Honors GeometryChapter 1 & 2 Syllabus

Section 1.1 Terminology

terms: undefined terms - point, line, plane definitions - geometry, deductive reasoning, postulates, theorems

Review existing knowledge of geometric terms

hw: pp. 4-5

Section 1.2 Patterns

Review solving quadratic equations using examples

Use of the Pythagorean theorem (“rope stretchers”)

define: irrational/ rational number, inductive reasoning, generalization

hw: pp. 9-12, read about Euclid - We are studying “Euclidean geometry”

Section 2.1 Logic

define: sets, elements, contain, subset, union, intersection, logic, the null set (danish letter Ø)

pp 18-20 (focus on #9)

Section 2.2 Algebraic Background info.

define: natural numbers, whole numbers, counting numbers, rational numbers, irrational numbers, one-to-one correspondence

memorize properties of equality/ inequalitymemorize all other fundamental algebraic properties

review order of operations (“Please excuse my dear Aunt Sally”)

pp. 25 - 26

Section 2.3 Distance

define: absolute value (arithmetic, algebraic, and geometric)

Note: absolute value(sum) ≠ sum(absolute value)

p.28

Section 2.4 Distance (continued)

define: measure

memorize distance postulate (how to organize postulates and theorems)

Use ruler/ tape measure

p. 30 #1,10

Section 2.5 Distance (continued)

review one-to-one correspondence

memorize the ruler postulatememorize the ruler placement postulate - “it’s O.K. to move the ruler”

discuss various coordinate systems

pp. 35-36 #3 - 6,8

Section 2.6 Mathematical Definitions

note: Watch notation...

define mathematically: between, determine, contain, segment, endpoints, length, ray, opposite rays, midpoint, bisect

memorize: line postulate (Euclidean geometry)point plotting theorem(note: on a ray)

pp. 42 - 43

Test - Chapter 1 and 2

(approximately 7-8 days)

Honors GeometryChapter 3 Syllabus

Section 3.1 Adding the third dimension - “perspective”

define: edge, lateral face, base

pp. 51-52

Section 3.2 Properties of Lines/Planes pay special attention to “queries” and “notes”

review point, line, and plane

define: collinear, coplanar

memorize line postulatememorize plane- space postulate

pp.54-56 #1-18

Section 3.3 More Properties of Lines/Planes

Differentiate between metric (or measuring) properties incidence properties (occurence)

Do this section inductively

memorize: flat plane postulate plane postulate intersection of planes postulate

List definitions,theorems, postulates in a central location!!(If computerized on a database, consider having one field for name, one for goal of theorem, one for ordered numbers)

pp. 59-61 except #19

Section 3.4 Separation Postulates (regions)

define: convex (inductively), half-plane, half-space

What would a Line Separation Postulate state?

Paraphrase the Plane Separation Postulate.

note: answer query on p. 64

pp. 65-67 except #15-16

Section 3.5 Topology (extension)

mobius strip, Bridges of Koenigsberg

time to analyze problems individually before giving certain properties of topology. Consider other discrete topics such as Euler and Hamiltonian Circuits...(Reports?)

Chapter Review

Chapter Test

Honors GeometryChapter 4 Syllabus

preparation - define interior of an angle on your own. define interior of a triangle on your own.

Section 4.1 Angles and Triangles

define: angle (watch notation), triangles, interior of a triangle,exterior of a triangle

pp. 81-83 all

Section 4.2 Comparison of Angle Measurement to Segment Measurement

define: linear pair, supplementary

measuring angles (watch notation) in degrees, radians, or gradients, mils

memorize: the angle measurement postulate angle construction postulate angle addition postulate supplement postulate

determine a one to one correspondence between metric postulates for angles and those for segments

pp. 87-90 #1-6,8,11-21 (day 1) #22 - 25 (day 2)

Section 4.3 Coterminal Angles

discuss briefly

Section 4.4 Angle Definitions

define: right/obtuse/acute angles congruent/complementary/ supplementary angles perpendicular lines

Begin to organize your postulates, definitions in your notebooks on your computer

pp. 96-97 all

Section 4.5 Equivalence Relations

Is congruence for segments an equivalence relation?...

Satisfy properties...

a) transitive postulate (remember: these are properties of b) reflexive postulate numbers)c) symmetric postulate

pp. 98-100 (“wanna be proofs”)

Section 4.6 Angle Theorems

define: properties of complements, supplements

Explain why each theorem must be true...

4.2 - 4.5 intuitively based on definitions4.6 - 4.8 thorough 2 column proof (memorize the logic)

pp.103 - 104 all

Section 4.7 Angle Theorems (continued)

Prove these while books are not open

pp. 106-108 all

Section 4.8 Conditional Statements (form)

define: hypothesis, conclusion

note: In any definition using the word “if”, the hypothesis and the conclusion are reversible (a biconditional statement)

p. 109and preparation for section 4.9

Section 4.9 Proof!!

For each assertion there must be support.Support comes from???

This section includes a detailed review of many postulates and theorems preceding this section.

Problem Solving!

hand out notes on analyzing proofs

pp.112-117 #1 - 18 all (group work)

Chapter Review pp. 117-

Honors GeometryChapter 5 Syllabus

Section 5.1 Correspondence

Identify corresponding parts (those which can be superimposed on each other) of two figures. When listing a congruence be sure these are in order. -- Use overheads

define: congruence, identity

hw: pp. 126-128 # 1-11 except #9

Section 5.2 Congruence

“ = “ real numbers are being compared

“ = “ segments, angles, and triangles are being compared

define:included side or angle

show congruence is an equivalence relation

hw: pp. 133-135 #1-10,12-14

Preparation for section 5.3 (on sketchpad)

Section 5.3 Congruence Postulates -

treat these as postulates - accept quickly and utilize them...

hw: pp.139 - 140 all

Section 5.4 Proof (do-it-yourself)

analyze methodology in detail

notes on abbreviating reasons

hw:pp.143-146 more practice (Use your “hints” page)

Section 5.5 more practice on Proofs

hw:pp.149-151 except #25

Section 5.6 Angle Bisector Theorem (Application of Congruence)

define: existence (incidence) theorems uniqueness theorems

Application of congruent triangles

Show how constructions may be a useful tool in analyzing a proof

hw: pp.153-154 concentrate on #7-9

Section 5.7 Isosceles and Equilateral Triangles

Application of congruent triangles

define: isosceles, base, base angles, legs, vertex angle, equiangular, equilateral, scalene, corollary

Look at proof of isosceles triangle theorem very carefully

hw: pp.157-158

Section 5.8 Converses

define: converse, conditional, biconditional

How to use the phrase “if and only if” to your benefit

note: All definitions can be rewritten as biconditional statements!

hw: pp. 160-161

Section 5.9 Overlapping Triangles

The key is to make them non- overlapping so that they are very similar to every other proof you’ve experienced.

group work

hw: pp. 164-166 all (on overheads)

Section 5.10 Quadrilaterals, Medians, and Bisectors

define: quadrilateral, diagonal, rectangle, square, median

Differentiate between an angle bisector and the angle bisector of a triangle.

hw: pp. 168-169 #1-12,14

Review for test - do supplementary problems if you wish for more practice

Test

Chapter 6 SyllabusMore on Proofs

Section 6.1 define: axiomatic system - a logical progression from initial statements and definitions to other statements which are based on those initial statements.

assignment: reread Chapter 1

Section 6.2 Logic and “Indirect” proofs

review converse

define: inverse and contrapositive

deal with truth tables and logical equivalence (worksheet)

discuss how this leads to the formation of indirect proofs

hw: pp.179-181 all (discuss possible legal uses)

Sect. 6.3 Review/Classification of theorems-lines @ planes

define: existence(incidence) - “at least one” uniqueness - “at most one”

put together: “exactly one” or “one and only one”

review: line and plane postulates (try to name unnamed ones)

then, for each of the line and plane theorems, analyze the “indirect” proofs which are given to prove uniqueness.

hw: pp185 - 186 all

Section 6.4 Perpendiculars

read pp. 187-191 classify each theorem as an existence or uniqueness theorem

(or both)

analyze proofs of existence and uniqueness

add these to your list to memorize

pp.192-193 #1-14 , 16(bonus)

Section 6.5 Auxiliary Lines (Sets)

back up any sets introduced with a postulate/theorem!

read examples carefully and supply a second proof for the first example

hw: pp 198-200 #1-19

Section 6.6 Added Information for future use

read pp. 201 - 204

hw: pp205-206 #1-4,6

quiz? prove the “Crossbar theorem” #7

Chapter Test

Chapter 7 SyllabusInequalities of one and two triangles

Section 7.1 Making reasonable conjectures based on observation

Draw conclusions inductively - give each of the “theorems” a name -that can be remembered! (Don’t use Kovak’s rule, Brian)

hw: pp 212-213 #1-10

Section 7.2 Inequality properties for numbers (segment lengths, and angles measures)

memorize these!

Paraphrase Thm 2.2:

What is(are) the difference(s) between Thm 2.2 and the Parts theorem?

Add Parts Theorem to your List!

hw: pp215-216 #1-15:try to prove the exterior angle theorem

without looking at the proof provided in the book.

Section 7.3 The Exterior angle theorem

define: exterior angle, remote interior angle, adjacent interior angle

Add Ext. < Thm. to your list!

hw: pp.219-221 #1-13

Section 7.4 More Congruence Theorems

Add them to your list!

Read proofs very carefully

hw: pp. 223-224 #1-9

Section 7.5 Single ∆ ≠ Thms. (What had you named them?)

Add it to your list!

pp. 227 -228 #1-17

Section 7.6 Distance between a line and a point

Read proofs carefully

make a note of the definition given.

What did you call the ∆ ≠ thm?

hw: pp.231-232 #1-11

Section 7.7 Two ∆ ≠ theorems

hw: pp. 234-236 all

Section 7.8

define altitude: (we’ll talk more about this later)]

Chapter Review and TEST

Chapter 8Perpendicular Lines and Planes in Spacein Space

writing assignment - Have students outline this chapter, stressing the relationships among the various theorems

Section 8.1 define: a line perpendicular to a planedefine: necessary - prerequisite whose falsity assures the falsity of another

statement sufficient - an adjective used to describe a situation where all of the

necessary conditions are met to assure the truth of another statement

Objective: to use these terms in describing the Basic Theorem on Perpendiculars

review: Chapter 3 and othershw: pp.244-245 all

Section 8.2 Analyze theorem 8.1 Use triangle congruence to prove that if two points are equidistant from endpoints of a

segment then every point betwee those two given points is equidistant from the endpoints of the segment (Used in the Basic Theorem on Perpendiculars)

Review theorem 6.2 If two points are equidistant from the endpoints of a segment, then they determine the

perpendicular bisector.

Carefully detail the proof of the Basic Theorem on Perpendiculars.

hw: pp 247-249 all

Section 8.3

Theorem 8.3 - Through a given point of a given line there passes a plane perpendicular to the given line.

The proof of almost all of the theorems regarding space depend on their counterparts in a plane. This is true of theorems 8.3 (auxiliary planes drawn) and 6.1 ( the usefulness of having a line perpendicular to a given line in a plane)... The remainder of the proof is basically contingent on the Basic Theorems on Perpendiculars

Theorem 8.4 - If a line and a plane are perpendicular, then the plane contains every line perpendicular the the given line at its point of intersection with the given plane.

Again, this is contingent on the idea that in a plane, there is only one line perpendicular to a given line (Auxiliary plane used)

Theorem 8.5 Uniqueness of the plane perpendicular to the given line

Theorem 8.6 Perpendicular bisecting Plane Theorem - extension of Perpendicular Bisector theorem (prove this!)

hw: pp251-253 #1-14,18,19

Section 8.4

Theorem 8.7 - Two lines perpendicular to the same plane are coplanar. Justify the main steps which are given in the bookDiscuss the “method of wishful thinking”.

Theorem 8.8 a composite of theorems 8.3 and 8.5 (However, these only dealt with a given point on the given line)

Theorem 8.9 refer to #18 and 19 of the previous section(However, these only dealt with a given point on the given line)

The second minimum theorem - The shortest segment to a plane from an external point is the perpendicular segment -- Again, relate this theorem to its two - dimensional counterpart.

define: distance from a point to a plane.

hw: p. 256 all, Chapter Review through #14

Test

Chapter 9 Parallel Lines in a Plane

Section 9.1 Sufficient Conditions for Parallel Lines

define: parallel, skew

query: What is the difference between the definition of parallel and theorem 9.1?

Analyze proofs carefully - especially the “Parallel Postulate”

hw: pp266-268 #1-10 in class #11-15

Section 9.2 More of the Same

define: corresponding angles, same-side interior angles

hw: pp.271-272 #1-8

Section 9.3 Formal intro. to the “Parallel Postulate”

query: How many of these theorems are converses of those theorems found earlier in this chapter?

hw: pp.275-276 #1-11#12-17, 19 (overheads)

Section 9.4 Triangle Angle Measures

Note the relationship between the Parallel Postulate, the sum of the interior angles of a triangle, and hyperbolic geometry.

hw: pp.279-280 #1-15

Quiz Possible

Section 9.5 Intro. to Quadrilaterals

define: convex, opposite, consecutive, diagonal, parallelogram, trapezoid, base, median

differentiate between the definition of parallelogram and other properties of parallelograms (make note of properties sufficient to prove parallelograms)

Note that there are theorems which deal with the converses of each other.

hw: pp. 285-288 #1-10 individual #11-27 group work (individual responsibility)

Section 9.6 Rhombus, Rectangle, and Square

hw: pp.289-291 #1-10 individual#11-15 group work (individual responsibility)

Quiz Possible

Section 9.7 Right Triangle Theorems

Analyze 30° - 60° - 90° triangles

note difference between median of a triangle and median of a trapezoid.

hw: pp. 292-293 #1-12

Section 9.8 Transversals to many parallel lines

define: intercept, proportionality

hw: p.296 all

Section 9.8 Analyze Concurrence Using the Sketchpad

define: concurrence

Make a note of the special property(ies) of the point of intersection in each case.

medians --

angle bisectors --

perpendicular bisectors --

hw: p.299 #1-5

Review Chapter Set B #1-20

Chapter Test

Honors Geometry- Chapter 10 SyllabusParallel Lines and Planes

Note: it will be very helpful to you to analyze this material in light of the information considered in Chapters 8 and 9. For many of the proofs, the introductionof an auxiliary plane will become essential.

Section 10.1Read pp. 307-311 very carefully, making note of all theorems and any counterparts found in

earlier chapters. Analyze proofs carefully!

Theorem 10-1 -

Theorem 10-2 -Note the use of theorems from Chapters 8 and 9 (Review these often!)

Theorem 10-3 - (converse of 10.2?)

Theorem 10-4 - Does this belong in this chapter?

How does the corollary 10-4.1 relate to the theorem?

How does the corollary 10-4.2 relate to the theorem? How is similar to a theorem in the previous chapter? How does it differ?

Theorem 10-5 - Relate this to the previous chapter.

hw: pp.311 - 313 #1-11,13,14

Section 10.2 Dihedral Angles and Perpendicular Planes

read pp.313-316 very carefully paying particular attention to the proofs

define: dihedral angle, edge, face, plane angle (Why is a plane angle defined this way?)

Know: how to measure a dihedral angle

define: the interior of a dihedral angle:

hw: pp 317-319 #1-12, Desargues’ Theorem

Section 10.3 Projections

define: locus, projection...

hw: p322 #1-10

Chapter Review - all

Polygonal Regions and Their AreasChapter 11 SyllabusSection 11.1 Area of Polygonal Regions (Part 1) define: polygonal region, triangulation

Complete the analogy: Area Postulate: __________ = area: distance

Fill in the blank: Congruent figures have ___________ areas.

State the converse of the above theorem; is this true or false.

State and memorize the Area Addition Postulate.

Areasquare = ________________

Arearectangle = ________________ (pay attention to proof)

hw: pp.334 #1-20

Section 11.2 Area of Polygonal Regions (Part 2)

derive the formula for: Areatriangle = _______________

Construct an acute, right, and obtuse ∆ and label the base and height of each.

derive the formula for: Areaparallelogram = _______________

Areatrapezoid = _______________

Theorem 11.6 is actually an immediate corollary of _______________

What theorem similar to 11.7 could be derived using the same process?

Pay particular attention to the theorems described in #15,18 (memorize these!)hw: pp. 342-344 #1-18,27

#19-26Section 11.3 The Pythagorean Theorem

Why do you think the authors decided to introduce this theorem at this point in the book?

Locate and copy one other proof of the Pythagorean theorem other than the one(s) given in the book.

State the two cases which would represent the contrapositive of the Pythagorean theorem. How do you know these are true? Where might these theorems be utilized:

Case 1:

Case 2:

hw: pp347 - 350 #1-27 odd (memorize your chart for #7b)#2-26 even

Section 11.4 “Special” Right Triangles

Derive the length of the hypotenuse of an isosceles right triangle.

Derive the length of the longer leg of a 30° - 60° - 90° triangle.

hw : pp. 353 - 355 #1-28Chapter ReviewSimilarity and ProportionsChapter 12 Syllabus

Section 12.1 define: similarity (informally), proportion -- note likeness to analogies

, geometric mean

review: correspondenceObjective 1) recognize notation used in stating proportionality / similarity

~ symbol has two related meanings 2) Theorem 12.1 Proportionality is an equivalence relation 3) recognize patterns -> properties of proportions

hw: pp365-367 #1-19

Section 12.2define: similarity (formally) -- CASTCSP

hw: pp 370-372 #1-19

Section 12.3

Objective 1) familiarize yourself with the “Basic Proportionality Theorem” -- complete the details of the proof and

Memorize!

Also, state all of the proportions which could be easily derived from the result of this theorem:

2) familiarize yourself with the converse also. analyze the proof and memorize

Find one other theorem which had a similar method given in the proof (from the book in an earlier chapter)

hw: pp 375 - 379 #1-20

Section 12.4 “Shortcuts to proving similar triangles”

AA Similarity Theorem:

“Triangle Chop” Corollary:

hw: pp382 - 384 #1-21

Prove: Similarity is an equivalence relationSimilarity Substitution Corollary

Memorize and Analyze the Proofs of the SAS and SSS Similarity Theorems

hw: pp388 - 390 #1-11 and Honors problem

Section 12.5 Geometric Mean Theorems

Memorize these (be sure you know what the words are saying)

hw: pp 392 - 394 #1-10

Section 12.6 Ratio of Areas/ Volumes

if scale factor of two similar figures is a:b, then ratio of areas is a2 : b2, then ratio of volumes is a3: b3

hw: pp396 -398 #1-17, 21,22,24

Review for Test

Test

Coordinate Geometry - Ch 13 Syllabus

Read entire Chapter

Section 1 and 2define: ordered pair, x and y coordinate, Cartesian planedo pp.406-407 #4, 10, 16

Section 3 do pp. 411-412 #8, 9, 12

Section 4 define: slope

do pp. 417 -418 #6,9,12

Section 5 properties of parallel and perpendicular lines

do pp. 422 -423 #6, 9, 16

Section 6 define: write a formula for... the distance between two points,

the distance from a point to a line

do pp. 425 #10, 18

Section 7 write a formula for midpoint as a function of the endpoints

do p. 430 #6,10

Section 8 properties of the midpoint of the hypotenuse of a right triangle

do p.435 #2, 3

Section 9 graphing inequality conditions

do pp. 438-439 #6,9,13

Section 10 forms for linear equations

do pp. 444 #4, 10, 12

Chapter 14 SyllabusCircles and Spheres

Many of the theorems in this Chapter are founded upon congruent or similar triangles.Therefore, you should become familiar with most of them.

Section 14.1

define: locus, circle, sphere, center, radius, concentric, diameter, secant, chord,great circle

Objective: to construct a circle given: a) the center and radius b) the center and a point on the circle

pp. 452-453 # 1-11

Section 14.2

define: tangent, interior, exterior, point of tangency, internally tangent, externally tangent, equidistant

Objectives: 1) to create a diagram which will remind you of the necessary theorem from the section

2) construct a circle given any three points that lie on the circle

pp.455-457 #1-18pp. 460-462 #1-13

Section 14.3 This section corresponds to Section 14.1 and 14.2, incorporating the 3rd dimension.

pp. 465 - 466 #1-10

Section 14.4

define: central angle, minor arc, major arc, semicircle, degree measure

Objectives: 1) To determine the measure of an arc, given the necessary information2) To determine the measure of an angle, given the necessary arc info.

pp.469-470 #1-9

Section 14.5

define: inscribed angle, intercepted arcs, inscribed polygon, circumscribed polygon

Objectives: 1) To determine the measure of an arc, given the necessary information 2) To determine the measure of an angle, given the necessary arc info.

(similar to section 14.4)

pp.474-476 #1-19

Section 14.6

define: congruent arcs, congruent circles

Objectives: 1) To determine the measure of an arc, given the necessary information 2) To determine the measure of an angle, given the necessary arc info.

(similar to section 14.4)

pp. 478-480 #1-18, 19-28

Section 14.7

define: secant segment, the “power” of a circle and a point

Objective: 1) to determine the length of a given chord or segment, based on given info.

pp.485-488 #1-29

Section 14.8

Objectives: 1) to determine the equation of a circle, given the center and radius2) to determine the equation of a circle, given the center and a point of the circle.3) to graph a circle, given its equation

pp. 492-495 #1-33

Chapter 15 SyllabusCharacterizations or Loci

Section 15.1 and 15.2

definition: a “characterization” or “locus” is the set of all points which satisfy given characteristics or requirements.

We have dealt with some of these in previous chapters. For each of the following, specify what characteristics determine the following loci:

perpendicular bisector

angle bisector

circle

sphere

interior of a circle

note --To test if your description is specific enough: If you would read your description to someone else, would they be able to conceptualize (or draw) what you have stated? -- also try not to restate the conditions in the name of the locus

note -- If the conjunction “and” is used in the description, consider thinking of the locus as the intersection of two simpler loci.

hw: pp. 503-505 #1-27 excluding #22and pp. 506 #1-9 (to be done over a period of two days)

Sections 15.3 - 15.5

define: concurrent

Complete the following chart:

intersecting lines name of point characteristics of point Perpendicular bisectors of ∆

Altitudes of ∆

Medians of ∆

Angle bisectors of ∆Distribute “sketchpad” overview

Lab work (two days)p. 509 - 510 #1-8p. 512 - 513 #1-8p. 515 - 516 #1-5p. 527 #1-10

Test on Chapter 15

Areas of Circles and SectorsChapter 16 Syllabus

Section 16.1define: polygon

objectives: 1) to use all prefixes denoting polygons correctly 2) to differentiate between a concave and convex polygon 3) to determine the number of diagonals in a convex “n-gon” 4) to determine the sum of the measures of the interior and exterior angles in a convex “n-gon”

Arrive at formulas “discreetly”

pp. 537-539 #1-17

Section 16.2define: regular polygon, apothem

objectives:1) to determine the measure of each interior (or exterior) angle of a regular polygon

2) to determine the area...

pp.540 - 541 all

Section 16.3

C = π discuss the “method of exhaustion”d discuss “limits”

pp.544-545 #1-14

Section 16.4’define: annulus

Continue discussion of “ the method of exhaustion and limits”

pp. 547-549 #1-21

Section 16.5 define: arclength (as opposed to arc measure), sector

Continue discussion of “ the method of exhaustion and limits”

pp.552-553 #1-18 allCh. 16 TestSolid GeometryChapter 19 Syllabus

note: unlike a sphere, these figures are solid (include interior points)

Section 19.1define: prism (right and oblique), altitude, base, lateral face, cross - section, lateral edge, base edge, parallelepiped, rectangular parallelepiped, cube, lateral area, base area

objective: 1) to determine lateral area of a prism or parallelepiped 2) to determine various cross - sectional areas

pp. 629-630 #1-14 excluding #9

Section 19.2define: pyramid, vertex, altitude, ... slant height

objective: 1) to determine lateral area of a right pyramid2) to determine various cross - sectional areas

pp. 634 - 636 all

Section 19.3Discuss Cavalieri’s principle, limits, and the method of exhaustion (in determining volumes)

Note the differences among the given proofs.

read this section carefully

pp. 641-643 #1-16worksheet on Cavalieri’s principle

Section 19.4 and 19.5 Discuss relationship between sections 19.3 and 19.4

pp.647-649 all , 652-653 #1-12,16-18

Chapter 19 Test