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# Geometry

Here's a mind map of my geometry course - all the content in a 5-10 minute glance. You can put your arms around it.

by

Tweet## Mark Vasicek

on 8 June 2012#### Transcript of Geometry

GEOMETRY Points Lines and Planes Triangles Polygons

Quadrilaterals Perimeter, Area,

Surface Area and

Volume Circles LENGTH MIDPOINT SLOPE EQUATIONS SYSTEMS Distance Formula Pythagorean Theorem http://ed.voicethread.com/share/619970/ Endpoint http://ed.voicethread.com/share/625153/ Any Point Vectors and Scalars Find a point 2/3 of the way from

A (0,2) to B (23, 5). Positive rising to the right. Negative falling to the right Zero slope flat Undefined a vertical line Perpendicular Lines Perpendicular Lines have

slopes that are

opposite reciprocal

(multiplied they = -1)

OR

the slopes are

zero (horizontal)

AND undefined (vertical).

. . Parallel Lines have the same slope 3-D: Skew two or more lines that

aren't parallel and

don't intersect. Slope-Intercept Form

(also called "function form") y = mx + b Point-Slope Form y - y1 = m (x - x1) Other Forms: Standard Form ax + by = c Intercepts Form px + qy = 1 Parametrics Substitution Elimination Graphing Two Lines Cross at a point May never cross

(parallel) May cross infinite times

(coincidental) solve one equation for one of the variables

then plug it into the other equation multiply either or both equations so that

when you add the equations, one of the variables goes away.

Solve for the remaining variable,

then plug it in to solve for the other equation. Find two points on each line. Graph them.

Find out where the lines intersect (if they intersect) Segment Addition Easy How can we make this harder? Angle Addition Segment Bisector Angle Bisector Angle Relationships Congruence Classify Angles Linear Pair Vertical Pair Complementary, Supplementary, etc. Parallel Lines and Transversals Proof If this... Then that... statements Converse If that... then this... Counterexample Steps:

1. Given / Prove Statement

2. Draw the Figure

3. Mark the figure

a. First mark the given information

b. Second mark what is implied

4. Statement and Reason Table Rules:

1. Given

2. Definitions

3. Properties

4. Postulates(Axioms)

5. Theorems

6. Corollaries Properties

1. Associative

2. Commutative

3. Identity

4. Zero

5. Properties of Equality

6. Distributive

7. Reflexive

8. Substitution Things to watch out for:

Vertical Pairs

Linear Pairs

Parallell Lines

Radii of a circle Transformations Translation Vectors Parametrics Reflection Rotation Dilation Similarity Tesselations M.C. Escher Nomenclature: Say Words Write Symbols Draw Figures Model

Real World Point

Line

Plane

Segment

Ray

Angle

Plus or Minus

Congruent

Parallel

Perpendicular

Collinear

Coplanar Classify by Sides By Angles Perpendicular Bisector Theorem The perpendicular Bisector is the locus of points

equidistant from two points. Triangle inequality Theorem The longest side is opposite the largest angle. The smallest side is opposite the smallest angle. Special Right Triangles: 30-60-90 Pythagorean Theorem Pythagorean Triples Geometric Mean Thales Theorem Trigonometric Ratios Sine Cosine Tangent Special Right Triangles: 45-45-90 Proving Triangles Congruent Side-Side-Side (SSS)

Side-Angle-Side (SAS)

Angle-Side-Angle (ASA)

Angle-Angle-Side (AAS)

Hypotenuse-Leg (HL) CPCTC Corresponding parts of congruent triangles are congruent. INVALID: AAA (auto insurance) SSA/ASS ("Donkey Theorem") Some things you should know about a triangle: Triangle Angle Sum Theorem The sum of the measures of the angles is 180 degrees. Triangle Exterior Angle Sum Theorem The measure of an exterior angle is the sum of

its remote interior angles. Triangle Inequality Theorem The longest side is opposite the largest angle;

the shortest side is opposite the smallest angle. Is It a Triangle? The longest side is shorter than

the sum of the measures of the other two sides. Hinge Theorem Is it Right, Obtuse, or Acute? If two sides of one triangle are congruent to two sides of another, and

the included angle of the first is larger than that of the second,

then the third side is larger on the first triangle than on the second triangle. Points of Concurrency Perpendicular Bisectors Circumcenter Circumcircle Isosceles Medians Centroid 2/3 - 1/3 ratio from vertex Midsegment Theorem Sides of the medial triangle are

parallel and 1/2 the length of the opposite side. Angle Bisectors Incenter Incircle Angle Bisector Problems Altitudes Orthocenter Excircles Euler Segment 9-Point Circle Similarity Proportion / Ratio Proving Triangles Similar Angle-Angle Similarity (AA-Sim) Side-Side-Side Similarity (SSS-Sim) Side-Angle-Side Similarity (SAS-Sim) Side Splitter Theorem Angle Bisector (splitter) Theorem Five Likely Configurations Polygons Definition Polygon Exterior Angle Sum

(walk-around theorem) Polygong Interior Angle Sum Quadrilaterals Classify Properties Sides Angles Diagonals Perimeter, Area Triangles Quadrilaterals Regular Polygons Apothem Circles Area Circumference Surface Area,

Lateral Surface Area,

and Volume Prisms Cylinders Pyramids Cones Cavalieri's Principle

(B*h) Spheres Affect:

Change in perimeter affects area... how? volume? Nets Polyhedra Platonic Solids and patterns Archimedian Solids and patterns Schlegel Diagrams and the 4th dimension Flatland Euler's Formula: E = V + F - 2 Circles Arc Measure Arc Length Sector Area Pie Charts Arc Segments Concentric Circles and Probability

(dartboards) Angles Where is the vertex? Center On Circle Inside Circle Outside Circle Segments Where is the vertex? Chords Tangents Inside Outside Funky Areas How can we make this harder? Easy Similarity Centroid Theorem pons asinorum Isosceles trian gle theorem:

base angles of an isosceles triangle are congruent The Delian Problem Patterns: # Edges # Faces # Cubes Dip the cube in paint. How many cubes have... 3 Faces Painted? 2 Faces Painted? 1 Face Painted? 0 Faces Painted? Size 1 x 1 x 1 12 6 8 1 8 0 0 0

2 x 2 x 2 12 6 8 8 8 0 0 0

3 x 3 x 3

4 x 4 x 4

5 x 5 x 5

6 x 6 x 6

n x n x n # Vertices Flagpole Project Apollonius' circle "Star Trek: Theorem In a circle, the measure of an inscribed angle

is half the measure of its intercepted arc.

Full transcriptQuadrilaterals Perimeter, Area,

Surface Area and

Volume Circles LENGTH MIDPOINT SLOPE EQUATIONS SYSTEMS Distance Formula Pythagorean Theorem http://ed.voicethread.com/share/619970/ Endpoint http://ed.voicethread.com/share/625153/ Any Point Vectors and Scalars Find a point 2/3 of the way from

A (0,2) to B (23, 5). Positive rising to the right. Negative falling to the right Zero slope flat Undefined a vertical line Perpendicular Lines Perpendicular Lines have

slopes that are

opposite reciprocal

(multiplied they = -1)

OR

the slopes are

zero (horizontal)

AND undefined (vertical).

. . Parallel Lines have the same slope 3-D: Skew two or more lines that

aren't parallel and

don't intersect. Slope-Intercept Form

(also called "function form") y = mx + b Point-Slope Form y - y1 = m (x - x1) Other Forms: Standard Form ax + by = c Intercepts Form px + qy = 1 Parametrics Substitution Elimination Graphing Two Lines Cross at a point May never cross

(parallel) May cross infinite times

(coincidental) solve one equation for one of the variables

then plug it into the other equation multiply either or both equations so that

when you add the equations, one of the variables goes away.

Solve for the remaining variable,

then plug it in to solve for the other equation. Find two points on each line. Graph them.

Find out where the lines intersect (if they intersect) Segment Addition Easy How can we make this harder? Angle Addition Segment Bisector Angle Bisector Angle Relationships Congruence Classify Angles Linear Pair Vertical Pair Complementary, Supplementary, etc. Parallel Lines and Transversals Proof If this... Then that... statements Converse If that... then this... Counterexample Steps:

1. Given / Prove Statement

2. Draw the Figure

3. Mark the figure

a. First mark the given information

b. Second mark what is implied

4. Statement and Reason Table Rules:

1. Given

2. Definitions

3. Properties

4. Postulates(Axioms)

5. Theorems

6. Corollaries Properties

1. Associative

2. Commutative

3. Identity

4. Zero

5. Properties of Equality

6. Distributive

7. Reflexive

8. Substitution Things to watch out for:

Vertical Pairs

Linear Pairs

Parallell Lines

Radii of a circle Transformations Translation Vectors Parametrics Reflection Rotation Dilation Similarity Tesselations M.C. Escher Nomenclature: Say Words Write Symbols Draw Figures Model

Real World Point

Line

Plane

Segment

Ray

Angle

Plus or Minus

Congruent

Parallel

Perpendicular

Collinear

Coplanar Classify by Sides By Angles Perpendicular Bisector Theorem The perpendicular Bisector is the locus of points

equidistant from two points. Triangle inequality Theorem The longest side is opposite the largest angle. The smallest side is opposite the smallest angle. Special Right Triangles: 30-60-90 Pythagorean Theorem Pythagorean Triples Geometric Mean Thales Theorem Trigonometric Ratios Sine Cosine Tangent Special Right Triangles: 45-45-90 Proving Triangles Congruent Side-Side-Side (SSS)

Side-Angle-Side (SAS)

Angle-Side-Angle (ASA)

Angle-Angle-Side (AAS)

Hypotenuse-Leg (HL) CPCTC Corresponding parts of congruent triangles are congruent. INVALID: AAA (auto insurance) SSA/ASS ("Donkey Theorem") Some things you should know about a triangle: Triangle Angle Sum Theorem The sum of the measures of the angles is 180 degrees. Triangle Exterior Angle Sum Theorem The measure of an exterior angle is the sum of

its remote interior angles. Triangle Inequality Theorem The longest side is opposite the largest angle;

the shortest side is opposite the smallest angle. Is It a Triangle? The longest side is shorter than

the sum of the measures of the other two sides. Hinge Theorem Is it Right, Obtuse, or Acute? If two sides of one triangle are congruent to two sides of another, and

the included angle of the first is larger than that of the second,

then the third side is larger on the first triangle than on the second triangle. Points of Concurrency Perpendicular Bisectors Circumcenter Circumcircle Isosceles Medians Centroid 2/3 - 1/3 ratio from vertex Midsegment Theorem Sides of the medial triangle are

parallel and 1/2 the length of the opposite side. Angle Bisectors Incenter Incircle Angle Bisector Problems Altitudes Orthocenter Excircles Euler Segment 9-Point Circle Similarity Proportion / Ratio Proving Triangles Similar Angle-Angle Similarity (AA-Sim) Side-Side-Side Similarity (SSS-Sim) Side-Angle-Side Similarity (SAS-Sim) Side Splitter Theorem Angle Bisector (splitter) Theorem Five Likely Configurations Polygons Definition Polygon Exterior Angle Sum

(walk-around theorem) Polygong Interior Angle Sum Quadrilaterals Classify Properties Sides Angles Diagonals Perimeter, Area Triangles Quadrilaterals Regular Polygons Apothem Circles Area Circumference Surface Area,

Lateral Surface Area,

and Volume Prisms Cylinders Pyramids Cones Cavalieri's Principle

(B*h) Spheres Affect:

Change in perimeter affects area... how? volume? Nets Polyhedra Platonic Solids and patterns Archimedian Solids and patterns Schlegel Diagrams and the 4th dimension Flatland Euler's Formula: E = V + F - 2 Circles Arc Measure Arc Length Sector Area Pie Charts Arc Segments Concentric Circles and Probability

(dartboards) Angles Where is the vertex? Center On Circle Inside Circle Outside Circle Segments Where is the vertex? Chords Tangents Inside Outside Funky Areas How can we make this harder? Easy Similarity Centroid Theorem pons asinorum Isosceles trian gle theorem:

base angles of an isosceles triangle are congruent The Delian Problem Patterns: # Edges # Faces # Cubes Dip the cube in paint. How many cubes have... 3 Faces Painted? 2 Faces Painted? 1 Face Painted? 0 Faces Painted? Size 1 x 1 x 1 12 6 8 1 8 0 0 0

2 x 2 x 2 12 6 8 8 8 0 0 0

3 x 3 x 3

4 x 4 x 4

5 x 5 x 5

6 x 6 x 6

n x n x n # Vertices Flagpole Project Apollonius' circle "Star Trek: Theorem In a circle, the measure of an inscribed angle

is half the measure of its intercepted arc.