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PLANE GEOMETRY

THEOREMS AND POSTULATES

GRAPHING CALCULATOR AND ONLINE APPLICATIONS BRIEF GUIDE

CHAPTER 1

BASIC DEFINITIONS, ANGLE RELATIONSHIPS AND ANGLE PAIRS.

1.- Basic Geometric Definitions: Lines, Angles, Planes, Segment Addition Postulate, Midpoint, Distance Formula.

2.- Complementary, Supplementary, Linear Pair, and Vertical Angles: Definitions and Algebraic Proofs with Geometry.

3.- Angle Pairs: Complementary, and Supplementary Angles as Word Problems.

4.- Angle Pairs with Geolegs (manipulatives hands on approach): Complementary, Supplementary and Vertical Angles.

5.- Angle Pairs in Parallel Lines: Corresponding, Alternate Interior and Alternate Exterior, and Consecutive Interior Angles.

CHAPTER 2

ANGLE RELATIONSHIPS IN PROOFS, INDUCTIVE AND DEDUCTIVE REASONING. SLOPE.

1a.- Inductive and Deductive Reasoning: If - Then Statements, Law of Syllogism and Law of Detachment.

1b.- Introduccion al Razonamiento Inductivo y Deductivo: Enunciados Condicionales Si...Entonces sus enunciados conversos, reciprocos, contra positivos e inversos (si usamos proposicion todos son femeninos: Conversa, inversa, etc)

2.- Two Column Proofs Involving Segment and Angle Relationships: Two Column Proofs, Flow Proofs and Paragraph Proofs with Segment Addition Postulate and Angle Pairs.

3.- Slope Introduction: Slope of a Linear Equation. Horizontal, Vertical, Falling to the Right or to the Left. Parallel vs Skew Lines. Parallel and Perpendicular Planes.

CHAPTER 3: TRIANGLES

PROVING TRIANGLES CONGRUENT AND TRIANGLE INEQUALITY.

1.- Proving Triangles Congruent: Algebraic - Geometric Proofs to Prove Triangles Congruent.

2.- Proving Triangles Congruent: SSS, ASA, SAS, AND AAS.

3a.- Two Column Proofs Involving Triangle Congruence: Two Column Proofs and Flow Proofs using Angle Relationships and SSS, ASA, SAS, and AAS.

3b.- Demostraciones Con Tablas de Dos Columnas con Enuanciados y Razones, Demostraciones de Flujo y Demostraciones de Parrafo involucrando Triangulos y Quadrilateros: Usando los teoremas y postulados de LLL, LAL, ALA y AAL.

4.- One and Two Triangle Inequality Theorems: Ordering sides and Angles Using Triangle Inequality.

5.- Drilling in Geometric Statements: Congruent Geometric Statements and How to Write them and Read them Properly.

6.- Introduction to proofs by contradiction in paragraph format and two-column format

CHAPTER 4

QUADRILATERALS AND THEIR PROPERTIES IN GEOMETRIC AND ALGEBRAIC PROOFS.

1.- Properties of Quadrilaterals: Squares, Rhombi, Rectangles, Parallelograms and Trapezoids.

2.- Proving Properties of Special Quadrilaterals using Congruent, or Isometric, or Rigid Transformations.

3.- Segments and Angles in Parallelograms: Problems Involving Angles and Segments in Regular Parallelograms and Rectangles.

4.- Segments and Angles in Squares and Rhombi: Problems that Include Angles and Segments in Squares and Rhombi.

5.- Segments and Angles in Regular and Isosceles Trapezoids: Problems Deal with Segments and Angles in Trapezoids.

6.- Two Column Proofs with Quadrilaterals and Triangles: Solving Two Column Proofs and Flow Proofs Using Properties of Parallelograms and Quadrilaterals and Properties of Triangles to Prove Congruence Statements and Segment or Angle Relationships.

INSIDE EACH LESSON YOU MAY DO AS THIS GRAPHIC SHOWS

CHAPTER 5

PROPORTIONAL REASONING SIMILARITY AND TRIGONOMETRY.

1.- Similarity Theorems AA, SAS and AAA. Similarity Applied to Two Triangles and Lines Cut by Transversals.

2.- Similarity Theorems Involving Special Segments In Triangles: Atltitudes, Medians, Angle Bisectors and Perpendicular Bisectors Similarity Relationships in Triangles.

3.- Similarity Applied in a Right Triangle: Drawing a height to the Right Angle of A Right Triangle.

4.- Proofs in Two Column Format involving similarity in triangles. Two Column Proofs and Flow Proofs for Similarity.

5.- 30°-60°-90° and 45°-45°-90° Triangles: This lesson deals with special right triangles from the geometric informal proof to solution of some examples and very soon we are going to add suggested application problems. It has a set of videos to review concepts used in the proofs like special lines and segments in triangles, Pythagorean Theorem, and since browsers dropped the support for applets a few videos of applets than by themselves constitute a very robust dynamic geometric proof.

6.- Similarity in Special Right Triangles using Dilations Enrichment Lesson. See dilation lessons at bottom of this webpage before visiting this.

7.- Right Triangle Trigonometry: Right Triangle Ratios of Sine, Cosine and Tangent Applied to Solutions of Problems Involving Segments and Angles.

8.- Laws of Sines and Cosines: Trigonometry Applied to Acute and Obtuse Triangles Using the Laws of Sine and Cosine.

9.- Learning about angle measurement units: Radians and degrees in coterminal sides using special right triangles 30°- 60°-90° and 45°-45°-90°

CHAPTER 6

ANGLES, ARCS AND SEGMENTS IN CIRCLES.

1.- Basic Definitions For Circles: Arcs, Chords, Central Angles, Inscribed Angles and Arc Addition Postulate.

2.- Theorems Involving Congruent Chords and Intersecting Chords in Circles. Applied to Individual Circles or Several Circles.

3.- Inscribed and Central Angles: Problems that Focus in Inscribed and Central Angles Usings Triangles and Parallelograms.

4.- Arcs and Angles Formed by Secants, and Tangents in a Circle: Segments that Give Place Secant and Tangents in Circles and the Angles and Arcs that They Form in the Circle Intersecting Inside, On or Outside the Circle.

5.- Segment Relationships Created by Chords, Secants and Tangents. Chords, Secants and Tangents and the Segments that They Create When Intersecting Inside, On or Outside the Circle.

6 a1.- Geometric Two Column Proofs Involving Circles with Chords, Tangents and Secants involving angle and arc relationships with segments in circles.

6 a2.- Demostraciones Geométricas con rayos, cuerdas, tangentes, y secantes en relaciones angulares de ángulo y arco en círculos.

6 b1.- Geometric Two Column Proofs Involving Circles with Chords, Tangents and Secants involving segment relationships in circles.

6 b2.- Demostraciones involucrando tangentes, secantes, cuerdas y relaciones entre segmentos en círculos

6 c1.- Proofs with circles involving chords, secants and tangents. Part 1 performed using Ti-nspire CX graphing calculator from Texas Instrument

6 c2.- Proofs with circles involving chords, secants and tangents. Part 1 performed using Ti-nspire CX graphing calculator from Texas Instrument

6 c3.- Proofs with circles involving chords, secants and tangents. Part 3 performed using Ti-nspire CX graphing calculator from Texas Instrument

CHAPTER 7

AREAS IN POLYGONS (REGULAR AND IRREGULAR)

1.- Angle Measure in Polygons: Interior and Angles in Polygons.

2.- Calculating Areas in Parallelograms and Rectangles.

3.- Calculating Areas in Rhombi, Trapezoids and Triangles.