The goal of the course MAT 515 Algebra for Teachers is to provide the first part of geometric material relevant to the everyday work of a high-school teacher both in contents and methods of teaching, and form rigorous foundations for geometry of Common Core State Standards.

Students should be able to

• describe the basic objects of Euclidean plane geometry (points, lines, rays, segments, angles, circles, etc.) and relations between them;
• formulate and prove the theorems about
  • vertical angles,
  • isosceles triangles and their properties,
  • congruence tests for triangles,
  • inequality between non-adjacent exterior and interior angles in a triangle,
  • relations between sides and opposite internal angles in a triangle,
  • triangle inequality and its corollaries,
  • existence and uniqueness of perpendicular to a line from a point,
  • slant and perpendicular;
• describe the techniques of compass-and-straightedge constructions and solve using it the standard construction problems such as
  • from a point on a line erect a perpendicular to this line,
  • drop a perpendicular to a given line from a given point,
  • bisect a segment or an angle,
  • construct an angle with a prescribed side congruent to a given angle,
  • construct a triangle given segments congruent to its sides;
• formulate the parallel postulate and its reformulations, prove the equivalence of reformulations;
• formulate, prove and apply tests for parallel lines and converse theorems;
• formulate and prove the theorem about the sum of interior angles in a triangle and its corollaries;
• formulate and prove theorems about sides and diagonals in a parallelogram and parallelograms of special types (rectangles, rhombuses and squares);
• formulate and prove theorems about midlines in triangle and trapezoid;
• formulate and prove the theorem about inscribed angle and its corollaries and apply them to solving construction problems;
• construct inscribed and superscribed circles of a triangle, prove their existence and uniqueness;
• identify and prove existence of the classical concurrency points in a triangle;
• formulate and prove theorems about the structure of plane isometries:
  • restoring an isometry from its restriction to 3 points,
  • representing an isometry as a composition of at most 3 reflections,
  • classification of isometries: reflections, translations, rotations and glide reflections;
• calculate compositions of plane isometries and solve problems using the calculation of compositions (e.g., the Napoleon theorem);
• formulate and prove Thales' theorem;
• formulate definition of similarity transformations and decompose any similarity transformation into isometry and homothety;
• formulate, prove and apply similarity tests for triangles;
• use similarity for solving construction problems;
• use similarity for proving proportionality theorems, geometric means in right triangle, Pythagoras' theorem and its corollaries, etc.;
• give definition of inversion (reflection in a circle) and describe its properties, images of lines and circles;
• solve geometric construction problems using inversions, formulate and prove the Steiner theorem;
• outline the Poincaré model for the hyperbolic plane;
• demonstrate knowledge of the historical development of Euclidean and non-Euclidean geometries, including contributions from diverse cultures.