GEOMETRY

Students will:

•Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning

•Write geometric proofs, including proofs by contradiction

•Construct and judge the validity of a logical argument and give counterexamples to disprove a statement

•Prove basic theorems involving congruence and similarity

•Prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles

•Know and are able to use the triangle inequality theorem

•Prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles

•Know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures

•Compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders

•Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids

•Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids

•Find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems

•Prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles

•Prove the Pythagorean theorem

•Use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles

•Perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line

•Prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles

•Know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x))2 + (cos(x))2=1

•Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side

•Know and are able to use angle and side relationships in problems with special right triangles, such as 30º, 60º, and 90º triangles and 45º, 45º, and 90º triangles

•Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles

•Know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections