GEOMETRY NOTESSemester 2Unit 11 Solids GeoUnit11Unit 10 Area GeoUnit10Unit 9 Circles GeoUnit9Unit 8 Trig GeoUnit8Unit 6.5 "Trinequalities"GeoUnit6Similarity "Similarity"Semester 1Unit 1 "Basics"Unit 2 "Logic"Unit 3 "Transversals"Unit 4 "Triangles"Unit 5 "Quadrilaterals" Geo5Notes0CONSTRUCTIONSSSS-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:Using AEA Converse:Using Same-Side Interior Angles (supplementary) Converse:Using Perpendiculars:How to construct perpendicular through a point on a line, and not on a line.CONSTRUCT PERPENDICULARS VideoBISECT ANGLE VideoBISECT SEGMENT VideoCOPY ANGLE VideoCOPY SEGMENT Video
Pyramid total Surface Area:Use the formulas for Lateral and Surface Area to copy and complete the table:Lateral Area, Surface Area, and Volume formulas for prisms:Cut out nets and fold into these prisms:Measure each dimension (in cm) and use these formulas to:Copy and complete the table for prisms' perimeter, Lateral Area, Surface Area (with Base), and Volume.
Geometry Spring Semester NotesMay 22, 2012"Platonic Solids"Count the number of faces in each net. Then name each polyhedron accordingly.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.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?"Similar Solids"There is a typo on the second side. The Volume of the tetrahedron, or triangular pyramid, is incorrect.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"CONES"And its Volume from a pyramid V = Bh/3 is V = pi r^2 h / 3.May 14, 2012"CYLINDER L.A., S.A., and Volume"May 10, 2012"PYRAMID Surface Area and Volume"May 8, 2012May 1, 2012Polyhedra = "many faced solids"face = polygonal BaseNote: 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. QEDPRISMS: Vertices + Faces - Edges =(2n) + ( n + 2) - (3n) = 3n + 2 - 3n = 3n - 3n + 2 = 0 + 2 = 2. QEDThis sum is also 2 for the Platonic solids : tetrahedron, hexahedron, octahedron dodecahedron, icosahedron.ACTIVITY: Cut out and fold the following nets into polyhedra.For first side of Similar Area worksheet:
REVIEW ACTIVITY: Find the area of each polygonal shape in the nets to find the total surface area of each pyramid and prism.
Unit 10 "Area"Formulas for Review(1/2) apothem * perimeter = (1/2)*8.66*51.96 = 129.9PENTAGON: radius = 10, Area = (1/2) apothem * perimeterHEXAGON radius = 10, Area = ap/2
April 26, 2012"Similar Areas"April 19, 2012"Circumference Area"April 17, 2012April 5, 2012
"Regular Polygon Area"For a regular triangle, we must first calculate the height from a 30-60-90 triangle:And since there are six regular triangles in a hexagon, we can get that formula by multiplying by 6:In general, for any regular polygon (the example was an octagon), the area is: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:
#3 Trapezoid DerivationSo another formula for the are of a trapezoid is Area = median * height (A = mh)Deriving area of a kite: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 trapezoid area:Practice test question:"Parallogram Area"April 2, 2012Confirming with Pick's Theorem:Applying Pick's Theorem to a triangle:Guesstimating the triangle area formula from its grid area estimate:April 2, 2012Area packet "rectangles and squares" page. These are the answers to #1 on:https://www.tamdistrict.org/cms/lib/CA01000875/Centricity/Domain/539/Geo%20Area/Area%20of%20Rectangle.pdf.pdfAnd these are for #2:So 1 rectangle has area 2bh/2 or bh. A = bh (base * height).Unit 9 Circles"Arc Angles"
The Inscribed Angle Formula provides an alternate methodof calculating polygonal interior angle measures:triangle => 360/3 = 120 => 1(120) / 2 = 60square => 360/4 = 90 => 2(90) / 2 = 90pentagon => 360/5 = 72=> 3(72) / 2 = 108hexagon => 360/6 = 60=> 4(60)/ 2 = 120septagon => 360/7 = 51 3/7=> 5(360/7)/ 2 = 128 4/7 = 128.57octagon => 360/8 = 45 => 6(45)/ 2 = 135nonagon => 360/9 = 40=> 7(40)/ 2 = 140decagon => 360/10 = 36=> 8(36)/ 2 = 144hendecagon => 360/11 = 32 8/11 => 9(360/11)/ 2 = 147 3/11 = 147.27dodecagon => 360/12 = 90 => 10(360/12)/ 2 = 150n-gon => 360 / n => (n-2) * (360 / n) / 2 => 360*(n - 2)/(2n) = 180(n - 2) / n
Tangent CirclesMarch 21, 2013More on Tangents and Inscribed AnglesCommon External and Internal TangentsThe 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.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.Angle-ArcsCircle Segments
"Inscribed Angles"front page
"Tangent Theorems""Diameter-Chord Theorems"front page"Chord Corollaries"back pageMarch 5th, 2012"Arcs and Chords"Monday, February 27th, 2012"Circle Vocabulary"
CIRCLE VOCABULARYUnit 9 Lesson 1
ARCS & ANGLESCircle SegmentsChord-ChordSecant-SecantSecant-Tangent
Special Right Triangles
The Isosceles 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
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
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.
Special right triangles have side length ratios which you can use to quickly find missing side lengths.
We are going to look at two different types of special right triangles:
- 30-60-90 Triangle
- 45-45-90 Triangle (Isosceles Right)
The 30-60-90 triangle has angles of 30°, 60° and 90° and is the same shape as half of an equilateral triangle.
The sides of the 30-60-90 triangle have the ratios: 1 : √3 : 2
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
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:
PYTHAGOREAN TRIPLESHow to generate them? One way is the Babylonian method.Another way is the Pythagoreans' method:This formula starts with an odd number.How many of these Pythagorean tripled triangles can you generate?Can 6, 7, and 8 form sides of a triangle?Hinge TheoremsUnit 7 SimilarityThe Golden RatioDonald in Mathmagic LandGolden ratio covered from 7 minute mark.To calculate the ratio, setup a proportion:Letting a = 1, one can calculate the solution:So a golden rectangle has an internal ratio of 1 to 1.618.Also, 1 / 0.618 = 1.618.Such rectangles can be nested within each other to create a golden spiral:You can approximate a golden rectangle by adding larger and larger squares:Here is how to fold one:
- 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
SIDE-SPLITTING THEOREM:A line parallel to a side of a triangle splits the other two sides proportionally.
PROPORTIONAL PARALLELS (TWO TRANSVERSALS) THM:Transversals split parallel lines proportionally.TRIANGLE ANGLE BISECTOR THEOREM:An angle bisector in a triangle splits the opposite side in proportion to the other two sides.\
Project Mathematics! Video "SimilarityPROPORTION PROPERTIES(see Downloads for Answer Key)
TRANSFORMATIONS UnitLesson 1 ReflectionsGeo6Notes2 TranslationsGeo6Notes3 RotationsGeo6Notes4 Glide ReflectionsGeo6Notes5 DilationsGeo6Notes6 Games (billiiards & mini golf)Geo6Notes7 SymmetryTRANSLATIONS LessonROTATIONS LessonGLIDE REFLECTIONS Lesson
Dilation #2Dilations #3Dilated Graph 3Dilation #4COORDINATE TRANSFORMATIONS
SYMMETRYWeird Al Yankovic palindrome video "Bob":Each verse is the same forward and backward (subtitled):Unit 4Unit 4 Notes 4#1-6 on Medians & Altitudes worksheetLesson 4-1 "Congruent Triangles"
Unit 3 Notes #5APOLYGON INTERIOR ANGLE SUMS AND REGULAR POLYGON MEASURESUnit 3 Notes #5BWhat about a polygon's exterior angle sum? Get your pencil out:
The exterior angle sum = 360.
Exterior Angle Measure = 360/n.POLYGON EXTERIOR ANGLE SUMS & REGULAR POLYGON EXTERIOR ANGLE MEASURES
The interior angle sum formula is 180(n-2).
Interior Angle Measure = 180(n-2)/n.
How to name a polygon:
Can we show that congruent corresponding angles are a direct consequence of Euclid's Postulate?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?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.Unit 3 Notes #4TRIANGLE SUM THEOREMDraw any triangle and cut off its three angles.Then arrange those three angles into a line by joining the vertices.What conclusion can be made about the angles of a triangle?Here's why this works:
Prove this conjecture for all cases--acute, obtuse, right, and equiangular.The Triangle Sum Theorem is true for all classifications of triangles.Here's an example of the triangle exterior angle theorem:See PROOFS link to complete the answers to the proofs in part II on the back.
Proving Parallel Lines Converse TheoremsGeometry Unit 3 Notes #2If lines (cut by a transversal) are parallel, then five conclusions can be made: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.If same-side interior angles or same-side exterior angles are supplementary, then lines are parallel.
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.Each person in the group is responsible for one color proof, however, ALL proofs must be copied.
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:TRANSVERSALSGeo3Notes1 Parallel Lines TheoremsGeo3Notes2 Proving Parallel LinesGeo3Notes3 Perpendicular and Parallel CorollariesGeo3Notes4 TRIANGLE SUM TheoremsGeo3Notes5 POLYGONS(Interior Angle Sum/Measure, Exterior Angle Sum/Measure)CORRESPONDING ANGLESALTERNATE INTERIOR ANGLES (AIA)ALTERNATE EXTERIOR ANGLES (AEA)SAME-SIDE INTERIOR ANGLESaka CONSECUTIVE INTERIOR ANGLES(make forwards and backwards C's)SAME-SIDE EXTERIOR ANGLES (SSE)Unit 3Transversals and AnglesWhen a transversal line (AB) cuts across two other lines (PQ & RS),eight angles are formed.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 theconsecutive interior angles being less than 180 degrees.Intersect on left side where 70 + 104 < 180. Intersects on right side wjere 110 + 64 < 180.CORRESPONDING ANGLES POSTULATE:If a transversal cuts across parallel lines ,then corresponding angles are congruent,alternate interior angles are congruent,and same-side exterior/interior angles are supplementary.
#2 on last page of Parallel Packet#3 on last page of Parallel Packet#4 on last page of Parallel PacketUnit 2Geo2Notes1 If-then StatementsGeo2Notes3 Properties of CongruenceGeo2Notes3A Two Column ProofsGeo2Notes4 Theorems of AnglesGeo2Notes5 Flowchart ProofsGeo2Notes6 Paragraph Proofs - PerpendicularParagraph ProofsFlowchart ProofsUnit 2 Notes #5Use 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.Now translate the two column proof for the Angle Bisector Theorem into a flowchart proof. Click on right pic for solution.You can use the Midpoint Theorem Flowchart to help answer the Angle Bisector Theorem Flowchart proof, and vice-versa.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.But with a flowchart, both directions can be proven with one proof.See if you can complete the Common Overlapping Angle Flowchart proof from using the Common Segment flowchart.Use the completed two-column proofs below to check answers in the flowchart above.
Now use the two-column proofs above to complete the flowcharts below:The completed flowcharts for all the Common Overlapping figures are below:Unit 2 Notes #3Properties of EqualityApply these properties to justify each step in the proof of an algebra solution.Unit 2 Notes #1September 11th, 2012
Unit 2 "Logic"
Lesson 5 "Paragraph Proofs"Prove that perpendicular lines make four congruent, right anglesin two-column and then paragraph form.Translate the two column proofs above into the paragraph proof below:Most books cite a "Linear Pair Theorem" or "Property",and there is a sister theorem for perpendicular lines.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 ProofsSupply the missing Justifications in the proofs of Right Angle, Linear Pair, and Vertical Angle TheoremsGeometry Unit 2Notes 2 -- ProofsThe 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.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.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.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.Note how the biconditional if-and-only-if statement in the Segment Congruence Definition is true in both directions.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)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:Lessons 1-2 "If Then Statements"P = hypothesis (or antecedent) = the if-part, Q = conclusion = the then-part.Euler diagrams Names of If-Then statements Sentences T or F?EULER DIAGRAMSCONDITIONAL CONVERSE BICONDITIONAL INVERSE CONTRAPOSITIVEIf 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 BasicsUnit 1 "Basics"Geo1Notes1 - Konigsberg BridgesGeo1Notes2 - Undefined TermsGeo1Notes3 - Measuring LengthGeo1Notes4 - Angle MeasuresGeo1Notes5 - Angle PairsUnit 1 Notes #5 Angle Pairs
Unit 1 Notes #4 "Angle Measures"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,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.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.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:We could add a zero angle to this list, and remove the reflex angle since it is greater than 180 degrees.We will only consider the first four--acute, right, obtuse, and straight--along with the zero angle.A full rotation is also called a complete angle.Unit 1 Notes #3"Measuring Length"Review of Lines, Rays, and Segments: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?Find any congruent segments on this number line.
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.An ant walks along the sidewalk to demonstrate the segment addition postulate:Applying an algebra example in geometry:Unit 1 Notes #2
"Undefined Terms"Points, Lines, PlanesUndefined vocabulary terms: point, line, planeDefined terms: ray, segment, space (depend on point, line, plane)DEFINITIONS: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 lineCoplanar -- points, lines, and figures in the same planeA true assumption in geometry is called a postulate.(Also, an axiom accepted as fact.)Postulates:"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?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.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 answersAll 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
NETWORKSAre the Bridges of Koenigsberg traversable? Draw a network to find out.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 RatiosMonday 2/13--Tangent RatioTuesday 2/14--Sine and Cosine RatiosThursday 2/16--Solving Right TrianglesFebruary 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: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: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, 2012Unit 7 "Similarity"January 23rd, 2012front pageback pageJanuary 19th, 2012front pageback pageUnit 6 "Triangle Inequalities" (period 4)Lesson 3January 9th, 2012Lesson 2January 5th, 2012Lesson 1January 4th, 2012"Indirect Reasoning Proofs"Geometry Class NotesUnit 5 "Quadrilaterals"Quadrilateral Quest programGeo5Notes1 KITESGeo5Notes2A PARALLELOGRAM PROPERTIESGeo5Notes2B PARALLELOGRAM PROPOSITIONSGeo5Notes3 MIDSEGMENT THEOREMGeo5Notes4A TRAPEZOID MEDIAN THEOREMGeo5Notes4B ISOSCELES TRAPEZOIDSGeo5Notes5 SPECIAL PARALLELOGRAMS(rhombus, rectangles, square)Geo5Notes6 Quadrilateral HierarchyUnit 5 Notes 1
PARALLELOGRAMSIf a quadrilateral is a parallelogram, it has the following properties:
PARALLELOGRAM PROPERTIES:PARL paragraph1 proof , PARL paragraph2 proof , Quad Quest Quiz per7The converses are also true. These are sufficient conditions to be a parallelogram.Also, if one pair of sides is both parallel and congruent, then the quadrilateral is a parallelogram.
QUADRILATERAL MIDSEGMENTSThe quadrilaterals formed by joining consecutive midpoints of other quadrilaterals are:PARALLELOGRAMS
TrapezoidsTrapezoid Median Theorem TransparencyClass Puzzle ProofFind the length of the median by adding each midsegment length:
Unit 4 "Congruent Triangles"
Unit 4 "Triangles"Geo4Notes1 Congruent Triangles DefinitionGeo4Notes2 Congruency ConditionsGeo4Notes3 Triangle Congruence TheoremsGeo4Notes4 Special SegmentsGeo4Notes5 Isosceles TrianglesGeo4Notes6 Midsegment TheoremLesson 5 "Midsegment Theorem Proofs"Unit 4 Notes 6Perpendicular Bisector/Bisected Angle Sides Theorems(see PROOFS link for completed proofs)PERPENDICULAR BISECTOR BICONDITIONAL THEOREMA point is on the perpendicular bisector of a segmentif and only ifit is equidistant to both endpoints of that segment.BISECTED ANGLE SIDES BICONDITIONAL THEOREMA point is on the angle bisectorif and only ifit is equidistant to both sides of the bisected angle.
Lesson 4 "Isosceles Triangles"Unit 4 Notes 4Equilateral Corollary , Equiangular Corollary , Isosceles Vertex Angle Flowchart Proofs
Isosceles Example Find x , Find length , Found lengths , Isosceles Example Find angles , Isosceles ProofCPCTCCorresponding Parts of Congruent Triangles are CongruentIf 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.This applies to medians, altitudes, perpendicular bisectors, and angle bisectors as well.
RIGHT TRIANGLE CONGRUENCEThere 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.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.Now write congruence statements and justify why each pair of triangles is congruent.Answer Key:1. RWI by SAS 2. ESH by AAS 3. GNT by SSS4. ARE by ASA 5. DST by AAS 6. HEY by SAS7. ILH by SSS 8. AGN by AAS 9. HAT by SAS10. KPG by AAS 11. YDE by ASA 12. SAK by AASA REASON SNAKES SHED IS THAT THEY KEEP GROWING UNTIL THEY DIE
CONGRUENCY CONDITIONSUnit 4 Lesson 2Click 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?Note that they do not list all six pairs of parts (congruent angle pairs and congruent sides)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.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.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. ASAGoing around clockwise gives us side-angle-angle.Going counterclockwise yields ANGLE-ANGLE-SIDE, aka AAS.Note that these two triangles are different:So there is no Angle Side Side condition.To wrap up:
Go around each triangle and determine the information given:
Determine whether the triangles in each pair are forced to be congruent:Lesson 1 "Congruent Polygons"Unit 4 Lesson 1Definition of Congruent TrianglesMatch 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.Check diagrams for like markings.Now draw two triangles to match this:So it's important to match up the vertices in the same order:And also that congruent triangles can be flipped or rotated and still be the same shape and size.
SSS, SAS, ASA PostulatesClick this link for photos of all cases:(SideSideSide, SideAngleSide, AngleSideAngle, AngleAngleSide, SideSideAngle, AngleAngleAngle)Unit 3 Review1.e) SSE angles: 2 & 3 and 6 & 7