same eice and taape.

The properties of eiuedrilaterals aretreated very briefly by some texts.Theee. prepertlee can very profitablybe treated as a separate short unitof study. Many of the exercises offer-ed by texts coeld be treated as theo-rema by the class since they willbeeome useful in later work.

V. 4 Chapter 6

1 Chapter 6, 7, 8

2 Chapter 2, 43 Chapter 2, 8

6 Chapter 5, 6, 11

7 Chapter 6, 9

8 Chapter 5, Appendix VI9 Chapter 8

11 Chapter 4, 7

Filmstrips:

1122 - Congruent Triangles (EOM)

GEOMETRY


VI. Similarity of Polygons

A. Ratio and Proportion1. Definition of ratio2. Definition of proportion

a. termsb. meansc. extremes

3. Algebraic applications

B. Special Properties of a Proportion1. Means-extremes product pro-

perty2. Evivalenteform property3. Daneninator-addition property4. Denominator- subtraction

property5. Numerator-denomiaator sum

property of equal ratios

C. Similar Polygons1, Definitiens2. Ratio of peraeters

D. Similar Tria ngles1. AA postulate2. CeroLleries3. AppLieetiees theorems about

one trianglea. line parallel to a side and

intercectieg til*, sides of atrisegLe

b. angte biser,,tor

4. Corellar7 on baronsl linesand trans iersels

5. CeerespeedIng altitedes

E. Propertiee leading ?...o the proof of

the. Pythagerean Theorem1. Definitions

a. mean proportionalb. projections

2. Properties determined b7 thealtitude to the hypotenuse

F. Pythageeean Theorem1. Proof2. Converse theorem3. 30-60-90 triangle4. IsosceAes eight triangle5. Applieatians Ls. solid figures6. Projections

VI. 1. Write ratios. in simplest anddifferent forms.

2. Use ratios of three or morenumbers to compute measures.

3. Use the means-extremes productproperty and its converse towrite equivalent equations.

4. Solve proportions for specifiedvariables.

5. Compute measures of sides andperimeters of similar polygons.

6. State conclusions based uponsimilar triangles and otherpolygons.

7. Preeve triangles similar.8. Compute measures of segments as

determined by a line parallel toone side of a triangle.

9. Find measures of segments formedon transversals intersected by}rallel lines.

10. 2i!ed the point of intersectionof the bisector of an angle ofa triangle with the oppositeside by using the proportionalproperties of the angle bisec-tor.

II. Compute the measure of an alti-tude in one of two similar tri-angle.s.

12, Write a statement about the.similarity, of the three trianglespresent when the altitude isdrawn to the hypotenuse of aright triangle.

13. Compute measures of segments ofa right triangle when the alti-tude is drawn to the hypotenuse.

14. Describe the projection of afigure on a line or plane.

15. Compute the measure of the pro-jection of a given segment.

16. Use the Pythagorean theorem tocompute the measure of a side0

of a right triangle..17. Given the measure of three

eider of a triangle, determineif it in a right triangle.

13. Given the measure of one sideof a 30-,$0-90 or a 45-45-90triangle, state the measureof the other two sides.

uomErRy

OBSERVAXION'S AND SUGGESTIONS REFERENCES AND FILMS

VI. A ratio is the quotient of to numbersIt must be clear to the student thatif the discussion i3 centered aroundsegments for comparison, that thecomparison is always between theirmeasure.

To keep students from making the in-valid conclusion .21.-at polygons are

similar if their corresponding anglesare equal, use several physical draw-ings to demcms,trate otarwise; i.e. asquara and resta'Agle. It should bekept in mmci, ho4ler, that drawingsare used to exp,71,flits. or enhance

u2:tderstanding !and constitute no con-dition for accepti:.d qalidity.

It should be emphasised that theproperties of a proportion all

consequIces of the properties ofequality are not assumed proper-ties of a proportion.

As %.ith congruency, in writing state-ments iltout simfbirity, it is helpfulto write vertices in order of corres-pondence. Since length of asegment .ce.:::not be saro, division by

serf II:: teaching ratio and proportiondoes not c'notit rl..-: pro7:.1.7>m when

appliel to te meaning of segmentsof similar pcl7g=1. The teacher,howeTer, thocV pint o.rt IttIt makingeij..?ler or c't'7.: denominators zero is

equivalent by mero.

VI. SIGNIFICANT OUTCOMES (continued)CLIMImapipr.

19. Compute the measure of the dia-gonal of a rectangular solid andthe measure of segments of reg-ular pyramids.

VI. 4 Chapter 71 Chapter 15, 162 Chapter 73 Chapter 66 Chapter 127 Chapter 8, 13, 14, 168 Chapter 711 Chapter 8, 12, 13

Films:

Similarity. (M-H) Color, 14 min.

Filmstrips:

1106 - The Pythagorean Theorem (FUN)1116 - Similar Triangles -

Experiment and Deduction (FON)

-12-

17

GEOMETRY

MAJOR AREAS SIGNIFICANT CUTCDMES

VII. Trigonometry

A. Definitions of trigonometricfunctions1. Tangent2. Sine3. Cosine

B. Table of Functions

C. Applications1. Angle of elevation2. Angie of depression3. Law of sines

VII. 1. State trigonometric ratiosin terms of the sides of theright triangle or in terms ofthe unit circle.

-13-

18

2. Read values of trigonometricfunctions from a table.

3. Given the value of a trigono-metric function determine themeasure of the angle to thenearest integer.

4. Compute measures of segmentsof right triangles using trigo-nometric ratios.

5. Compute measures of acute anglesof right triangles.

6. Use meanuree of angles of ele-vation or depression.

7. Compute measures of angles orsides in acute triangles usingthe law of sines.'

GEOMETRY


VII. I n most geometry texts, the algebraicstudy of trigonometry is largely byimplication. But the concepts drivenhome here may very well affect thestudent's desire to pursue mathematicson a higher level -- positively ornegatively. The introduction of trigonometry should lead to a deeper appre-ciation of the importance of the studyof similar triangles. However, theteacher might very well wish to widenthe scope of algebraic applicationthrough the use of the unit circle.The coordinates in the plane which aresets of points on the circle can beshown to be such that (x,y)=(cosine a,sine a) where a is the measure of theangle at the center of the circle.This approach can lead to the funda-mental trigonometric identity sin2a+cos2a=1. Here the Theorem of Pythago-ras has new application and new powersfor the alert student. A furtherjustification for introducing thecircle is the later consequence ofperiodic functions. It is expectedthis would be a brief introduction.

The Law of Sines and especially theLaw of Cosines become useful to thestudy of physics and the physicalproperties of chemistry. The Law ofCosines has not been included in theoutline of this major area. However,the teacher should feel free to presentthis idea, either to the entire classor as supplementary material to betterstudents.

Since understanding of the concept offunction is of such importance inmathematics, the teacher might usethis term at this time. For example,the tangent function could be definedas the set (a, tan a) such that tan ais defined as the ratio of the lengthof the opposite leg to the length ofthe adjacent leg.

VII. 4 Chapter 81 Chapter 162 Chapter 83 Chapter 136 Chapter 127 Chapter 169 Appendix X

11 Chapter 15

-14-

19 ;-

Films:

Trigonometry.(EBF) B&W, 30 min.

Filmstrips:

1108 - Indirect Measurement:TangentRati1775'0M

1143 - The Law of Sines (FON)

GEOMETRY

MAJOR AREAS SIGNIFICIANT OUTO3MES

VIII. Circles

A. Definitions1. Equal circles2. Central angle3. Arcs4. Measure of arcs

B. Arc-Addition Postulate

C. Angle Measure in Relation to ArcMeasure1. Inscribed angle2. Angles formed' by SLeecant and

tangent raysa. vertex on the circleb. vertex in the interiorc. vertex in the exterior

D. Lines and Segments1. Bisectors of arcs2. Diameter perpendicular to a

chord3. Length of segments

a. intersecting chordsb. two secants from an external

point

c. a tangent and a secant froman external point

VIII. 1. Use proper names and symbols todescribe lines, segments andarcs in relationship to a circle.

-15-

20

2. Identify the intercepted arc orarcs for various angles.

3. Compute the measures of anglesfrom their intercepted arcs.

4. Compute the measure of arcs fromthe measure of angles which inter-cept them.

5. State the equality of centralangles or inscribed anglesintercepting equal arcs.

6. State the cases where inscribedangles are right angles, acuteangles and obtuse angles, interms of their intercepted arcs.

7. State whether a circle can becircumscribed about a quadri-lateral when the measure of itsangles are given.

8. Draw conclusions about the rela-tionships between the tangent,point of tangency and the per-pendicular diameter.

9. Recognize where equality existswhen arcs are formed by parallellines.

10. Draw conclusions about the rela-tionships between the diameterperpendicular to a chord andits intersection with the arcsof the chord.

11. State conclusions about compara-tive distances of chords fromthe center of a circle.

12. Compute measures of segments ofchords, secants, or tangentsformed by intersections with oneanother and with a circle.

GEOMETRY

OBSERVATIONS AND SUGGESTIONS

VIII. The formal definition of a circleshould be fully and readily compre-hensive to every student in the class.The set of points in a circle couldbe illustrated by the motion, in aplane, of one end point of a segmentwhere other end point is fixed.

It should be emphasized that a state-ment such as 4! AOB= AB is ridiculousin terms of the definitions of anangle and an arc. Therefore, thistype of statement is actually per-taining to their measures and not totheir sets of points.

The theorem, "The measure of an angleformed by a secant ray and a tangentray drawn from a point in a circleis one-half the measure of the inter-cepted arc", may be presented by anintuitive use of limits based uponthe measure of an inscribed angle.However, it would be well to alsoestablish this theorem by a fonnaldeductive proof. This would helpconvince the student of the validityof the limit argument. Since thistheorem occurs relatively early in theoutline, its usual proof by deductioncan not be done. It can be proved ifthe theorem, "A diameter drawn to a.tangent at the point of tangency isperpendicular to the tangent", isavailable.* This can be proved byuse of an indirect proof based on thedefinition of the distance from apoint to a line.**

The outline of this major area con-tains the mimimum number of essentialtheorems. Many other important rela-tionships should be covered as exer-cises and may, if the teacher sodesires, be treated as theorems.Included among these are tangent anddiameter relationships, equal arcsintercepted by parallel lines, radiuland chord relationships, and compara-tive distances of chords from thecenter of the circle.

REFERENCES AND FILMS

VIII. 4 Chapter1 Chapter2 Chapter3 Chapter6 Chapter7 Chapter9 Chapter

11 Chapter

9

11,

6

9

14,

2,

12

4,

12

15, 167, 10,

9, 10,

11,

11

12, 13, 15

Films:

IChordsandIEEDletsofgas...E.

(KB) S&W, 10 min.*** D namics1 of the Circle. (MR)

Color, 14 min.

Filmstrips

1151 - D ameter Perpendicular toa Chor (FOM

1123 - A c and Angle Measurement(FOM)

*** To be available February, 1968.

* See Smith and Ulrich p. 266** See Moise and Downs p. 426

-16-

GEOMETRY

MAJOR AREAS SIGNIFICANT' OUTCOMES

IX. Constructions and Loci

A. Constructions1. Definitions

a. drawingb. construction

2. Basic constructionsa. an angle equal to a given

angle with a given ray asside

b. angle bisectorc. perpendicular bisector of

a segmentd. perpendicular line to a

given line through a givenpoint

e. a line parallel to a givenline through a given pointnot on the line.

3. Other constructionsa. bisector of a given arcb. determination of the center

of an arc or of a circlec. tangents to a given circled. circumscribed and inscribed

circlese. dividing a segment into

any number of equal partsf. segments of proportional

measure

B. Locus of Points1. Definition2. Fundamental loci

a. at a given distance from agiven point

b. at a given distance from afixed line

c. equidistant from two par-allel lines

d. equidistant from two pointse. equidistant from the sides

of an anglef. equidistant from two inter-

secting linesg. vertex of the right angle

of a right triangle with agiven hypotenuse

3. Intersection of loci4. Loci in the development of a

construction (optional orenrichment)

IX. 1. Differentiate between a drawingand a construction.

-17-

22

2. Construct an angle equal to agiven angle.

3. Construct an angle bisector.

4. Construct perpendicular lines.

5. Construct angles of measure 30,45, 60, etc.

6. Construct a line parallel toa given line by at least twomethods.

7. Construct a tangent to a circleat or from a given point.

8. Circumscribe a circleabout a triangle or a polygonif possible.

9. Inscribe a circle in a triangleor a polygon if possible.

10. Divide a segment into two, three,four, or five equal segments.

11. Construct a fourth proportionaland a mean proportional.

12. Illustrate and describe thelocus of points fitting funda-mental loci.

13. Illustrate and describe the inte-section of loci.

GEOMETRY


IX. The extent tp which the teacher maychoose to emphasize part A of this unitwill be determined by whether theemphasis earlier in the course hasbeen on drawing or on construction offigures. If constructions have beentaught earlier, concentration on theproof of the validity of these methodswill be in order.

The order of the construction of per-pendiculars in this outline may becontrary to some texts. However, ifthe method of perpendicular bisectingof a segment is taught first, the othetwo perpendicular constructions can betaught as a special case of the per-pendicular bisector where the end-points of the segment are to be deter-mined as the initial step.

The methods for circumscribing a circlabout a triangle and inscribing acircle in a triangle may be extendedto polygons in general. If the per-pendicular bisector of the sides ofany polygon are all concurrent, acircle can be circumscribed about it.If the bisectors of all of the anglesof any polygon are concurrent, a circlmay be inscribed in it. The teachercan emphasize that triangles and regu-lar polygons have both an in-centerand a circumcenter

The fundamental loci should be de-scribed by two different sets of points,depending upon whether the conditionsare to be met in a plane or in space.

The similarity between the locus ofpoints equidistant from the sides ofan angle and the locus of pointsequidistant from two intersecting linesjustifies the treatment of the firstas a special case of the second.

IX. 4 Chapter 101 Chapter 5, 12, 142 Chapter 2, 3, 6, 7, 10, 113 Chapter 1, 2, 3, 7, 86 Chapter 4, 8, 157 Chapter 2, 7, 10, 11, Appendix9 Appendix XII

11 Chapter 1, 6, 11, 17

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23

Film:

Locus. (1411) Color, 13 min.

GEOMETRY


X. Ordered Pair and Ordered Triples

A. Correspondence between Pointsand Numbers1. On a line2. In a plane3. In a space

B. Symmetry1. With respect to a point

2. With respect to a line

3. With respect to a plane

C. Graphs and Formulas of AlgebraicConditions1. Open sentence of one variable

a. intersectionb. union

2. Distance between two points3. Circle4. Midpoint of a segment

D. Lines1. Slope

a. formulasb. parallel linesc. perpendicular lines

2. Linear equationsa. Ax+By=Cb. y-y1=m(x-x1)

3. Graphing by the two intercepts4. Solution of intersecting lines

E. Proofs1. Choice of axes.2. Lines, segments, and points3. Polygons

X. 1. Draw graphs representing setsof numbers.

24'

2. Identify corresponding speci-fied sets and graphs.

3. Draw graphs of solution sets forequalities and inequalities onthe Cartesian plane.

4. Find coordinates of pointssymmetrio- with respect tothe origin, x-axis, and y-axis.

5. Compute the distance between twopoints on the coordinate planeand in coordinate space.

6. State the center and radius of acircle from its equations.

7. Write the equation of a circlewhen given its center and radius.

8. Find the midpoint of a segmentwhen given the coordinate ofits endpoints.

9. Find the slope of the linedetermined by two points.

10. State readily whether the slopeof an observed line is positive,negative, zero, or undefined.

11. Determine by their slopeswhether lines are perpendicularor parallel.

12. Write equations of lines by usingthe point-slope form.

13. Given the point-slope form of theequation of a line, draw itsgraph.

14. Given the equations of two lines,compute the intersection

15. Place geometric figures on co-

ordinate axes to the best advan_tage.

GEOMETRY


X. The student's concept of symmetry canbe tested by the teacher posingquestions about the sphere, hemisphere,tangent spheres and tangent hemi-spheres. Can he find a point (lineor plane) with respect to which asphere has symmetry? Can he do thesame for a hemisphere, for two tangentspheres? What condition is requiredbefore two tangent hemispheres havesymmetry with respect to a line orplane? The obvious answer to thislast question is when the bases areparallel, but symmetry also occurswhen the bases form equal dihedralangles with the tangent plane.

Up to the present, algebraic applica-tions have been limited to the solu-tion of problems involving measures ofgeometric figures. This major arealends itself to the feasibility as wellas to the advantages of expandingalgebraic.applications'to geometricprinciples. Algebraic manipulationsmay often be simplified by the choiceof geometric interpretation. Whetheror not time can be spent on developingthe ability to write proofs in co-ordinate geometry will have to bedetermined by the time left for compre-hensive treatment of the areas whichfollow.

X. SIGNIFICANT OUTCOMES (continued)

16. Compute the coordinates of pointsof tri-section of a given segment

-20-

on5

X. 4 Chapter 11, 121 Chapter 202 Chapter 113 Chapter 3, 7, 8, 9

6 Chapter 13, 15

7 Chapter 7, 8, 9, 10

B Chapter 39 Chapter 8, 9

11 Chapter 17

Films:

Graphing Linear Equations. (Coronet)B&W, 11 min.

Filmstrips:

1107 - An Introduction to CoordinateGeometry (F01713----

1115 - The Slope of a Line (FOM)1124 - Geometric Proof Usingaorli-nates (FOM

1132 - Perpendicular Lines in Coor-dinaa...aomela (FOM)

GEOMETRY


Xl. Area of Geometric Regions

A. Polygons1. D(linitions

a. polygonal regionb. area

2. Postulatesa. congruent triangles have

equal areasb. area-additionc. area of a rectangle

3. Area theorems and corollariesa. squareb. parallelogramc. triangled. trapezoid

4. Similar regionsa. triangleb. polygons

B. Regular Polygons,1. Special properties

a. a circle can be circum-scribed about any regularpolygon

b. a circle can be inscribedin any regular polygon

2. Definitionsa. centerb. radiusc. apothemd. central angle

3. A central angle of a regularn-gon has measure of 360/n

4. Area formula5. Ratio of the areas of similar

polygons

C. Circles1. Circumference2. Area3. Length of an arc4. Sectors and segments

a. definitionsb. areas

D. Area Constructions

XI. 1. State when the area - additionpostulate is applicable.

-21-

26

2. Compute areas of polygons.

3. Use properties of proportion tocompare the measures of similarpolygons.

4. Compute the measure of a centralangle of a regular polygon.

5. Compute the radius and apothemfor an equilateral triangle,square, and regular hexagon.

6. State the definition fort.

7. Recognize that Ms irrational.

S. State the decimal approximationfor7to five significant digits.

9. Compute the circumference andarea of circles of given radiior diameter.

10. Given the circumference or areaof a circle, compute the radius.

11. Compute the measure of an arcwhen given its measure andradius.

12. Compute the measure of an arcwhen given its length andradius.

13. Compute the area of a sector.

14. Compute the central angle fora sector.

15. Construct the square with areaequal to the sum of the areasof two given squares.

GEOMETRY


XI. This major area serves as the founda-tion to many areas of application notnecessarily purely mathematical. Someprinciples will find their way intophysics, chemistry, or other relatedfields and the emphasis must be onunderstanding rather than memorization.The student should understand that thecomputation of area involves the pro-duct of one or more pairs of perpendi-cular segments. Proofs of moreadvanced formulas rest on this basicidea.

A practice that is sometimes used is towrite 6 oq. in. as 6 in. This prac-tice can be confusing to the student,as he may interpret 6 in.2 as 6 inchessquare. It may not be obvious to thestudent that a figure of 6 squareinches and one 6 i1.7,4'..7ass square are not

the same.

In most of the exercises about thecircle, answers should be left in termsof filo increase the students recogni-tion of this symbol as a number. Theteacher should stress that the numeri-cal values used forVare approximations

Two circles can be thought of as beingsimilar figuree in tae sense of havingthe same shape.. Therefore, the proper-ties of similarity concerning lengthsof corresponding segments, circumfer-ences, and areas will also apply to twocircles.

XI. 4 Chapter 131 Chapter 17, 182 Chapter 9, 103 Chapter 9, 106 Chapter 11, 167 Chapter 14, 15, Appendix9 Chapter 11, 1211 Chapter 8, 14

Films:

The Meaning of Pi. (Coronet)

B&W, 10 min.Areas. (KB) B&W, 10 min.The Meaning of Area. (MH) Color,

14 min.

-22-

27

GEOMETRY


XII. Measures of Solids

A. Prisms and Pyramids1. Definitions2. Prisms

a. Lateral areab. Total areac. Volumed. Diagonal of a right rectan-

gular prism

B. Cylinders and Areas1. Definitions2. Lateral area3. Total area4. Volume

C. Sphere1. Area2. Volume

D. Similar Solids1. Ratios of corresponding seg-

ments2. Ratios of areas of correspond-

ing surfaces3. Ratios of volumes

XII. 1. Associate names of spacefigures with drawings andmodels.

2. List vertices, faces, andedges of given polyhedrons.

3. Compute lateral areas, totalareas and volumes of prisms,pyramids, cylinders and cones.

4. Compute the area and volume ofa sphere.

5. Compute the diagonal of a rightrectangular prism.

6. Compute corresponding measuresof similar solids.

-23-

2.8

GEOMETRY


XII. A cross-check of mathematical writersshows much disparity between defini-tions for the term geometric solid.Some definitions include the interiorpoints in union with the surfacepoints, while others exclude theinterior points altogether. At leastone author defines a sphere in thesecond manner and other solids in thefirst. The consensus, as formed bystudying several definitions, seemsto favor the inclusion of the interiorpoints. A otatement which seems todefine a solid well, according to thisviewpoint, is: "If a set of pointsseparates space into two distinctregions, one of w'ninh is finite inextent, the union of this oeparationset of points and the set of wAnts ofthe finite reginn i2 a solid."

For any finite region and any linewhose intersection with the region isnot empty, the union of the points ofintersection have a greatest lowerbound and least upper bound.

Stress should be placed upon the factthat similarity implies proportional it'.

of corresponding segments. Therefore,areas, which are basically a productof two perpendicular segments, will beproportional to the squares of themeasures of onee.dimensional elements.Following the same reasoning, volumeswill be proportioned to the cubes ofthe measures of one dimensional ele-ments. While formulas can state theproportionality between volumes, itwould be well to avoid giving thestudent the formulas until ample dis-covery exercises have been offered.Development might start with the cubeand right rectangular prism and con-tinue into other solids as his compre-hension grows.

XII. 4 Chapter 141 Chapter 13, 192 Chapter 2, 4, 123 Chapter 8, 9, 10, 116 Chapter 177 Chapter 9, 179 Chapter 11, Appendix VII

11 Chapter 8, 14

-24-

2'9

Films:

Surface Area of Solid I. (CEF)Color, 15 min.

Surface Area of Solid II. (CEP)Color, 15 min.

Volumes of Cubes, Prisms, ar2al.in:ders. (CEF) Color, 30 min.

Volumes of P ramids, Cones, andSpheres. (CEF Color, 15 min.

SUGGESTED EVALUATIVE TEST ITEMS

The test items included in this guide are considered examples of some

criteria to be used to determine whether outcomes have or have not been

achieved. Some might be used for diagnosis of learning difficulties. The major

reason for their inclusion was to place emphasis upon important outcomes.

The items may be incorporated in tests of major areas or units in the

form shown or in alternate forms. Possibly they could be used as part of semes-

ter or final examinations. They could be suggestive of items for possible

city-wide surveys of achievement in geometry. By no means should these items

be considered as covering every facet of evaluation of achievement in geometry.

* * * * * * * * * * * * * * * * * * * * * * * * * * *

80

In the drawing, if BOC is a straight)line ZA0B and LAOC can be describedas:

c)d)e)

2.-I.

adjacent andequaladjacent andcomplementarynone of these

supplementary

complementaryand vertical

.. I Ica. .3 y 5 I

The length of AB is:

a) -646

d) -4e) none of these

3.-I.

From the drawing, select the co r ctstatements:

I h3 a.-ad Z6 are verticalII 45 and Z4 are supplementaryIII Ray XD and Ray XC are opposite Rays

a) I and II are correctb) II and III are correct

I and III are correctd) all three statements are correcte) all three statements are not

correct

4.-1.

A on a numberB on the sameordinate - 5The length

a) -60 4

c) 6

d) -e) none of these

-25-

20

has coordinate - 1number line has co-

of AB is:

5.-I.

If set x contains 6 elements,set y contains 10 elements,

and xny contains 3 elements,then x Uy contains:

13

b) 16c) 19d) 7

e) none of these

6.-I.

If a point X is between the pointsR and S, which of the following must 11-IIbe true?

10. -II.

A

Using the figur own, which of thefollowing is not a correct name forthe polygon?

a) ABODEb) DEABC

BCAEDAEDCB

e) CBAED

a) RX = SXb) RS + SX = RXc) R + RS = S

CI RX + XS = RSe) none

7.-11.

AUsing the figure shown in QABC, CDis perpendicular to AB. Which of thefollowing must be true?

a) CD bisects ABCD is an altitude

d) CD is a mediand) CD bisects4ACBe) none of these

8.-11.

If Joe goes to church every Sundayand today is Sunday, we can concludeJoe will go to church today. This isan example of:

a) inductive reasoningb) indirect reasoning

deductive reasoningintuition

e) none of these

9.-11.

A statement accepted without proofis called:

Oa postulateb) a theoremc) a corollaryd) a conversee) none of these

Using the figure shown, circles A and B:

a) have common centersb) have a common internal tangentc) have a common chord

have common radiie) have none of these

12.-11.

Given that 3x = 12, select the propertywhich proves that x = 4;

a) transitive property of realnumbers

b) subtraction property of realnumbers

Odivision property of realnumbers

d) symmetric'property of realnumbers

e) addition property of real numbers

13.-111.

Given L1 and L3 are complementaryGI and 42 are supplementary

1'42 =II 42 is obtuse

III 42 = 43 + 90

Select the correct answer:

a) I and II are trueII and III are true

c) only I is trued) only III is truee) none of the statements are true

-26-

31

14.-III.Given: 4a and Lb are supplementaryand Lc and Lb are supplementary.Which of these statements must betrue?

a) La and Lc are supplementaryb La is a right anglec La = Lcd) La and Lb are adjacente) none of the above are true

15.-111.The supplement of an angle is threetimes its ccmplement. The measureof the angle is:

a) 60b) 30c) 270d no such anglee) 45

16.-111.

0Given T-.)t 175S) and CVJ:6-0, OMB bis ects

LYOX. Using the figure, the measureof LAOX is:

a) 120b) 130c) 90

45135

17.-111.

If ray AM lies between ray AC andray AD in a half-plane, which of thefollowing must be true:

a) LMAC = ZMAD + LDACCI LC.AD = GCAM + 4MADc) ray AM bisects LCADd) AC and AD are opposite rayse) none of the above are true

Line a Mine b cut by transversal t.The measure of Ll = (x-15). Fran thesefacts the correct measure fort 8 is:

a) (90 - x)b) (180 - x)c) (165 - x)

e)105(195 - x)

19.-IV.

Using the figure shown, all b, which ofthe following pairs of angles need notbe equal?

a) Ll and L6L3 and 44

(c;?4 1 and 45d) L3 and L5e) 45 and 46

20.-IV.The measures of the angles of a triangleare in the ratio of 1:3:5. The smallestangle has a measure of:

a) 10b) 1

c) 30CY 20e) none of these

21.-IV.AB, CD, and XY are co-planar and AB.LXYat X and CD+XY at Y. Which"- statementmust De

C5 AB CD

b) AB_LCDc) AB =CDd) AB =CDe) none. of these are true

-27-32 .,

AIn the figureshown,L l'=:la° and

LA = x° . The measure of G B is:

a) m° + x°b) x° - m°Qm° - x°

180 -

e) m + x2

23.-V.

I. SAS26.-V.II. SSS

III. SSAIV. AAS

a) are peqpendicularV. AAAb) bisect ithe angles of the rectangle

Which of the above correspondences areare equlal in lengthare not acceptable for triangle d) are-parallelcongruence: e) none og these

25.-V.

Using the drawing to the right, AD andBD are perpendicular to MD at D. IfBD = AD, which statement must be true:

a) L4 r:

b) Ll =.Z5c) = 43

e)

Ll =0 and AM = MDLl =L2 and AM = MB

The diagonals of a rectangle:

a) I and IIIb) II and IV

CD III and Vd) IV and Ie) II and V

24.-V, O.

27.-V.

A 8Given AC = BC and A = 80 , the measure.of LC is:

i

a) 80:. nR o 6

:) o20

An auxiliary line is drawn from vertex d) 50C of AABC and intersects AB at point e) none of theseD. Which of the following conditions,in general, would be improper: 28.-V.

a) ' bisects LC In parallelogram ABCD, LA = 80o. Theb CD is a median of A ABC measure of 4 B is:

CD is the perpendicular bisec-tor of AB

d) AD:DB =:2:3 .

b) 50c) 10e) CDIABd) 80e) none of these

I

-28-

33

29.-VI.

o2.

0.

In the figure shown, DERAB and seg-ments have the measures shown. Themeasure of BE is:

a) 18

b) 2

c) 10

0 8e) none of these

30. -VI.

BC parallel to DE and makes AABCequilateral. The statement whichmust be true:

a) AB = BDb) BD CEc) BC = BD

ABCE-rliCBDB is midpoint of AD

31.-11. or VI.

AIn the figure show , which of thefollowing statements must be true?

a) DE= BCb) 1 =42

/-1 +1.2 = 180E is the midpoint of AB

e) none of the above must be true

32. -VI.

In triangle ABC, LC = 900,AC = 6 andBC = 8. The measure of AB:

a) is 10is 14

c) is 7

d) is TNe) is not given above

33.-VI.

In the figure shown,Z A = 30', LC = 90°,and AB = 6. Which of the followingstatements are true?

a) AC = 3

b) BC = 12AC = 6BC = 3

BC = 316--

34. -VI.

11110

C.Alik

1:4.!In the figure shown, MI...LAB and ITCJ

If AD = 9 and DB = 4, the measure of CDis:

a) Ogb) greater than 9c) less than 4d) 7C3 6

35. -VI.

If a c then :'

ad = beb) ac = bdc) ab = cdd)a+b=c+ de)a-b=c- d

36. -VI.' 14

I`.I

14

E

The figure represents a rectangular boxwith dimensions shown. Find the lengthof the diagonal AE:

4

37.-.VII. 40. -VII.

A (I Sin A=Using the figure shown, 3 would be thevalue of: 5

a) TJ:il

a) tan A b) BCb) tan B ACc) in C

sin A c) AC

eS cos A AB

//41 41.-VII.

d) BCAB

e) ABAC

38.-VII.

Use the figure shown and the ratios,Sin x <cos x if the measure of angle x

sin 266 = .44cos 26° = .90

is:

tan 26° = .49 a) 0 4:x 5..45

Which of the following would be the (0)0 Sx <45closest approximation to BC? c) 0 4cx <90

d) 45 5.1.x 4:90

e) 45 4::x 15.90a) 8.8(19 9.8c) 18.0d) 22.2e) 40.8

39.-VII.

\11) Given semi-circle ABC, with 'BD perpendi-

6 cular to the diameter AC,

Use the figure shown and the ratios:I-Triangle ABC is an isoceles triangle

sin 410 .66II-Triangle ADB is similar to AABC

=

sin 820 = .99III -BD2 = (AD) x DC or AD BD

Which of the following would be theB9 DC

IV-AC2 = AB2 + BD

2+ CB

best approximation for AB?

a) 8

b) 12 a) I only is true

920 TI and III are true18 b) I and II are true

e) 24 d) III only is truee) II and IV are'true

Select the correct statement:

-30-35

In the figure shown A and tt aretangents to circle. 0 at points B andC, respectively. Which one of thefollowing statements are not true?

a) A6.; bisects L BACb) AB = ACc) AOB is a rht triangled) AO bisects BCe5 points B, C, and C might be

coll innar

H44.-VIII.

A

AC is diameter of the circle andpoints A, B, and C are on the circle.Angle 1 has a measure of 55. Usingthe figure, choose the only correcttrue statement:

a) Z3 = 35 aze equal to ABb) LI = Z2 str.d! A ABC is isoscelesc.1) /..4 :-.1 50 altd Z..2 -15

0) G4 = 90 and measure of AB = 70e) All angles are acute

Using the figure shot,m, theAasure of Z 0 is 15. The measure ofCB is:

a) 25b) 32c) 28.0 30

e) none of these

46.-VIII,

1111Using the figure shown, AE=6, EC=8, aandBE=4. The measure of ED is:

a) 3

b) 5 2/3c) 10

12.

16

47. -VIII.

DUsing the figure shown, LB = 70 . Themeasure of LC is:

a) 140b) 35e) 110

20none of the above

In the figureAhown, AB is a diameter ofcircle (1) and AC = 100. The measure of

LCAB is:

40b) 50c) 80d) 100e) none of the above

49.-VIII.

In a circle, chord AB is nearer to thecenter than chord CD. Which of the follow-ing statements must be true?

a AB =CDb) AB> CDc) AB <CDd) AB II CD

e) no conclusion can be drawn

50.-VIII. 54.-IX.

In a circle, chord AB is 6" long and Which of the following are permiLisible

is 4" from the center of the circle. _in making a construction:The radius of the circle is: .C: ruler

a) 2"

b) 4"

6"

e) 10"

51.-IX.

The construction method for circum-scribing a circle about a scalenetriangle requires finding its centerby constructing the:

Operpendicular bisector of thesides

b) bisectors of the anglesc) medians of the triangled) altitudes of the trianglee) parallel lines bisecting two

sides of the triangle

52.-IX.

The locus of points lying in a givenplane and equidintart from points Aand B in the. pi' ne is:

a) a circle with AB as diameterb) a.circie with A as center and

AB as radiusc) the midpoint of ABd) a segment that is theperpen-

dicular hisetor of A3the line that is the perpendi-cular bisector of .4.B

53.-IX.

The locus of porn ::s space that are5" from point A is:

a) a circle with A as center anda 5" radius

b) two parallel planes 5" on eitherside of A

(gthe surface of a_ sphere with Aas center and a 5" radius

d) two sphere, tong:ant at A andwith 5" radii

e) none of the above

II: straight-edgeIII: protractorIV: compass

a) I and IIb) I and IIIc) I and IV

&11 and IIIII and IV

A line segment OP is drawn from the point(10, 0) to the point 6, 4). What are thecoordinates of the midpoint of OP?

ed)(6, 8)

(8, 2)

c) (8, 4)d) (16, 2)e) (12, 8)

56.-X.

A triangle which has vertices (0, 0),(2, 3), and (3, -2) belongs to whichof the following sets:

I - Isosceles trianglesII - Right triangles

III - Equilateral triangles

a) none of theseb) onlyc) YT only

'Fa onlyI ane, II only

57.-X.

On the coordinate plane, three verticesof a rectangle have coordinates of (0, 0),(a, 0), and (0, b), respectively. Thefourth 17ertex will be:

a) (0, a)(b, 0)

b)

d) (a Jr b, a 4- b)


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37

58.-X.

(3, 6) and (-2, 3) are two points onthe coordinate plane. The distancebetween these points is:

(a2),V34

b)N/175--c) 8

d) 4


59.-X.

62.-X.

Two lines on the coordinate plane areperpendicular, If the slope of one lineis 2 the slope of the other is

3

a) 2

3

b) 3

2

c) 2

3

a3

Circle A is drawn on the coordinate e) none of the aboveplane. Circle A is symmetric withrespect to:

63.-X.

a) the x-axis, the y-axis, and theOn the coordinate plane, a circle isdrawn with center at (3, 2) and radius of5". The equation of the circle is:

originb) the x-axis, the y-axis and

point Ac) the x-axis and point Ad) the y-axis and point Ae) the y-axis and the origin

60.-X.

Given the following points on acoordinate plane:

I. - (3, -2)II. - (-2, -3)

III. - (-4, 0)IV. - (2, -2)V. - (-3, 0)

The points which are in the fourthquadrant are:

a) I and IIb) III and IVc) III and V(DI and IVe) I, III, IV, and V

. 61.-X.

On the coordinate plane, a line isdrawn with slope 1/2 through the point(1, -2). An equation for this lineis:

(2y + 2 = 1/2(x-1)

y - 2 = 2(x-1)c) y 2 = 1/2,(x+1)

d) y - 1 = 1/2(x+2)


a) 3x + 2y2= 5

b) (x + 3)2

+ (y + 2)2

2= 25

c) (x - 2)2 + (y - 3)2 = 25

d (x - 3)2 - (y - 2) = 25

(x - 3)2 + (y - 2)2 = 25

Using the figure, the best fitting state-ment below is:

The curve is symmetrical to:

a) the x-axisb) the y-axisc) a point on x-axis

a point on the y-axisthe origin

38-33-

1

Using the figuretrue statement is

The coordinatesof BA are:

a) (-1, 3 )

2

b) (-1, -1/2)

c)( 1, -1)

d) (-2, 3)2

(E)(-1, 3)2

66.-X.

shown, the only

of the midpoint

Given that -4 a x and 4 :Sy, thewhole numbers excluded from theunion of these sets are:

a) -4, -3, -2, -1, 0, 1, 2, 3

b) -3, -2, -1, O. 1, 2, 3, 4-3, -2, -1, 0, +1, +2, +3

d) -16, -12, -3, -5, -1

e) none of these

67.-X.

0

(x1,y1) are coordinates of m. Using

the figure above, the only truestatement below is:

a) the measure of OM =2 2

x1

+ y1

5

68.-XI.

A rectangle has sides of 8" and 6", Whichof the following measures would give aright triangle an area equal to that ofthe rectangle?

legs of 3" and 4"legs of 8" and 12"

c) legs of 16" and 12"

d) a leg of 8" and a hypotenuse of 12"e) legs of 8" and 16"

69.-XI.

The shortest sides of two similartriangles are 4" and 8", respectively.If the area of the larger triangleis 36 sq. in., the area of the smalleris:

a) 18 sq. in.b) 72 sq. inc) 9 sq. in.

144 sq. in.e) none of the above

70.-XI.

A circle has radius of 12". An arc ofthe circle has measure of 120. The

length of the arc is:

a) 24171nchesb) 1217"inches

c) 6 inches

81/Inches24 inches

71.-XI.

The exact ratio of the circumference ofa circle to its diameter is:

a) 22

3.1415926535a) 3.1416e) 2

4E) if m moves at constant distancefrom 0 its equation is

72.-XI.

--x2

+ y2

= OM2

1 1

c) OM2 =f

d)4NOA must equaltMAOe) none of these

A parallelogram has an angle of 30and sides of 10" and 8" Its area is:

-34-

3.9

a) 80 sq. in.b) 40 IT sq. in.

c) 40r sq. in.

d) 80 tr sq. in.

40 sq. in.

73.-XI.

The diagonals of a rhombus are 10"and 6" in length, Its area is:

a) 120 sq. in.

b 60 sq. in.

c 30 sq. in.

d) 16 sq. in.e) 8 sq. in.

74.-XI,

A 6In the figure shown, the area of AABDis 12, the area of ABC is 12, and The figure shown is a right prism with

Aa

the area of BABE is 7. The area ofrectangular base. Its volume is:

the polygonal region ABCED is:

a) 31b) 24c) 19(D17e) none of the above

77.-XII,

Two solids are similar. The larger has avolume of 54 cu. in. and a side of 6".If the corresponding side of the smalleris 4", the volume of the smaller is:

0)16 cu. in.

b) 36 cu. in.

c) 24 cu. in.

d) 30 cu. 'n.


78. -XII.

75. -XI.

The measur= of central szle AOB is120. M is midpoint of AME and isendpoint of radius OM. Using thefigure and dimensions shown, thearea of AAOM is:

a) equal to 2 AMDOb) equal to AM X MD

equal to OHMDd) equal to 2 X 41Te) none of these

76.-XI.

a) 13 Cu. in.b) 72 in.c) 72 sq. in.

72 Cu. in.e) 30 sq. in.

79. -XII.

A cube has an edge of 5". Its totalarea is:

a) 40 sq. in.

b) 25 sq. in.

c) 100 sq. in.

d) 125 sq. in.

(::)150 sq. in.

80.-XII.

A right circular cylinder has a radiusof 3" and an altitude of 5". Its volumeis:

a) 151-itu. in.

b) 451? in.

c) 301ru. in.d) 75Mu. in.e) none of the above

81.-XII.A circle has area of 620 square units.The area of the circle having a If only planes are used to form the sur-

radius one-half that of the given face of a solid, the minimum number

circle is: needed is:

a) 320 sq. in.

b) 420 sq. in.1240 sq. in.

155 sq. in.

e) none of these

a) 2b) 3

d) 5e) 6

-35-

40

BIBLIOGRAPHY FOR GEOMETRI

I. Goodwin, Wilson A., and others. Geometry: A Unified Course. Columbus,

Ohio: Charles E. Merrill Books, Inc. , 1967.

2. Bart, Waiter W., and others. New Plane Geometry and Supplements. Boston,

Massachusetts: D. C. Heath and Company, 1964.

3. Henderson, Kenneth B,, and others. Modern Geometry: Structure andFunction. Chic ago, Illinois: McGraw-Hill, 1962,

4. Jurgensen, Ray C., and others. Modern Geometry: Structure and Method.Geneva, Illinois: Boughton. Mifflin Company, 1965.

5. Moise, Edwin E., and Floyd L. Downs. Geometry. Reading, Massachusetts:Addison-Wesley Publishing Company, 1964.

6. Schacht, J., P., and others. Contemporary Geometry. New York, New York:Holt, Rinehart, and Winston, Inc., 1962,

7. Smith, R. R. and J, V. Ulrich, Geometry: A Modern Course. New York,

New York: Harcourt, Brace, and World, Inc., 1964.

8. Welchons, A. M. Plane Geometry. New York, New York: Ginn and Compapy,1956.

. 9. Wichura, Gerhard. Geometry. Reading, Massachusetts: Addiso%-Wesley Pub-lishing Company, 1958.

10. School Mathematics Study Group. Geometry with Coordinates, Part 7. NewHaven, Connecticut: Yale University Press, 1962.

11. School Mathematics Study Group. Geometry with Coordinates, Part II. NewHaven, Connecticut: Yale University Press, 1962.

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