Download Geometric Relationships

Geometry Outline
I.Geometric Relationships ≈ 4 days
1. Lines and Planes
a. Undefined terms: point, line, plane
b. Notation for point, line, plane, parallel, perpendicular
c. Parallel Planes
d. Identify parts of postulate: hypothesis, conclusion
e. Collinear, coplanar
f. Vertical angles, complementary, supplementary
g. Application of line perpendicular to plane
h. Review & Assessment
II. Informal and Formal Proofs ≈ 78 + days
1. Logic – 12 days
a. Logic Concepts
(1) translate written expressions to symbolic representation (p, q, V, →, etc)
(2) negation
(3) conjunction
(4) disjunction
(5) conditional, hypothesis, conclusion, hidden conditional
(6) biconditional
(7) truth values – tables & words
(8) converse (change order)
(9) inverse (insert negation)
(10)contrapositive (change order and negate)
(11)logically equivalent
b. Logic Proofs
(1) contrapositive
(2) detachment
(3) syllogism/chain rule/modus tollens
(4) DeMorgan’s
(5) disjunctive inference
c. Review & Assessment
2. Triangles- Congruent ≈ 25 days
a. Review & expand (see resources)
b. Corresponding parts (sides, angles, appropriate naming)
c. Identify included side, included angle
d. Congruent () – define & recognize
e. Establish the format of a two-column proof
f. SSS
g. SAS
h. ASA
i. AAS
j. HL – right triangle
k. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
l. Review & Assessment
3. Triangles – Similar ≈ 15 days
a. Similar triangles – dilation
b.
c.
d.
e.
f.
g.
h.
AA Similar Proofs
SSS Similar Proofs
SAS Similar Proofs
Pythagorean theorem & converse
Mean proportional in Right Triangles (Geometric Mean)
Corresponding Sides of Similar Triangle are in Proportion (CPSTP)
Review & Assessment
4. Triangle Properties ≈ 7 days
a. Angle sum in a triangle (measuring angles with protractors & software)
(1) Review classification using acute, obtuse, right, scalene, isosceles &
equilateral
b. Isosceles Triangle and Properties
(1) Investigate & apply
(2) Proof & indirect proof
c. Exterior Angle Theorem
(1) Investigate & apply
(2) Proof & indirect proof
d. Triangle Inequality Theorem
(1) Investigate & apply
(2) proof & indirect proof
e. Review & Assessment
5. Polygons (  3 days)
a. Exterior angle of a triangle = sum of remote interior angles of a triangle
b. Sum of the measures of the interior angles of a polygon
c. Sum of the measures of the exterior angles of a polygon
d. Measure of an interior angle of a regular polygon
e. Measure of an exterior angle of a regular polygon
f. Classify polygons by the number of sides and/or angles
6. Quadrilaterals (  16 days)
a. Properties of special quadrilaterals
(1) Parallelogram
(2) Rectangle
(3) Rhombus
(4) Square
(5) Trapezoid
(6) Isosceles Trapezoid
(7) Solving linear equations using props of quads
(8) Solving quadratic equations using props of quads
b. Given a specific quad (parallelogram, rectangle, square, rhombus, trapezoid),
prove a specified congruence (angle  angle, segment  segment)
c. Given specific properties, prove a quad is a parallelogram, rectangle,
rhombus, square, trapezoid, isosceles trapezoid)
d. Review and Assessment
III.
Coordinate Geometry ≈ 15 Days
1. Linear: Parallel and Perpendicular ( ≈3 days)
a. Parallel lines have equal slopes
b. Perpendicular lines have negative reciprocal slopes
c. Vertical and Horizontal lines are perpendicular
d. Point-Slope form of a line
2.
3.
4.
5.
e. Slope-Intercept form of a line
Linear: Distance and Midpoint ( ≈3 days)
a. Length (distance) of a line segment
b. Midpoint of a line segment
c. Find endpoint given one endpoint and the midpoint
d. Perpendicular bisector of a line segment
Linear: Applications/Informal Proofs ( ≈3 Days)
a. Justify algebraically the properties of quadrilaterals and triangles
b. Using the distance formula to find lengths of sides and diagonals to classify
triangles and quadrilaterals
c. Using the midpoint formula:(e.g. show that the diagonals bisect each other)
d. Using the slope formula to find parallel/perpendicular sides/diagonals
Circles (≈2 days)
a. The center of a circle is an ordered pair
b. The relationship between the distance formula and the equation of a circle
c. Equation of a circle
d. Graphing a circle
Quadratic-Linear Systems ( ≈1 day)
a. Linear Equation
b. Quadratic Equation(including the Circle)
c. Systems of Equations solved Graphically
d. Checking Solutions
e. Review and Assessment ( ≈3 days)
IV. Transformational Geometry (≈12days)
1. Reflections (≈2days)
a. Line reflections (x-axis, y-axis, x=a, y=b, y  x , y   x )
b. Point reflections
c. Symmetry
d. Orientation, preservation of distance, preservation of angle measure,
parallelism, perpendicularity, collinearity, midpoint
e. Opposite (indirect) isometry
2. Rotations (≈2days)
a. About the origin of 90  and 180  , clockwise and counterclockwise
b. Orientation, preservation of distance, preservation of angle measure,
parallelism, perpendicularity, collinearity, midpoint
c. Point reflection, half-turn
d. Direct isometry
3. Translations (≈2days)
a. Orientation, preservation of distance, preservation of angle measure,
parallelism, perpendicularity, collinearity, midpoint
b. Direct isometry
4. Dilations (≈2days)
a. Similarity (Corresponding sides are proportional, corresponding angles are
congruent)
b. Factor of dilation(positive/negative, enlargement/reduction)
5. Composition of Transformations (proper notation required) (≈2days)
a. Glide Reflection
(1) Orientation, preservation of distance, preservation of angle measure,
parallelism, perpendicularity, collinearity, midpoint
b. 2 or more of a reflection, rotation, translation, or dilation
6. Review and Assessment (≈2days)
V. Circles (  20 Days)
1. Relationships of angles and arcs of a circle
a. Central
b. Inscribed
c. Interior angle formed by 2 intersecting chords
d. Exterior angle formed by secants and/or tangents
e. Angle formed by tangent and chord on the circle
f. Angle formed by tangent and diameter/radius
2. Relationship of segments that intersect circles
a. Intersection of two chords in the circle
b. Find the measures of secants and tangents
(1) (including problems involving quadratic equations)
c. Two tangents to the same circle from the same
(1) exterior point
3. Relationship of Arcs and Chords
a. Arc measure
b. Congruent chords and arcs
c. Chords equidistant from the center of the circle
d. Perpendicular bisector of a chord
4. Common tangents of two non-intersecting or tangent circles
5. Inscribed and Circumscribed polygons
6. Review and Assessment( ≈3 days)
VI. Geometric Relationships(~7 days)
1. Solids
a. Use nets to describe prisms, pyramids and cones
b. Analysis of 2 and 3 dimensional units of measure
c. Properties, area, volume, surface area, lateral area of:
(1) Cone
(2) Cylinder
(3) Prisms
(4) Pyramids
(5) Sphere
d. Analyze and solve verbal problems
e. Review & Assessment
VII. Constructions and Locus (≈12days)
1. Constructions (≈3days)
a. Angle bisector
(1) definition of angle bisector
(2) congruence
(a) Radii
(b) Angle
(c) Triangle (CPCTC)
(3) measurement
(a) Tools of construction and their use
(b) Protractor (validating conjectures)
b. Perpendicular bisector
(1) definition of perpendicular lines
(2) bisect a segment
(3) right angles
(4) properties of isosceles triangles (median, altitude, angle bisector to base)
(5) measurement
(a) tools of construction and their use
(b) protractor (validating conjectures)
c. Parallel or perpendicular lines thru a given point
(1) 2 points determine a unique line
(2) Method of proving lines parallel
(a) Copy an angle (construct congruent angles)
(b) If alternate interior angles are congruent, then the lines are parallel.
(3) Construct lines perpendicular using:
(a) Point on the line
(b) Point NOT on the line
d. Equilateral triangles
(1) Interior angle sum of a triangle
(2) Types of triangles
(3) Radii of the same circle are congruent
(4) Equilateral vs. equiangular
2. Locus (≈7days)
a. Centers related to a triangle
(1) Complete appropriate constructions to locate the:
(a) Incenter (angle bisectors)
(b) Centroid (medians)
(c) Orthocenter (altitudes)
(d) Circumcenter (perpendicular bisectors)
(2) Identify Euler’s line (optional)
b. Compound loci
(1) 5 Fundamental Loci
(a) Given distance from a point
(b) Equidistant from 2 points
(c) Given distance from a line
(d) Equidistant from 2 parallel lines
(e) Equidistant from 2 intersecting lines
(2) Solve problems involving 2 or more fundamental loci, including centers of
triangles (reference G.G.21).
3. Review and Assessment (≈2days)