Semester 2
    Unit 11 Solids GeoUnit11
    Unit 10 Area GeoUnit10 
     Unit 9 Circles GeoUnit9
    Unit 8 Trig GeoUnit8
     Unit 6.5 "Trinequalities"
     GeoUnit6Similarity "Similarity"
    Semester 1
    Unit 1 "Basics"
    Unit 2 "Logic"
    Unit 3 "Transversals"
    Unit 4 "Triangles"
    Unit 5 "Quadrilaterals" Geo5Notes0


    compass straightedge  
    SSS-CONSTRUCTING Side-Side-Side:
    SAS-CONSTRUCTING Side-Angle-Side:
    SSA-CONSTRUCTING Angle-Side-Side (How many triangles possible?):
    ASA-CONSTRUCTING Angle-Side-Angle:
    AAS-CONSTRUCTING Angle-Angle-Side:
    Using Corresponding Angles Converse:
    Using AIA Converse Theorem:
    constructing parallel   constructing parallels
    Using AEA Converse:
    Using Same-Side Interior Angles (supplementary) Converse:
    Using Perpendiculars:
    PERPENDICULAR LINES (Perpendicular , Perpendiculars)
     How to construct perpendicular through a point on a line, and not on a line.
    Perpendicular through points  perpendicular twice
    BISECT (Congruence Copy, Bisect) worksheets
    congruence construction copy Bisecting
    COPY ANGLE Video

    3D Formulas
    Pyramid total Surface Area:
    pyramid  pyramid formulas
    Use the formulas for Lateral and Surface Area to copy and complete the table:
    pyramid Base pyramid lateral surface areas  

    Lateral Area, Surface Area, and Volume formulas for prisms:
    LA and SA  prims net
    Cut out nets and fold into these prisms:
    net prisms
    Measure each dimension (in cm) and use these formulas to:
    prism formulas Volume Formula  
    Copy and complete the table for prisms' perimeter, Lateral Area, Surface Area (with Base), and Volume.
    prism perimeter and LA   prism SA and Volume
    Geometry Spring Semester Notes
     May 22, 2012
    "Platonic Solids"
    Count the number of faces in each net.  Then name each polyhedron accordingly.
    tetrahedron hexahedron octahedron dodecahedron icosahedron
    Which one folds into a pyramid? What kind?  Which one folds into a prism? What kind?
    Count the number of Vertices, Faces, and Edges of each.
    Confirm that V + F = E + 2 for each solid.
    Platonic VFE  
    Which one has the same number of faces and vertices?  This means it can be inscribed inside itself.
    Of the four solids remaining, pair them up so that their number of Vertices equals the other's number of Faces (F = V).
    These pairs of solids can be inscribed inside each other. Each can also be inscribed into a sphere, and a sphere inscribed into each.
    Why are there only five Platonic solids?
    Why five only  
    "Similar Solids"
    Similar Solids   
    There is a typo on the second side.  The Volume of the tetrahedron, or triangular pyramid, is incorrect.
    Similar Solids  
    The correct tetrahedron volume in #8 should be 18 root 2, not 8 root 2.  Then 1152 / 18 is indeed 64, or 4 cubed.
    May 17, 2012
    "Sphere Volume and Surface Area"
    May 15, 2012
     cone's surface area
    And its Volume  from a pyramid V = Bh/3 is V = pi r^2 h / 3.
    Cone Lateral & Surface Areas  
    Cone Volume  
    Cone Example 1  Cone Examples 2-3
     May 14, 2012
    "CYLINDER L.A., S.A., and Volume"
     Circle based prism
    Example cylinder  
    cylinder solutions  
    May 10, 2012
     "PYRAMID Surface Area and Volume"
     Finding Total Surface Area
    Finding surface area  folded oblique pyramid
    pyramid stencil areas  
    three pyramids  
    pyramid formulas  pyramid solutions
     May 8, 2012
    rectangular triangular prisms  rectangular triangle volumes
    polygonal sa  polygonal prism volumes
    octagonal trapezoidal prisms  octagonal trapezoidal volumes
     May 1, 2012
    Polyhedra = "many faced solids"
    face = polygonal Base
    Note:  Square Prisms are Rectangular Prisms and both are Tetragonal Prisms (each base is a four-sided quadrilateral)
    Proof of Euler's Formula that V + F - E = 2 always:
    PYRAMIDS: Vertices + Faces - Edges =
    (n + 1) + ( n + 1) - (2n) = n + n + 2 - 2n = 2n - 2n + 2 = 0 + 2 = 2.  QED
    PRISMS:  Vertices + Faces - Edges =
    (2n) + ( n + 2) - (3n) =  3n + 2 - 3n = 3n - 3n + 2 = 0 + 2 = 2.  QED
    This sum is also 2 for the Platonic solids :  tetrahedron, hexahedron, octahedron dodecahedron, icosahedron.
    ACTIVITY:  Cut out and fold the following nets into polyhedra.
    triangular prism net tetragonal prism net pentagonal prism net hexagonal prism net octagonal prism net octagonal pyramid net
    hexagonal pyramid net pentagonal pyramid net  square pyramid net triangular pyramid net
     For first side of Similar Area worksheet:
    Square Ratios

    REVIEW ACTIVITY:  Find the area of each polygonal shape in the nets to find the total surface area of each pyramid and prism.

     area formulas
    Unit 10 "Area"
     Formulas for Review
    square  rectangle and parallelogram
    hexagon trigon  
     regular trigon
    (1/2) apothem * perimeter = (1/2)*8.66*51.96 = 129.9
    PENTAGON: radius = 10, Area = (1/2) apothem * perimeter
    pentagon angles pentagon sides pentagon area  
    HEXAGON radius = 10, Area = ap/2
    hexagon angles hexagon sides hexagon area  
    hexagon area

    kite  rhombus tetragon
    arclength  sectorarea
     April 26, 2012
    "Similar Areas"
    #1A-E & 2a-e  Number 3-4 a-e
    April 19, 2012
    "Circumference Area"
     Circumference #1
    Circumference #2  
    Circumference #3  
    Circle #8  
    April 17, 2012
    apothem and perimeter  
     April 5, 2012

    "Regular Polygon Area"
    For a regular triangle, we must first calculate the height from a 30-60-90 triangle:
    triangle grid area  regular triangle formula
    And since there are six regular triangles in a hexagon, we can get that formula by multiplying by 6:
    Hexagons Areas  
    In general, for any regular polygon (the example was an octagon), the area is:
    polygon area  
    where the apothem is the perpendicular from the polygon center to its side.
    In this case, the side was multiplied by 8 to get the perimeter of the octagon;
    however, in general multiply by n (number of sides) so that the perimeter p = ns.
    April 3, 2012
     "Trapezoid Area"
     #2) Trapezoid area triangle derivation:
    Trapezoid triangles
    #3 Trapezoid Derivation
    Trapezoid Derivation 3  
    trapezoid base substitution  trapezoid median area
    So another formula for the are of a trapezoid is Area = median * height (A = mh)
    Trapezoid Test Question  
    Deriving area of a kite:
    Kite Area proofs  
    Also works on rhombi and squares.  For a square, it is d^2/2 (d squared divided by 2.)
     April 2-3, 2012
     "Triangles and Trapezoids"
    Deriving acute triangle area formula:
    deriving acute bh/2  
    Deriving trapezoid area:
    derive trapezoid area
    Practice test question:
    triangle test question
    triangle solutions
     "Parallogram Area"
     April 2, 2012
     Reviewing parallelogram grid area:
    parallelogram grid  
    Confirming with Pick's Theorem:
    pick's parallelogram
     Applying Pick's Theorem to a triangle:
    pick's triangle  
    Guesstimating the triangle area formula from its grid area estimate:
    triangle grid  
    April 2, 2012
    Area packet "rectangles and squares" page. These are the answers to #1 on:
    rectangle squarea  
     And these are for #2:
     proving base times height
    So 1 rectangle has area 2bh/2 or bh.  A = bh (base * height).

     Unit 9 Circles
    "Arc Angles"

    Lost Arcs
    Arc Angles
    Angle Chord
    AC Example 1  AC Example 2
    Angle Secant
     AST ATS
    Inscribed Angles

    x = 40 m arc AB = 68  
    The Inscribed Angle Formula provides an alternate method
    of  calculating polygonal interior angle measures:
    Inscribed Polygons Circle
     triangle =>    360/3 = 120 => 1(120) / 2 = 60
     square =>      360/4 = 90 => 2(90) / 2 = 90
     pentagon => 360/5 = 72=> 3(72) / 2 = 108
     hexagon =>   360/6 = 60=> 4(60)/ 2 = 120
     septagon => 360/7 = 51 3/7=> 5(360/7)/ 2 = 128 4/7 = 128.57
     octagon =>    360/8 = 45 => 6(45)/ 2 = 135
     nonagon =>    360/9 = 40=> 7(40)/ 2 = 140
     decagon =>    360/10 = 36=> 8(36)/ 2 = 144
     hendecagon => 360/11 = 32 8/11 => 9(360/11)/ 2 = 147 3/11 = 147.27
     dodecagon =>    360/12 = 90 => 10(360/12)/ 2 = 150
    n-gon => 360 / n => (n-2) * (360 / n) / 2 => 360*(n - 2)/(2n) = 180(n - 2) / n

    Tangent Circles
    Tangnet Internal External
    Tangents External Internal  
    March 21, 2013
     More on Tangents and Inscribed Angles
    Practice 12-2 KEY
    Practice 12-4 KEY
    Common External and Internal Tangents
    Tangents Definitions
    The lines are internally tangent when the circles are tangent externally.
    When the circles appear internally tangent, the lines are externally tangent to the circles.
    Prove the common internal tangent segments are congruent and likewise for the external tangent segments.
    Tangents Theorems
    Apply the Pythagorean Theorem to find the radius BD of the small circle B.
     Use the scale factors between similar triangles to calculate CE, the radius of the large circle E.
    (Use the whole side AE, not just the partial side DE.  So AE = 8 + 4 = 12.)
    For finding BC, apply the Side-Splitting Theorem since BD // CE.
     Unit 9 "Circles"
     circle terms and theorems
    angle segements
     arc angles
    angle chord formula
    angle secant formula

    Circle Segments
    Circle Segments
    Segment Chord Thm  
    Segment Chord Proof
    SC Ex1

    Segment Secant Theorem
    Segment Tangent Thm  
    ST Ex1
    ST Ex2
    ST Ex3  

    Inscribed Angles

    x = 40 m arc AB = 68
    "Inscribed Angles"
     front page
    Inscribed Angle KEY
    back page
    Inscribed Corollaries KEY  

    "Tangent Theorems"

     "Diameter-Chord Theorems"
    front page
    Diameter Chord
    "Chord Corollaries"
     back page
     Chord Corollaries
     March 5th, 2012
    "Arcs and Chords"
    ArcsAnglesChords KEY
    Monday, February 27th, 2012
    "Circle Vocabulary"
    Circle   Radius
    Tangent Line   Secant Line
    Chord   Diameter
    Inscribed polygon   Circumscribed polygon
    Central Angle   Arc

     Unit 9 Lesson 1
    circle wordle

     Circle Basics Circle Vocabulary
    Circle Words
    Circle Segments

    angles formed by chords & secants  fill in the measures
    angle segements arc angles
    angle chord formula
    angle secant formula
    Circle Segments
     segment chord
     chord formula
    segment secant
    outer times whole   
    secant tangent  
    chord formula chord example  
    segment chords chord segments  
    secant secant formula
    secant tangent

    Unit 8 Trig
    trig words
    triangle HOA
    trig ratios
    sin cos tan
    sine cosine tangent

     sohcatoa example
    sohcahtoa banner
    sincostan divides  
    inverse sine  
    sohcah inverses  
    inverse tan  
    sohcahtoa trig  
    sohcahtoanotes trig2
     Trig Thms 1-2
    Trig Thms 3-4
    Trig thm 5
    Trig thm 6  

     Special Right Triangles
     special triangles


    The Isosceles Right Triangle

    Isoscesles right triangle

    The 45-45-90 triangle is another of the special right triangles. It is also known as the Isosceles Right triangle(it has two legs of the same length making it an isosceles triangle).

    It has angles of 45°, 45° and 90° and is the same shape as half of a square.

    The sides of the 45-45-90 triangle have the ratios: 1 : 1 : √2

    Isosceles right triangle ratios

    Opposite the 90° angle is the side with a ratio of √2

    Opposite the 45° angles are the legs, both with a ratio of 1

    We can now use these ratios to help us find the length of missing sides

    See how in the worked example below

    isosceles right triangle

    Before we go through the examples, it is very useful to know the different side relationships - they help you solve 45-45-90 triangle problems quickly:

    • Both legs are the same length, if you have the value of one you also have the other.
    • If you have the length of the legs and you need to find the length of the hypotenuse then you

      must multiply the length of a leg by √2.
    • If you have the length of the hypotenuse, to find the length of the legs you must divide the

      length of the hypotenuse by √2.
    45 45 90 triangle

    Special right triangles have side length ratios which you can use to quickly find missing side lengths.

    Half-Equilateral Triangle


    Half equilateral triangle

    We are going to look at two different types of special right triangles:

    The 30-60-90 triangle has angles of 30°, 60° and 90° and is the same shape as half of an equilateral triangle.
    short long hypotenuse
    The sides of the 30-60-90 triangle have the ratios: 1 : √3 : 2

    special 30 60 90  

    Opposite the 30° angle is the smallest side with the ratio of 1

    Opposite the 90° angle is the largest side with the ratio of 2

    Finally the 60° angle is opposite the ratio √3

    We can use these ratios to find the missing lengths of sides

    See how in the worked example below

    30 60 90 formula

    Before we go through the examples, it is very useful to know the different side relationships - they help you solve 30-60-90 triangle problems quickly:

    • When you have the length of the short leg, to find the length of the long leg you must multiply the short leg by √3.
    • If you have the length of the long leg and you need to find the length of the short leg then you must divide the long leg by √3.
    • If you have the length of the short leg, to find the length of the hypotenuse you must
      multiply the length of the short leg by 2.
    • If you have the length of the hypotenuse you can find the length of the short leg by dividing
      by 2.


    special examples  special triangles
    special right triangles  

    trip triangles  
    Triple Triangles
     How to generate them? One way is the Babylonian method.
    Babylonian Triple Babylonian triples  
    Another way is the Pythagoreans' method:
    Pythagorean triples method  
    This formula starts with an odd number.
    pythagorean triple table  
     How many of these Pythagorean tripled triangles can you generate?
    pythagorean triples triangles  

    Ext Angle threorem
    Ext Angle Contradictions
     angles opposite sides
    Opposite Angles Inequality
    Sides opposite angles  
    Opposite Sides Inequality
    Triangle Inequality Thm
    Can 6, 7, and 8 form sides of a triangle?
    Triangle Inequality Example  
    No triangle with sides 4, 8 & 2 No Triangle with sides 10, 4, and 3
    Sides not long enough
    makes a segment since 1+2 = 3  
    The Triangle Inequality  
    shortest distance  
    shortest distance proof  
     Hinge Theorems
     SAS Hinge Inequality
    SSS Hinge Inequality  

    Unit 7 Similarity
     similarity wordle
    The Golden Ratio
    the golden rectangle  
    Donald in Mathmagic Land
    Golden ratio covered from 7 minute mark.
    To calculate the ratio, setup a proportion:
      golden ratio definition
    Letting a = 1, one can calculate the solution:
    golden ratio calculation golden rectangles
    So a golden rectangle has an internal ratio of 1 to 1.618.
    gold rect 1618 rectangle ratio
    Also, 1 / 0.618 = 1.618.
    Such rectangles can be nested within each other to create a golden spiral:
    golden phi golden euclid golden spiral
    You can approximate a golden rectangle by adding larger and larger squares:
    fibonacci rectangles
    Here is how to fold one:
    golden fold
    A line parallel to a side of a triangle splits the other two sides proportionally.
    side splitting
    side splitter side splitter thm
    Transversals split parallel lines proportionally.
    proportional parallels
    two transversals  
    An angle bisector in a triangle splits the opposite side in proportion to the other two sides.
    Triangle Angle Bisector  
    triangle bisector thm  
    Practice 8-3 key 1-6
    Practice 8-3 key 7-13  
    Practice 8-5 key 1-9 \
    Practice 8-5 key 10-18  

    Project Mathematics! Video "Similarity
     Similarity video questions
    (see Downloads for Answer Key)
     Proportion Properties

    Lesson 1 Reflections
    Geo6Notes2  Translations
    Geo6Notes3 Rotations
    Geo6Notes4  Glide Reflections
    Geo6Notes5  Dilations
    Geo6Notes6 Games (billiiards & mini golf)
    Geo6Notes7  Symmetry
    translation 2  
    ROTATIONS Lesson
    Find rotation lines  
     rotation 3
    rotation 4-5
    GR 1
    GR 2
    GR 3
    GR 4
    GR 4-6  
    Dilation Dilation Dilation Dilation Dilation
    Dilation 1
    Dilation #2
    Dilation 2  
    Dilations #3
    Dilations 3a-e  
    Dilated Graph 3
    dilation graphs  
    Dilation #4
    dilation 4a-e
    dilation 4b
    dilations 4  
    Sir Putt a Lots golf lessons
     billiards 1
     simpsons billiards
    billiards 2  
    miniature golf  
    practice use for geometry
    golf 3  
    Dead Putting Society  

    Transformation #1
    Transformations 2-4
    Transformation 5
    Transformations 6-7

    Symmetry 1
    symmetry 2
    symmetry 3  
    symmetry 5  
    Weird Al Yankovic palindrome video "Bob":
    Each verse is the same forward and backward (subtitled):
    Unit 4
    medians 2/3s
    circumcenter circle  incenter circle
    incenter inradius incenter point
    orthocenter  orthocenters
    orthecenter point  
    "Special Segments and Bisectors"
    Unit 4 Notes 4
    #1-6 on Medians & Altitudes worksheet
    medians, altitudes, and bisectors  
    angle bisectors and isosceles  bisectors and equilateral
    Lesson 4-1 "Congruent Triangles"
    Triangle TRI  Triangle GON
    Congruent TRIGONs  

    Unit 3 Notes #5A
    polygon word cloud  
    polygon wordle
    polygons black
     Interior Angle Sums interior angle measure
    Unit 3 Notes #5B
    What about a polygon's exterior angle sum?  Get your pencil out:
    pencil proof  

    The exterior angle sum = 360.

    Exterior Sum = 360

    Exterior Angle Measure = 360/n.

    Exterior Angle Sum  exterior angle measure


    regular polygons

    The interior angle sum formula is 180(n-2).

    polygon angle measures  

    Interior Angle Measure = 180(n-2)/n.


    polygon names

     How to name a polygon:

































    Playfair's Axiom  

    Euclid's Fifth Postulate  
    Can we show that congruent corresponding angles are a direct consequence of Euclid's Postulate?
    Corresponding from Euclid
     If Angle 1 is larger, then Angle 3 is smaller (Linear Pair.)
     So Angles 3 & 5 are less than 180 and the lines intersect on their side by Euclid's postulate.
    If Angle 5 is larger, then Angle 6 is smaller (Linear Pair.)
    So Angles 4 & 6 are less than 180 and by Euclid's fifth postulate, the lines intersect on their side.
    Either case contradicts the given statement that lines l and m are parallel.
    This conclusion is a contradiction so our assumption was false, and corresponding angles must be equal.
    What about the converse?  Is it a postulate or a theorem?
    Corresponding Angles Converse  
    Let's say Angle 1 is 80 degrees.  Then so is Angle 3 by the Given. 
    But Angle 2, when formed, plus Angle 3 also must have a sum of 80 degrees for the Triangle Sum Theorem to be true.
    So either the measure of Angle 2 is zero, or Angle 3 is less than Angle 1.
    This conclusion contradictions the Given (Angle1 = Angle3), or lines l and m overlap (Angle 2 = 0.)
    Either case produces a contradiction, so the original assumption was false and the lines are indeed parallel.
    euclid's parallel postulate  

    Unit 3 Notes #4
    Draw any triangle and cut off its three angles.
    triangle three pieces
    Then arrange those three angles into a line by joining the vertices.
    three triangle angle line  
     What conclusion can be made about the angles of a triangle?
    acute ob rtuse  
     angles a+b+c=180
     Here's why this works:
    triangle alpha beta gamma
     Prove this conjecture for all cases--acute, obtuse, right, and equiangular.
    Triangle Sum Thm KEY
     The Triangle Sum Theorem is true for all classifications of triangles.
    triangle types
    triangle classify
    There are four important consequences, or corollaries, to the the Triangle Sum Theorem.
     equiangular corollary   complementary corollary  
    third angle theorem triangle exterior angle  
    Here's an example of the triangle exterior angle theorem:
    exterior angle example
    See PROOFS link to complete the answers to the proofs in part II on the back.
     acute complement


    Proving Parallel Lines Converse Theorems
     Geometry Unit 3 Notes #2
    If lines (cut by a transversal) are parallel, then five conclusions can be made:
    If parallel lines, then ...  
    The converses of each of conclusion for corresponding, alternate inteior/exterior, and same-side angles are:
    But each converse is not automatically true, they must be proved true.  We will prove the Corresponding Angles Converse at the end of this unit.
     Supply the missing Statement that matches each Justification in the proof of each converse theorem.
    If alternate interior angles are congruent, or alternate exterior angles are congruent, 
    then lines cut by the transversal are parallel.
    AIA Converse AEA Converse
    If same-side interior angles or same-side exterior angles are supplementary, then lines are parallel.
    SSE Converse  SSE Converse
    Parallel Planes Thm  Perpendicular to Parallel

    Two Perpendiculars  Three Parallels Transitivity

     transversal theoremms

    Unit 3 "Transversals"
    Lesson 3.3 "Parallel Lines Theorems"
    Match the Justifications with the Statements in each colored puzzle proof.  Work in groups of 4.
     Alternate Interior Angle Thm. Alternate Exterior Angle Thm.
    Each person in the group is responsible for one color proof, however, ALL proofs must be copied.
    Same-Side Interior Angle Thm.  Same-Side Exterior Angle Thm.
    Copy the theorem, diagram, and proof into page 3 of the Parallel packet.  Use same diagram for all theorems.
    Here's another way to prove these theorems:
    AIA Thm Proof  
    AEA Thm proof
    SSI Thm Proof
    SSE Thm Proof

    Geo3Notes1 Parallel Lines Theorems
    Geo3Notes2 Proving Parallel Lines
    Geo3Notes3 Perpendicular and Parallel Corollaries
     Geo3Notes4 TRIANGLE SUM Theorems
    Geo3Notes5 POLYGONS
    (Interior Angle Sum/Measure, Exterior Angle Sum/Measure)
    corresponding angle pairs
    d and h make F corresponding angles congruent  
    Corresponding F Angles
    Corresponding angles illustration
    math is a highway   window to corresponding angles
    aia pairs
    the Z test
    (make forwards and backwards C's)
    same side interior angles
    same side exterior angles
     SSE Highway

    Unit 3
     Transversals and Angles
    When a transversal line (AB) cuts across two other lines (PQ & RS),
    eight angles are formed.
    transversal cutting parallel lines
    If the lines are parallel, the angle pairs are either congruent or supplementary.
    If the lines are NOT parallel, they intersect on the same side as the
    consecutive interior angles being less than 180 degrees.
    non parallel  non parallel
    Intersect on left side where 70 + 104 < 180.                    Intersects on right side wjere 110 + 64 < 180.
    If a transversal cuts across parallel lines ,
    corresponding angles  corresponding angle pair
    then corresponding angles are congruent,
    Alternate Interior Angles
    alternate interior angles are congruent,
    same side interior
    and same-side exterior/interior angles are supplementary.


     #2 on last page of Parallel Packet
    Unique Parallel Line Threorem  
    #3 on last page of Parallel Packet
    Perpendicular Offline  
    #4 on last page of Parallel Packet
    Perpendicular Online Thm.  
    Unit 2
    Geo2Notes1 If-then Statements
    Geo2Notes3 Properties of Congruence
     Geo2Notes3A Two Column Proofs
    Geo2Notes4  Theorems of Angles
    Geo2Notes5 Flowchart Proofs
    (Geo2Notes5A-Bisectors, Geo2Notes5B-Common Overlaps)
    Geo2Notes6 Paragraph Proofs - Perpendicular
    Paragraph Proofs
    Paragraph Proof
    Flowchart Proofs
    Unit 2 Notes #5
    Use the two column proof of the midpoint to complete the flowchart proof on the right.  Click on the picture on the right for the answers.
    Midpoint Theorem  midpoint flowchart
    Now translate the two column proof for the Angle Bisector Theorem into a flowchart proof.  Click on right pic for solution.
    Angle Bisector Proof  Angle Bisector Flowchart
    You can use the Midpoint Theorem Flowchart to help answer the Angle Bisector Theorem Flowchart proof, and vice-versa.
    Bisector Flowcharts
     Flowchart Proofs are good for proving true biconditional theorems containing if-and-only-if statements. 
    Observe how there are two proofs required to prove the conditional and the converse directions of the Common Overlapping Segment.
    Common Segment Proof
    Common Segment Converse  
    But with a flowchart, both directions can be proven with one proof.
    Common Overlapping Segments
    See if you can complete the Common Overlapping Angle Flowchart proof from using the Common Segment flowchart.
    Common Overlapping Angles
    Use the completed two-column proofs below to check answers in the flowchart above.

     Common Angle Theorem
    common overlap anlges KEY
    Common Angle Converse
    Now use the two-column proofs above to complete the flowcharts below:
    The completed flowcharts for all the Common Overlapping figures are below:
    common overlaps flowchart


     Properties of Angles
     Two Column Proofs
    comp supp diagram
    Draw the angle that is supplementary to a 40 degree angle.
    congruent aliens
    Congruent Complements Supplements
    because reasons


    Unit 2 Notes #3
    Properties of Equality
    Addition  Subtraction  Multiplication  Division
    add equals  subtract equals  multiply equals  dividing nonzero equals
    Identity  additive and multiplicative IDs  Inverse Properties  Additive and Multiplicative Inverses
    Reflexive  Symmetric Property  Transitive Property
    a = a  a=b iff b=a  transitive equations
    Associative Properties   of addition and multiplication
    Commutative Properties  of addition and multiplication
    Distributive Property of multiplication over addition  
     Apply these properties to justify each step in the proof of an algebra solution.
    Justify each step  Transitive Poperty
    Unit 2 Notes #1

    September 11th, 2012
    Notes for 9/11/12

    page 2 of 3
    Unit 2 "Logic"

    Lesson 5 "Paragraph Proofs"
    Prove that perpendicular lines make four congruent, right angles
    in two-column and then paragraph form.
    Perpendicularity Thm Paragraph
    Converse  converse paragraph
    Translate the two column proofs above into the paragraph proof below:
     perpendicular paragraph
    Most books cite a "Linear Pair Theorem" or "Property",
    and there is a sister theorem for perpendicular lines.
    Perpendicular Pair Theorem  Perpendicular Pair Converse
    The yellow proof specifies "adjacent" complementary angles.  Is it true without that key word?
    Note that for this to be true, the angles must be acute.  Can you draw a counterexample with obtuse angles?

      Lesson 4 "Properties of Angles"
    with Two Column Puzzle Proofs
     Supply the missing Justifications in the proofs of Right Angle, Linear Pair, and Vertical Angle Theorems   
    Right Angle Thm  Linear Pair Thm
    Vertical Angle Thm  Supplementary Equals Thm

     Properties of Angles
     Two Column Proofs
    comp supp diagram
    congruent aliens
    Complements  Supplements
    because reasons

    Geometry Unit 2
    Notes 2 -- Proofs
    The REFLEXIVE Property of Equality states that a = a, a quantity is equal to itself.  To establish this property is also true for congruence of angles and segments, we list our Statements with Reasons justifying every step.
     Reflexivity Theorem
    To show that the SYMMETRIC Property of Equality applies to angle congruence, use the definition to show that congruence implies equality of angle measures.  Then apply the symmetric property by reversing the order of the equation.  Finally, equality implies angle congruence, so we translate the equation into the congruence statement we wanted to prove.
    Angle Symmetry
    The TRANSITIVE Property states if a = b and b = c, then a = c.  Given three angles, two of which are congruent to a third, substitute angle measures in place of a, b, and c to establish congruence of angles is transitive.
    Angle Transitivity
     Hence, congruence of angles is reflexive, symmetric, and transitive.
    To prove segment symmetry, start with two segments given to be congruent and work toward the final statement to be proven.
    Segment Symmetry
     Note how the biconditional if-and-only-if statement in the Segment Congruence Definition is true in both directions.
    Segment Transitivity
    Most of the time, but not 100%, we start a two column proof with a GIVEN statement. 
    Likewise, the statement to be proven is usually last.  There are some exceptions to this trend, however.

    Lesson 3 "Properties of Equality"
    (Geo2Notes2B from 2012)
    Properties of Equality  
    Which ones are Addition, Subtraction, Multiplication, and Division Properties of Equality?
    Identify the Associative, Commutative, Distributive, Inverse, and Identity Properties.
    Find the Reflexive (Duh!), Symmetric (Mirror), and Transitive (Transylvania Train) properties.
    Use the properties of equality to deductively prove this algebra solution:
    Justifying algebra steps  

    Lessons 1-2 "If Then Statements"
     P = hypothesis (or antecedent) = the if-part,      Q = conclusion = the then-part.
     If P then Q statements
    Euler diagrams                  Names of If-Then statements                        Sentences T or F?
    Names of if-then statements True of False
    EULER DIAGRAMS                                                                    
       CONDITIONAL            CONVERSE               BICONDITIONAL               INVERSE                 CONTRAPOSITIVE
    If P then Q If Q then P P if and only if Q If not P then not Q If not Q then not P
       If P, then Q.             If Q, then P.            P if and only if Q.      If not P, then not Q.      If ~Q, then ~P.
    The CONDITIONAL and CONTRAPOSITIVE are both true or both false.
    The CONVERSE and INVERSE are both false or both true.

    Unit 1 "Basics"
    Unit 1 Basics
    Unit 1 "Basics"
    Geo1Notes1 - Konigsberg Bridges
    Geo1Notes2 - Undefined Terms
    Geo1Notes3 - Measuring Length
    Geo1Notes4 - Angle Measures
    Geo1Notes5  - Angle Pairs
    Unit 1 Notes #5 Angle Pairs
     comp vs supp
    complementary example
    comp supplementary examples
    supplementary vs linear pair
    comp supp word problem  
     solve a math problem
    comp supp word problem  
     complimentary complements

    Unit 1 Notes #4 "Angle Measures"
     angle word cloud
    Ancient Babylonians knew the Earth revolved around the sun and took a guess at the number of days in one revolution.
    Based on keeping track of the stars at night and matching them with the change of seasons annually, 
    360 degree circle
    they figured it was about 360 days.  This conclusion fit in nicely with their sexagesimal (base 60) number system.
    Half a year means a straight line across the circle corresponds to 180 degrees.  Hence, we have the Straight Angle Assumption.
    Straight Angle Assumption
    From this history we arrive at the Protractor Postulate, that every angle in geometry can be measured from 0 to 180 degrees.
    This is why a right angle is 90 degrees instead of 100, which would match our base 10 Arabic numeral system.
     Protractor Postulate
    To find the measure of angle COD in this picture calculate 135-33.  It's 102 degrees.
    Likewise, measure of angle DOC is |33-135| = |-102| = 102°.
    Here are five types of angle classifications based on their measurements:
    Angle Classifications  
    We could add a zero angle to this list, and remove the reflex angle since it is greater than 180 degrees.
    Angles Types with Zero  
    We will only consider the first four--acute, right, obtuse, and straight--along with the zero angle. 
    Angle Types
    A full rotation is also called a complete angle.
    acute angle parking

    Unit 1 Notes #3
    "Measuring Length"
     Review of Lines, Rays, and Segments:
    Segment Ray Definitions
    Segment definitions  
    Note:  AB and BA are the same segment and distance,
    but ray RS and ray SR are not the same (different endpoints).
    Opposite rays have the same endpoint.
    How do we measure the distance or length of a segment?
    The Ruler Postulate
    Ruler Postulate Definition
    Ruler Formulas  
    Ruler Congruence
    Find any congruent segments on this number line.
    Ruler Postulate Practice
    AB = 2, DE = 2, FG = 2, GH = 2 and HI = 2 so all are congruent.
    Are CE and EF congrent?
    Name a segment congruent to FH.
    Name two segments whose lengths are 6.
    Segment Addition Postulate  
    An ant walks along the sidewalk to demonstrate the segment addition postulate:
     AB + BC = CA
    Applying an algebra example in geometry:
    Find x, AB, and BC.  
     Unit 1 Notes #2
    "Undefined  Terms"
    Points, Lines, Planes
    PLP title
    Undefined vocabulary terms:  point, line, plane
    Defined terms:  ray, segment, space (depend on point, line, plane)  
    geometry terms  
    Ray -- All colinear points to one side of a point on a line.
    Segment -- All points between two points on a line.
    Use CAPITAL BLOCK LETTERS to name a point.
    For a line, you can use a handwritten cursive lower case letter ( l ), 
    or the two points on it with the proper symbol above (line, segment, ray).
    A plane can also be named with any 3 or 4 non-colinear points in it. (e.g. plane ABCD or plane XYZ)
    Also you can name a plane with its own capital letter, but write it as italicized or cursive (plane M)
    DEFINED TERMS: (colinear and coplanar)
    Colinear -- points that lie together in the same line
    Coplanar -- points, lines, and figures in the same plane
    colinear definitions
    A true assumption in geometry is called a postulate.
    (Also, an axiom accepted as fact.)
    "Through any two points there is exactly one line."
    (Two points determine a line.)
    "Two lines intersect at a point."
    "Two planes intersect in a line."
    "Three non-colinear points determine a unique plane."
    (colinear points such as ... won't work because many planes go through them!)
    Identify two lines that are coplanar in the diagram.
    Points A and B are colinear.  Are points H and E?
    Name 3 coplanar points.  Are J, C and E coplanar?
    intersecting planes  
    Lines AB and MN are coplanar (both in horizontal plane.)
    Yes, H and E are colinear because any two points determine a line (even if not connected.)
    MAB are coplanar (in lane K), points B, D, and F are coplanar (any three non colinear points make a plane.)
    Are colinear points coplanar?  Yes, D and G make a line and are in the vertical plane (L).
    Are coplanar points colinear?  Not always, so false.  Points in a plane may not lie in the same line.
    plane box  
    The plane on the left side can be named ADFE, DFAE, ADF, and many other combinations.
    Is ABGF a plane?  Yes, imagine a diagonal plane from the top back (AB) through the bottom front (FG).
    Even if not drawn, any 3 non-colinear points make a plane.

    Traversable Networks answers
     #5 A-E #5 E-L #5 L-P
    All the traversable networks have 0 or 2 odd nodes.  If there are more than 2 odd notes, the network is not traversable.
    Unit 1 Notes #1
    Are the Bridges of Koenigsberg traversable? Draw a network to find out.
    seven bridges  bridges of koenigsberg
    Koenigsberg  konigsberg bridges
    Here's an excerpt from the TV show Numb3rs that discusses it:





     Unit 8 "Right Triangle Trigonometry"
    February 9th-14th, 2012
     "Trigonometric Ratios"
    Thursday 2/9--Trig Ratios
    Monday 2/13--Tangent Ratio
    Tuesday 2/14--Sine and Cosine Ratios
    Thursday 2/16--Solving Right Triangles
     February 7th, 2012
     "Special Right Triangles"
    There are some triangles where one only needs to know the length of one side,
    and the other two can then be calculated.  This is a slight improvement on the Pythagorean Theorem,
    where you need two sides to calculate a third.  Here is the first triangle:
    Isosceles Right Triangle  
    Note how there is always a square root of 2 as a factor in the hypotenuse length of an isosceles right triangle.
    Here is the second special right triangle:
    30 60 90 triangle  
    This time, there is a square root of 3, but as a factor of the long leg, not the hypotenuse. 
    Here's how to remember both formulas:
    The isosceles triangle has a root 2 because it has two equal sides.
    The 30-60-90 triangle has a root 3 adjacent to the angle with the 3 in it,
    and the longest side (hypotenuse) is twice the shortest leg. (or to travel from 30 to 60, multiply by 2!)
    You will see these triangles again and again not only on the quiz, test, and final in geometry,
    but also in future math courses of Trigonometry, Pre-Calculus, and Calculus.  Memorize these patterns.
    January 31st, 2012
    Unit 7 "Similarity"
    January 23rd, 2012
    front page
    Side-Splitting Theorem
    back page
    Proportional Parallels Thms.  
    January 19th, 2012
     front page
    SAS Similarity worksheet
    back page
    SSS Similarity worksheet
    Unit 6 "Triangle Inequalities" (period 4)
    Lesson 3
    January 9th, 2012
    Triangle Opposites Inequalities
    Lesson 2
    January 5th, 2012
     Triangle Inequality
    Lesson 1
    January 4th, 2012
    "Indirect Reasoning Proofs"
    Unique Lines
    Indirect Proofs
    Geometry Class Notes
    Unit 5 "Quadrilaterals"
     Quadrilateral Quest program
    Unit 5
    word cloud
    (rhombus, rectangles, square)
    Geo5Notes6  Quadrilateral Hierarchy
    quadrilateral euler diagram
     quadrilateral groups
    quadrilateral 8

      Unit 5 Notes 1
    kite angles
    kite perpendicular  
     opposite kite angles
    Kite Bisectors Kite Diagonals  

     If a quadrilateral is a parallelogram, it has the following properties:
    parallelogram properties

    The converses are also true.  These are sufficient conditions to be a parallelogram.
    Parallelogram Conditions  
    Also, if one pair of sides is both parallel and congruent, then the quadrilateral is a parallelogram.
    bisecting diagonals  
    congruent parallel pair  
    opposite sides  
    opposite angles
    consecutive supplementary angles  
    The quadrilaterals formed by joining consecutive midpoints of other quadrilaterals are:
     quad midsegment  quad midsegments

    Trapezoid Median Theorem Transparency 
    Class Puzzle Proof
    Find the length of the median by adding each midsegment length:
    trapezoid midsegments  
     trap midsegment
    trapezoid transparency
    isosceles trapezoid parts
    isostrap theorems  
     special parallelogram parallelogram chart
    square definitions  
    rhombus cloud
    rectangle properties  
    Identify whether the diagonals are bisecting, congruent, perpendicular or a combination:
    parallelogram diagonals
    special parallelograms
    parallelogram venn circles
     parallelogram checkbox
    quadrilateral table
    quadrilateral wordle
     quadrilateral parallels
    quadrilateral venn  
     quad class
    quadrilateral chart

    quadrilateral tree quadrilateral branches


    Unit 4 "Congruent Triangles"

    Unit 4 "Triangles"
    triangle wordle  
    Geo4Notes1 Congruent Triangles Definition
     Geo4Notes2 Congruency Conditions
    Geo4Notes3 Triangle Congruence Theorems
     Geo4Notes4 Special Segments
    Geo4Notes5 Isosceles Triangles
    Geo4Notes6 Midsegment Theorem
     Lesson 5 "Midsegment Theorem Proofs"
    Unit 4 Notes 6
     midsegment triangles
     Proving with triangles. Proving with parallels.
    Perpendicular Bisector/Bisected Angle Sides Theorems
    (see PROOFS link for completed proofs)
    A point is on the perpendicular bisector of a segment
    if and only if
    it is equidistant to both endpoints of that segment.
     Perpendicular Bisector Thm Perpendicular Bisector Converse
    Bisected Angles Sides Theorem Bisectoed Angle Sides Converse
    A point is on the angle bisector 
    if and only if
    it is equidistant to both sides of the bisected angle.

    Lesson 4 "Isosceles Triangles"
     Unit 4 Notes 4
    isosceles terms
    isosceles theorem   isosceles converse

    Corresponding Parts of Congruent Triangles are Congruent
    If two triangles are congruent because of one of the congruency conditions
    (SSS, SAS, ASA, AAS, HL, HA, LL, LA),
    then the rest of their parts--angles and sides not already marked congruent, are the same.
    Corresponding Parts  
    This applies to medians, altitudes, perpendicular bisectors, and angle bisectors as well.
    Special Segments
    Special Bisectors  

    There are special cases of SAS, ASA, AAS,
    and even an instance of S.s.A. for right triangles.
    In these cases, we call the perpendicular sides LEGS,
    and the HYPOTENUSE is the side across from the right angle.
     LL Thm  LL Thm
    HA Thm   HL
     call AAA to bail out SSA

     Lesson 3 "Angle Angle Side"
    While SSS, SAS, and ASA are postulates assumed to be true,
    Angle-Angle-Side (AAS) is a theorem whose truth we can prove.
    Transparency Puzzle Proof
    Now write congruence statements and justify why each pair of triangles is congruent.
    congruent triangle puzzle
    Answer Key:
    1. RWI by SAS      2. ESH by AAS      3. GNT by SSS
    4. ARE by ASA      5. DST by AAS      6. HEY by SAS
    7. ILH by SSS        8.  AGN by AAS     9. HAT by SAS
    10. KPG by AAS   11. YDE by ASA    12. SAK by AAS
    congruent triangle puzzle KEY  
    baby sas   when i prove triangleSSS   not sure if ASA or AAS
                                                               office angle side side

     Unit 4 Lesson 2
     Click on this link and create two triangles given the same three parts:
     (three sides, two sides and an angle between, an angle and then two sides,
    two angles followed by a side, two angles and a side in between, and three angles)
    Do these triangles appear to be congruent?
    congruent equal triangles  
    Note that they do not list all six pairs of parts (congruent angle pairs and congruent sides)  
    three parts needed  
    Turns out one only needs three pairs of parts in a certain order to get congruent triangles.
    So what are all the combinations of three pairs of parts we can list?
    Choose from A = Angle and S = Side and make six abbreviations with these two letters.
     Side Side Side SSS
    side angle side Side Angle Side
    Go around the triangle until you run into a part marked congruent. 
    The first part is a SIDE, the next is an ANGLE, and the final part is another SIDE.
    SAS and SSS Theory  
    angle side angle Go around the triangle until you see a marked part.
    The first is an ANGLE, followed by a SIDE, and the last is another ANGLE.  ASA
    angle angle side
    Going around clockwise gives us side-angle-angle.
    Going counterclockwise yields ANGLE-ANGLE-SIDE, aka AAS.
     Note that these two triangles are different:
    NOT Angle Side Side  
    So there is no Angle Side Side condition.
    To wrap up:

    Go around each triangle and determine the information given:
    Triangle Start Up  

     Determine whether the triangles in each pair are forced to be congruent:
    Triangle Forced Congruence  

    Lesson 1 "Congruent Polygons"
    Unit 4 Lesson 1
    congruent triangles
     Definition of Congruent Triangles
    Definition Picture
    Match the letters in the order they appear:  T G, R O, I N, not by where they are in the picture. 
    For example, note how triangle GON is flipped over from triangle TRI. They're mirror images of each other.
    TRI pic  GON pic
     Fill in the missing parts below with the matching equal angles and the corresponding congruent sides. 
    Check diagrams for like markings.
    congruent parts
    Now draw two triangles to match this:
    triangle congruents  
    So it's important to match up the vertices in the same order:
    Triangles ABC = XYZ
     And also that congruent triangles can be flipped or rotated and still be the same shape and size.

     Lesson #2
    "Congruent Conditions"
    SSS, SAS, ASA Postulates
    Click this link for photos of all cases:
    (SideSideSide, SideAngleSide, AngleSideAngle, AngleAngleSide, SideSideAngle, AngleAngleAngle)

    Unit 3 Review
    1.e) SSE angles:  2 & 3 and 6 & 7
    Review 3 Key page 1
    Review 3 Key page 2  
    Transversal Theorems  






    Lesson 4-2

    Lesson 5-3 Proving Parallelgrams
Last Modified on January 18, 2019