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Arcadia Unified School District Geometry Grades 9-12 Introduction The main purpose of the geometry curriculum is to develop students' geometric skills and concepts and their ability to construct formal logical arguments and proofs. The skills and concepts in this discipline are useful to all students. Throughout the course, students will continually integrate previously learned algebraic skills in geometric settings. Geometry and Measurement 1. Students will demonstrate understanding by identifying and giving examples of undefined terms, postulates, theorems, definitions and properties. Ρ Know how to add and subtract the measurement of angles and segments 2. Students will write geometric proofs. Spring 2009 1 Ρ Prove relationships between angles by using properties of complementary, supplementary, vertical, and exterior angles Ρ Use indirect reasoning and indirect proof to reach a conclusion 3. Students will demonstrate understanding of inductive and deductive reasoning. Ρ Construct and judge the validity of a logical argument and give counterexamples to disprove a statement * Use inverse and contrapositive in logical arguments 4. Students will prove basic theorems involving congruence and similarity. Ρ Prove that triangles are congruent or similar Ρ Use the concept of corresponding parts of congruent and/or similar triangles 5. Students will know and be able to apply properties of inequalities to triangles. Ρ Recognize and use relationships between sides and angles in a triangle Ρ Apply the triangle inequality theorem Spring 2009 2 Ρ Understand and write proofs with inequalities involving two triangles Spring 2009 3 6. Students will prove and use theorems involving the properties of parallel lines cut by a transversal. Ρ Prove lines are parallel from angle relationships Ρ Apply slope to parallel and perpendicular lines 7. Students will prove and use theorems involving the properties of quadrilaterals. Ρ Use properties of parallelograms (including rectangles, squares and rhombi) in proofs and in solving problems Ρ Apply trapezoid formulae 8. Students will prove and use theorems involving the properties of circles. Ρ Prove and solve problems regarding relationships among arcs, chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles 9. Students will know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. Spring 2009 4 Ρ Find and use measures of sides, interior and exterior angles of polygons (including regular polygons) to classify figures and solve problems Ρ Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids Ρ Compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres, committing to memory the applicable formulae Ρ Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids * Utilize area formulae in computing geometric probability 10. Students will prove and use the Pythagorean theorem to find missing lengths of sides of right triangles. Ρ Know and are able to use angle and side relationships in problems with special right triangles, such as 30o, 60o, 90o triangles and 45o, 45o, 90o triangles Ρ Know and are able to use the first four Pythagorean Triples 11. Students will perform basic constructions with straightedge and compass. Spring 2009 5 Ρ Construct an angle bisector Ρ Construct a perpendicular bisector Ρ Construct the line parallel to a given line through a point off the line Spring 2009 6 12. Students will demonstrate facility in using coordinate geometry. Ρ Graph linear equations using intercepts and point/slope methods Ρ Write linear equations with graphical information given Ρ Prove theorems by using coordinate geometry, including midpoint of a line segment, distance formula, and slope of a line * Perform operations with vectors, including magnitude and direction * Use distance and midpoint formulae to locate points in space * Sketch and describe the locus of a given set of conditions, as well as the intersection of loci 13. Students will know and understand the basic trigonometric functions defined by the angles of a right triangle. Ρ Use trigonometric functions to solve for an unknown length of a side draw unknown measure of an angle * Know and are able to use elementary trigonometric identities, i.e., tan x = (sinx/cosx), (sin x) 2 + (cos x) 2=1 Ρ Are able to solve a triangle using either the law of sines or law of cosines Spring 2009 7 14. Students will know and understand the effect of mapping on figures in the coordinate plane and in space. * * Identify rotations, translations, and reflections Transform figures using reflection, rotation, translation and/or dilation Spring 2009 8