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Geometry (textbook: Jurgensen/Brown)

Fall Semester

1. Deductive reasoning (Chapter 2)

a. Logic (connectives: negation, conjunction, disjunction, conditional, biconditional)

b. Truth tables, tautologies

c. Converse, inverse, contrapositive

d. Introduction to proofs

i. Direct proof (examples: odd + odd = even, odd [pic] odd = odd, odd + even = odd)

ii. Indirect proof (example: Prove [pic] is irrational.)

e. Logic proofs (optional)

2. Points, lines, planes, and angles (Chapter 1) [finding measures of angles by solving algebraic equations]

a. Vocabulary: definition, postulate, axiom, theorem, lemma, corollary

b. Definitions and notation: points, segments, rays, lines, planes, angles, polygons, circles, lengths, angle measures, congruence vs. equality [Ex: A line segment of length 36 is subdivided into two segments whose ratio is 4:5. Find the length of each segment.]

c. Angle pairs: vertical angles, supplementary and complementary angles, adjacent angles

d. Parallel and perpendicular lines (definitions)

3. Parallel and perpendicular lines (Chapter 3) [algebraic problems involving angle measures and lengths]

a. Properties of parallel lines

b. Proving lines parallel

c. Sum of the measures of the angles of a triangle is 180 degrees.

d. Exterior angle = sum of the two remote interior angles

e. Generalization for polygons (sum of interior and exterior angles)

f. Pythagorean Theorem and Its Converse [solving quadratic equations by factoring, completing the square, or using the Quadratic Formula (OPTIONAL)]

g. Pythagorean Theorem (proof using areas of squares and triangles) (Ch. 8-2)

h. Determining whether a triangle is obtuse, right, or acute (Ch. 8-3)

i. Distance formula (Ch. 13-1)

j. Equations of circles in the coordinate plane

k. Midpoint formula (Ch. 13-5)

l. Slopes, parallel and perpendicular lines (Ch. 13-2, 13-4)

m. Writing equations of lines ( Ch. 13-6) [finding the coordinates of the points of intersection of lines]

4. Congruent triangles (Chapter 4)

a. SSS, SAS, ASA, AAS, Hy-Leg criteria

b. Isosceles triangle theorems [algebraic problems]

c. Using more than one pair of congruent triangles, overlapping triangles

d. Medians, altitudes, angle bisectors, and perpendicular bisectors

e. Definition of similar polygons and AA similarity in triangles

5. Constructions (Chapter 10) *may be taught as a unit or spread throughout the semester!!!

a. Line segments, triangles, circles

b. Angles, angle bisectors

c. Perpendicular bisector of a line segment

d. Perpendicular lines through points on and not on a given line.

e. A line parallel to a given line through a given point

f. Equilateral triangle, square, regular hexagon

5. Rigid Transformations (NEW STUFF)

a. Represent transformations in the plane; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

b. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

c. Develop the definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

d. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.

e. Use geometric descriptions of rigid motions to transform figures and to predict the effect of of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

f. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

g. Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions.

6. Quadrilaterals (Chapter 5) [algebraic problems from the Regents Exam]

a. Parallelogram, rectangle, rhombus, square

b. Trapezoid, isosceles trapezoid

c. All properties

d. Writing proofs

e. Area formulas (including d1d2/2 and d2/2)

7. Coordinate Geometry

a. Proving properties of triangles

b. Finding areas of triangles ([pic]bh, [pic]absinC (optional), Pick’s theorem, Heron’s formula, using circumscribed rectangles) using a method derived from determinants (optional)

c. Proving properties of quadrilaterals

d. Writing equations of lines (slope-intercept and point-slope form)

e. Solving systems of linear equations algebraically and graphically with the graphing calculator

f. Find the coordinates of the centroid, orthocenter, and circumcenter. [solving systems of linear equations to find coordinates of these points]

8. Inequalities (Chapter 6)

a. Review properties of inequalities

b. Exterior Angle Inequality

c. Triangle inequalities involving sides and opposite angles

d. Hinge Theorem (SAS Inequality Theorem) (optional)

e. Indirect proofs

Geometry (textbook: Jurgensen/Brown)

Spring Semester

9. Similar Polygons (Chapter 7)

a. Ratios and proportions

b. Similar polygons

c. AA, SSS Similarity, SAS Similarity criteria

d. Parallel lines that cut off proportional lengths in a triangle [algebraic problems]

e. Triangle (interior) Angle Bisector Theorem

10. Right Triangles (Chapter 8) [algebraic problems from the Regents Exam]

a. Similarities in right triangles

b. Mean proportions in right triangles when altitude is drawn to the hypotenuse

c. Special right triangles

d. Right triangle trigonometry (sine, cosine, tangent), relationship between the sine and cosine of complementary angles

e. Laws of sines and cosines and applications (optional)

f. Areas of regular polygons using apothems

11. Circles (Chapter 9) [algebraic problems from the Regents Exam]

a. Basic terms

b. Prove all circles are similar

c. Properties of tangent lines to a circle, external and internal tangents

d. Arcs and Central Angles

e. Arcs and Chords

f. Inscribed Angles

g. Properties of angles of cyclic quadrilaterals

h. Other angles formed by tangents, secants, and intersecting chords

i. Lengths of segments, formed by tangents, secants, and intersecting chords (Power of a Point Theorem)

12. Constructions (page 701)

a. Centroid, orthocenter, incenter, and circumcenter

b. Define circumscribed and inscribed circles of a given triangle

c. Tangent to a circle at a point on the circle

d. Tangent to a circle from a point outside the circle (optional)

e. Find the center of a given circle.

f. Circumscribed and inscribed circles of a triangle

g. Regular polygons inscribed in a circle

h. Trisection of a line segment (optional)

i. Nine-Point Circle (optional)

13. Loci (Chapter 10) [Writing equations of loci in the coordinate plane (lines, circles, parabolas)]

a. Five fundamental loci

b. Loci in the coordinate plane [including finding equations of angle bisectors of (1) y = -3x + 7 and y = 3x – 5 (2) x = 3 and y = -4]

c. Intersections of loci

14. Circle area and circumference (Chapter 11)

a. Arc length

b. Area of a sector

15. Volumes and Surface Areas

a. Prisms

b. Pyramids

c. Cylinders and Cones

d. Spheres

e. Surface areas and volumes of similar solids

f. Volumes of revolution

g. Cavalieri’s Principle

h. Identify the shapes of 2-D cross-sections of 3-D objects, and identify 3-D objects generated by rotations of 2-D objects.

16. Transformations

a. Reflections

b. Translations and Glide Reflections

c. Rotations

d. Dilations (defining similarities as dilations)

e. Compositions of transformations

f. Determine which transformations preserve lengths, angle measures, areas, and orientation

g. Apply transformations to y = x2

17. Intersections of lines, parabolas and circles in the coordinate plane

a. Derive an equation of a circle of given center and radius using the Pythagorean Theorem

b. Complete the square to find the center and radius of a circle given by an equation

c. Derive equations of parabolas, given the focus and directrix.

d. Derive equations of ellipses and hyperbolas, given the foci, etc. (optional)

e. Solving linear-quadratic systems of equations algebraically and graphically

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