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MEA

MEA . Geometry 2011. COURSE OUTLINE. FOUNDATIONS FOR GEOMETY EUCLID & Van Hiele Definitions, Postulates, Proofs DEFINITIONS REVISITED & POLYGONS GEOMETRIC REASONING CONSTRUCTION TOOLS PATTY PAPER DYNAMIC SOFTWARE COMPASS & STRAIGHT EDGE TRIANGLES POLYGONS CIRCLES Model Building

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  1. MEA Geometry 2011

  2. COURSE OUTLINE • FOUNDATIONS FOR GEOMETY • EUCLID & Van Hiele • Definitions, Postulates, Proofs • DEFINITIONS REVISITED & POLYGONS • GEOMETRIC REASONING • CONSTRUCTION TOOLS • PATTY PAPER • DYNAMIC SOFTWARE • COMPASS & STRAIGHT EDGE • TRIANGLES • POLYGONS • CIRCLES • Model Building • Non-Euclidean Geometry (While there are many other topics that should be covered, there simply isn’t time.)

  3. 1. Euclidean Geometry • Euclid (~300 BCE) – Collected all the known geometric ideas • Building Bocks • Undefined terms • Point • Line • Plane • Postulates (5) • Definitions • Conjectures/Theorems (The rest of the story).

  4. Van HIELE MODEL

  5. Proof/Sturcture • Undefined terms • Point • Line • Plane • Defined terms • Line segment • Ray • Collinear/Coplanar • Midpoint & Congruent

  6. Postulates • Line Postulate – You can construct exactly one line through any two points – two points determine a line. Line intersection Postulate – If two lines intersect, then they intersect in exactly one point. • Any straight line segment can be extended indefinitely to form a straight line. • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. • All right angles are congruent. • If two lines are drawn which intersect a third line in such as way that the sum of inner angles on one side is less than two right angle, then the two lines intersect each other on that side if extended far enough. The Converse is also assumed. Although not stated directly. If the inner angles are 180˚ or more then the lines do not meet on that side. Parallel lines never meet, hence the angles on either side sum to 180˚.

  7. 2. Definitions revisited

  8. Polygons HOMEWORK: PAGE 90-94/ ALL

  9. 3. Geometric Reasoning • Inductive Reasoning • Observe Data • Recognize Patterns • Make a Generalization • Examples • Page 99 • Page 101 • Finite Differences • Page 102 • Number of Diagonals from one vertex • Number of Segments • Number of Diagonals.

  10. Deductive Reasoning • Algebra • 3(2x + 1) + 2(2x + 1) + 7 = 42 - 5x • Geometry • Vertical Angles Conjecture – Vertical angles are congruent. • Isosceles Triangle Conjecture • Sum of the angles in a triangle = 180˚ (Parallel Line Postulate HOMEWORK: PAGE 140-142/ALL

  11. 4. Construction Tools • Duplicating • Perpendiculars • Bisecting – Angles, perpendicular bisector • Parallels – • Points of Concurrency • Angle bisector • Altitudes • Medians • Perpendicular bisectors • Euler’s line • Western – Compass & Straight Edge • Eastern – Paper Folding • Modern - Technology HOMEWORK: PAGE 166/17, 193/1-18, 37-62

  12. 5. TRIANGLES • PROPERTIES • Triangle Sum/Polygon Sum • Isosceles Triangle Conjecture • CONGRUENCE CONJECTURES • SSS, SAS, ASA, AAS, SSA, AAA • CPCTC HOMEWORK: PAGE 203/8-9, 209/10, 247/12, 251/1-31

  13. 6. POLYGONS • PROPERTIES • Polygon Sum/Exterior Angle Sum • CONJECTURES • Kite • Trapezoid • mid-segment • parallelograms HOMEWORK: PAGE 260/12, 304-305/1-16

  14. 7. CIRCLES • Properties of Circles • Tangents • Chords • Arcs and Angles • Central • Inscribed • Interior & Exterior • Etc.

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