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Common Core Mapping for High School: Geometry


Geometry Overview


  • Experiment with transformations in the plane
  • Understand congruence in terms of rigid motions
  • Prove geometric theorems
  • Make geometric constructions
  • Similarity, Right Triangles, and Trigonometry

  • Understand similarity in terms of similarity transformations
  • Prove theorems involving similarity
  • Define trigonometric ratios and solve problems involving right triangles
  • Apply trigonometry to general triangles
  • Circles

  • Understand and apply theorems about circles
  • Find arc lengths and areas of sectors of circles
  • Expressing Geometric Properties with Equations

  • Translate between the geometric description and the equation for a conic section
  • Use coordinates to prove simple geometric theorems algebraically
  • Geometric Measurement and Dimension

  • Explain volume formulas and use them to solve problems
  • Visualize relationships between two-dimensional and three-dimensional objects
  • Modeling with Geometry

  • Apply geometric concepts in modeling situations

  • Related Topics: Common Core for Mathematics

    Common Core Mapping for High School: Geometry








    Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

    Geometry Definitions Perpendicular and Parallel Lines

    Geometric Definitions

    Geometry Games


    Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

    Transformations in the Plane




    Define Rigid Transformation



    Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

    Polygons & Symmetry

    Symmetry of two-dimensional shapes



    Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

    Define Rotations, Reflections, Translations

    Define Translations

    Define Rotations

    Define Reflections

    Define rigid transformations



    Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

    Sequence of Transformations


    Rotations 1

    Rotations 2

    Reflections 1

    Reflections 2



    Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

    Congruence & Transformations

    Congruent Triangles 1

    Congruent Triangles 2

    Define congruence through rigid transformations

    Explore Rigid Transformations and Congruence



    Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

    Rigid Motions & Congruent Triangles (CPCTC)

    Congruency Postulates

    Define congruence through rigid transformations

    Explore Rigid Transformations and Congruence


    Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

    Triangle Congruence

    Congruency postulates


    Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment?s endpoints.

    Prove Line and Angle Theorems

    Line and Angle Proofs

    Proofs on Lines and Angles


    Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180?; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

    Prove Triangle Theorems

    Triangle Inequality Theorem


    Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

    Prove Parallelogram Theorems

    Proof of Parallelograms


    Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

    Geometric Constructions

    Compass constructions



    Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    Construct Shapes

    Compass Construction 2

    Similarity, Right Triangles, and Trigonometry









    Verify experimentally the properties of dilations given by a center and a scale factor:

    A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

    The dilation of a line segment is longer or shorter in the ratio given by the scale factor.





    Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

    Similarity Transformations

    Defining similarity through angle-preserving transformations



    Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

    Similar Triangles (AA, SAS, SSS)

    Similar triangles 1

    Similar triangles 2

    Solve similar triangles



    Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

    Prove Triangle Theorems


    Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

    Using Congruent Triangles

    Using Similar Triangles

    Solve problems with similar and congruent triangles

    Solve similar triangles 2



    Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

    Similarity & Trig Ratios: Sin, Cos, Tan

    Trigonometric functions and side ratios in right triangles



    Explain and use the relationship between the sine and cosine of complementary angles.

    Sin and Cos of Complementary Angles

    Trig Functions in right triangles  


    Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

    Applying Trigonometric Ratios

    Special right triangles

    Apply right triangles



    (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

    Area of Triangle using Sine

    Area of Triangle using Sine

    Non-right triangle proofs


    (+) Prove the Laws of Sines and Cosines and use them to solve problems.

    Law of Sines and Law of Cosines

    Law of Sines

    Law of Cosines


    (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

    Apply Law of Sines

    Apply Law of Cosines

    Law of cosines

    Law of sines

    Law of Sines and Law of Cosines word problems









    Prove that all circles are similar.

    All Circles are Similar

    Define similarity through angle preserving transformations


    Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

    Inscribed angles, radii, chords

    Central, inscribed, and circumscribed angles

    Inscribed Angles



    Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

    Inscribed and Circumscribed Circles

    Inscribing and circumscribing circles on a triangle



    (+) Construct a tangent line from a point outside a given circle to the circle.

    Construct Tangent to Circle

    Construct tangent


    Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

    Arc Length & Sector Area

    Radians and Arc Length

    Area of circles and sectors

    Circles and Arcs

    Radians and Arc Length

    Expressing Geometric Properties with Equations







    Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

    Equation of a Circle

    Pythagorean theorem and the equation of a circle

    Equation of a circle in factored form

    Equation of a circle in non-factored form



    Derive the equation of a parabola given a focus and directrix.

    Equation of a Parabola

    Parabola intuition 1

    Parabola intuition 2

    Parabola intuition 3



    (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

    Equations of Ellipses and Hyperbolas

    Explore foci of an Ellipse

    Equation of an Ellipse

    Equation of a Hyperbola


    Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, ?3) lies on the circle centered at the origin and containing the point (0, 2).

    Coordinate Proofs

    Geometry problems on the coordinate plane



    Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

    Parallel and Perpendicular Lines

    Equations of parallel and perpendicular lines


    Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

    Partition Line Segment

    Dividing line segments

    Midpoint Formula



    Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

    Area and Perimeter on Coordinate Plane

    Coordinate Plane word problems

    Geometric Measurement and Dimension







    Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

    Geometric Formulas


    (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

    Geometric Formulas


    Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

    Using Volume Formulas

    Volume Word Problems



    Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

    Cross Sections and Solids of Rotation

    Cross sections of 3-D objects

    Rotate 2D to make 3D

    Modeling with Geometry







    Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

    Geometric Shapes word problems



    Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

    Density, Mass Volume

    Surface and volume density word problems



    Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

    Geometric Shapes word problems


    Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

    You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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