Honors Geometry
Chapter 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/ inequality
memorize 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 postulate
memorize 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 Geometry
Chapter 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 postulate
memorize 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 Geometry
Chapter 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 definitions
4.6 - 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. 109
and 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 Geometry
Chapter 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 Syllabus
More 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 Syllabus
Inequalities 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
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
book.
Section 7.8
define altitude: (we’ll talk more about this later)]
Chapter Review and TEST
Chapter 8
Perpendicular Lines and Planes in Space
writing assignment - Have students outline this chapter, stressing the relationships among the various
theorems
Section 8.1
define: a line perpendicular to a plane
define: 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 others
hw: 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 book
Discuss 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
theorem 9.1?
Analyze proofs carefully - especially the “Parallel
parallel and
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
this chapter?
theorems found earlier in
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,
trapezoid, base, median
parallelogram,
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
medians -angle bisectors -perpendicular bisectors -hw: p.299 #1-5
Review Chapter Set B #1-20
Chapter Test
intersection in each case.
Honors Geometry- Chapter 10 Syllabus
Parallel 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 introduction
of an auxiliary plane will become essential.
Section 10.1
Read 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 Areas
Chapter 11 Syllabus
Section 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
#19-26
Section 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 #1-28
Chapter Review
Similarity and Proportions
Chapter 12 Syllabus
Section 12.1
define: similarity (informally), proportion -- note likeness to analogies
, geometric mean
review: correspondence
Objective 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 12.2
define: 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 relation
Similarity 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 2
define: ordered pair, x and y coordinate, Cartesian plane
do 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 Syllabus
Circles 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 #1-18
pp. 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 information
2) 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 radius
2) 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 Syllabus
Characterizations 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 #22
and 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
Perpendicular bisectors of ∆
Altitudes of ∆
Medians of ∆
Angle bisectors of ∆
Distribute “sketchpad” overview
Lab work (two days)
p. 509 - 510 #1-8
p. 512 - 513 #1-8
p. 515 - 516 #1-5
p. 527 #1-10
Test on Chapter 15
name of point
characteristics of point
Areas of Circles and Sectors
Chapter 16 Syllabus
Section 16.1
define: 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 16.2
define: 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
d
discuss the “method of exhaustion”
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 all
Ch. 16 Test
Solid Geometry
Chapter 19 Syllabus
note: unlike a sphere, these figures are solid (include interior points)
Section 19.1
define: 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 19.2
define: pyramid, vertex, altitude, ... slant height
objective:
1) to determine lateral area of a right pyramid
2) to determine various cross - sectional areas
pp. 634 - 636 all
Section 19.3
Discuss 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 #1-16
worksheet 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