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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 × odd = odd, odd + even = odd)
ii. Indirect proof (example: Prove √2 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
** Bold-faced type indicates topics from the Common Core Standards
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
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
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b. Finding areas of triangles (2bh, 𝟐absinC (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
** Bold-faced type indicates topics from the Common Core Standards
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
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)
** Bold-faced type indicates topics from the Common Core Standards
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. Cavalieri’s Principle
g. 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
** Bold-faced type indicates topics from the Common Core Standards