This geometry lesson explains how angles are formed when a transversal intersects parallel lines.

Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence. The topic of transformations is introduced in a primarily experiential manner in Grade 8 and is formalized in Grade 10 with the use of precise language. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.

Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2.  To be able to discuss similarity, students must first have a clear understanding of how dilations behave.  This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria.  An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.

Module 3, Extending to Three Dimensions, builds on students’ understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids. The student materials consist of the student pages for each lesson in Module 3. The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.

In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. The module opens with a modeling challenge, one that reoccurs throughout the lessons, to use coordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the plane—the room in which it sits. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the coordinates of points that divide given segments in specific length ratios, and so on), students will describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.

This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year.  It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story.  This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.

This geometry lesson is Part 1 of 2 lessons that give a proof of Heron's Formula.

This geometry lesson is Part 2 of the Proof of Heron's Formula, showing that the expression derived in Part 1 is identical to Heron's Formula.

This geometry lesson is an introduction to the Pythagorean Theorem and shows how to calculate the length of the hypotenuse of a right triangle.

This geometry lesson gives more practice exercises using the Pythagorean Theorem and introduces 45-45-90 triangles.

This geometry lesson provides the proof that the diagonals of a rhombus are perpendicular bisectors of each other.

This geometry lesson provides a proof that a triangle inscribed in a circle and having the circle's diameter as one side is right triangle.

http://www.khanacademy.org/exercise/similar_triangles_1This geometry lesson introduces the concept of similar triangles and how to determine alternate interior angles.https://www.khanacademy.org/math/geometry/similarity/old_school_similarity/v/similar-triangles

This geometry lesson continues the discussion of similar triangles and the use of ratios to determine lengths of corresponding sides.

This geometry lesson discusses how to calculate the volume of triangular prisms and cubes.

Using parallel lines, transversals, and some identified angle measurements, Salman Khan asks students to determine missing measurements for corresponding and supplementary angles in an Angle Game.

This geometry lesson provides a quick review of triangle medians and shows that the centroid is always 2/3 of the way along every median.

This geometry lesson provides the proof that the centroid is 2/3 of the way along every median of a triangle.

A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.