Common Core High School Geometry Teaching and Learning Resources

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with CO-A, CO-B, CO-C and CO-D:

Imagine Learning Classroom (formerly LearnZillion) subscription needed: Lesson set: Define geometric terms precisely: Four video lessons: Definitions and representations are distinct, understand the importance of precise definitions, why point and line are undefinable; and define geometric terms precisely.  Aligns with CO-A.1.

Math.com: Quadrilaterals: Students learn about the five most common quadrilaterals: parallelogram, square, rectangle, rhombus, trapezoid.  Use with CO-A.3.

Mathwords.com: Index for Geometry: Terms from plane and solid geometry.  This interactive math dictionary has words, terms, formulas, pictures, diagrams, tables, and examples.  Use with CO-A.

Mathbits.com: Use with CO-A.2.

• Transformations: Lessons and practice problems include symmetry, reflections, translations, rotations, and dilations of a geometric figure.
• Geometry: Transformations: Lessons on using the TI-84+ family of calculators to perform transformations (reflect, translate, rotate, dilate).

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  The Pythagorean Theorem: Geometry's Most Elegant Theorem: "This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications."  Aligns with CO-A.1 and SRT-C.6, SRT-C.8.

National Library of Virtual Manipulatives: Transformations (Java required for online use)

• Reflection
• Rotation
• Translation
• Composition: Use reflection, rotation, and translation
• Effect of two transformations (Translations, Reflections, or Rotations) in any order.

Shodor Interactivate:

• Angles formed by parallel lines and one or two transversals.  This activity allows users to practice angle vocabulary.
• Translations, Reflections, Rotations: This lesson introduces learners to those concepts.
• Transmographer: Learners can explore translations, reflections, and rotations in the Cartesian coordinate system by transforming squares, triangles and parallelograms. Parameters: Shape, x or y translation, x or y reflection, angle of rotation.
• Transmographer 2: Learners build their own polygon and transform it in the Cartesian coordinate system. Experiment with reflections across any line, rotations about any point, and translations in any direction. Parameters: Shape, x or y translation, x or y reflection, angle of rotation.
• 3D Transmographer: Use the virtual manipulative to experiment with translations, reflections, rotations of polygons on the coordinate plane.  Learners can build their own polygons and also revolve them around any line to yield a 3D solid of revolution.

Walter Fendt's Sum of Angles in a Triangle, Virtual Manipulative

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 1 manipulative for CO-A.1, 29 manipulatives for CO-A.2, 14 manipulatives for CO-A.3, 43 manipulatives for CO-A.4, 105 manipulatives for CO-A.5. Among those:

• Reference angles: Aligns with CO-A.1.
• Composing rotations: Aligns with CO-A.2, CO-A.4, CO-B.6.
• Linear transformations of a polygon: Aligns with CO-A.2, CO-A.4, CO-A.5.
• Matrix transformations "F": Aligns with CO-A.2, CO-A.4, CO-A.5.
• Matrix transformation: applied to a circle: Aligns with CO-A.2, CO-A.4, CO-A.5,  CO-B.6.
• Classifying quadrilaterals: Aligns with CO-A.3, CO-C.11.
• Rotation about a point in a plane: Aligns with CO-A.3, CO-A.4, CO-A.5.
• 2D rotation using matrices: Aligns with CO-A.3, CO-A.4, CO-A.5.
• Reflection in two dimensions: Aligns with CO-A.3, CO-A.4, CO-A.5.
• Rotation in two dimensions: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
• Solve the Loculus of Archimedes by forming a square from given pieces using translations, reflection, and rotation. Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
• Rotation as product of two reflections: Aligns with CO-A.5.
• Reflection matrix in 2D: Aligns with CO-A.5.
• Quadrilateral slide: Aligns with CO-A.5.
• Understanding 2D reflection: Aligns with CO-A.5.
• Understanding 2D translation: Aligns with CO-A.5.
• Understanding 2D rotation: Aligns with CO-A.5.
• Parallelogram slide: Aligns with CO-A.5.
• Guessing the Frieze Group: Aligns with CO-A.5.  Note: A frieze group consists of elements (translations, rotations, reflections, and glide reflections) that map a frieze to itself. There are seven possible frieze groups.

YouTube video lessons:  Use the following with CO-A.1:

• Geometry: Some precise definitions
• Introduction to Geometry (1.1): Define Point, Line, Plane, Segment, Ray, Angle, Collinear, Coplanar.
• Point Line Plane Examples (1.1)
• Parallel lines, polygons, and the Pythagorean Theorem for finding values for alternate interior angles, finding values for the exterior angles of regular polygons, and using the Pythagorean theorem.  Video from the Ohio Resource Center Tutorials for High School Mathematics
• Frieze Patterns

CK-12: Dilations in the Coordinate Plane: Practice questions with review content and video.  Review content includes a Types of Transformations Study Guide, which is an overview of reflection, rotation, dilation, and translation.   Aligns with CO-A.2 and SRT-A.1.

Khan Academy: Practice questions with videos.

• Geometry: Introduction to Euclidean Geometry: Geometric definitions: Aligns with CO-A.1.
• Geometry: Transformations: Aligns with CO-A.2, CO-A.3, CO-A.4, CO-A.5.

MathBitsNotebook: Geometry:

• Segments, Angles and Triangles includes lessons, then practice problems on segments, segments in triangles, midsegments, angles, angles in triangles, isosceles triangles, triangle inequality, pre-proof reasoning (this latter is constructed response), and practice with proofs (this latter is constructed response).  Use with CO-A.1, and CO-C.9, CO-C.10.
• Parallels and Perpendiculars includes lessons, then practice problems on angles and parallel lines, perpendicular lines, and proofs involving parallel and perpendicular lines (this latter is constructed response).  Use with CO-A.1 and CO-C.9.
• Transformations includes lessons, then practice problems on transformations as functions, symmetry, reflections, translations, rotations, rigid transformations, and compositions (sequences) of transformations.  Use with CO-A.2, CO-A.3, CO-A.4, CO-A.5, and CO-B.6.

MathsIsFun.com: Transformations: Learners use virtual manipulatives to experiment with rotations, reflections, and translations of a triangle, rectangle, or L-shape, setting parameters for each experiment.  Following each section are multiple choice questions.  Aligns with CO-A.5.

Khan Academy: HSG-G-CO includes HSG-CO.A.1 through HSG-CO.A.5 Common Core aligned problems.

Also see Performing transformations: unit on translations, rotations, reflections, dilations. Practice questions with videos.

• Defining transformations: Aligns with CO-A.2, CO-A.4.
• Translate points: Aligns with CO-A.5.
• Translate shapes: Aligns with CO-A.5.
• Rotate points: Aligns with CO-A.5.
• Rotate shapes: Aligns with CO-A.5.
• Reflect points: Aligns with CO-A.5.
• Reflect shapes: Aligns with CO-A.5.
• Advanced reflections: Aligns with CO-A.5.

Illustrative Mathematics: Geometry:

• Defining Parallel Lines: Aligns with CO-A.1.
• Defining Perpendicular Lines: Aligns with CO-A.1.
• Reflected Triangles: Aligns with CO-A.5 and CO-D.12.
• Showing a Triangle Congruence: The general case: Aligns with CO-A.5.
• Showing a Triangle Congruence: The particular case: Aligns with CO-A.5.

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks.  Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5.  Choose the following within the module: 

• Task 1: Leaping Lizards! aligns with both standards CO-A.1, CO-A.2, CO-A.4, CO-A.5;
• Task 2: Is It Right? aligns with CO-A.1.
• Task 3: Leap Frog aligns with CO-A.4, CO-A.5.
• Task 4: Is It Leap Year? aligns with both standards CO-A.1 and CO-A.4.
• Task 5: Symmetries of Quadrilaterals; and Task 6: Symmetries of Regular Polygons align with CO-A.3.
• Task 7: Quadrilaterals-Beyond Definition aligns with both standards CO-A.3, CO-A.4.
• Task 8: Can You Get There from Here? aligns with CO-A.5.

NCTM's Reasoning and Sense Making Task Library: Taking a Spin includes the task overview, teacher notes for its use, and student activity sheet.  Aligns with CO-A.3, CO-A.4 and mathematical practice standards 3 and 8.

Math Open Reference: Congruent Triangles: Investigate the theorems for congruence.

National Library of Virtual Manipulatives: investigate methods for determining congruent triangles (Java required). Explore SSS, SAS, ASA, SSA.  Note SSA does not guarantee congruent triangles.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 32 manipulatives for CO-B.6. Among those:

• Solve the Loculus of Archimedes by forming a square from given pieces using translations, reflection, and rotation. Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
• Matrix transformation: applied to a circle: Aligns with CO-A.2, CO-A.4, CO-A.5,  CO-B.6.
• Rotation in two dimensions: Aligns with CO-A.2, CO-A.4, CO-A.5, CO-B.6.
• Composing rotations: Aligns with CO-A.2, CO-A.4, CO-B.6.
• Symmetries of a prism: Aligns with CO-B.6.

Khan Academy: Practice questions with videos.

• Congruent triangles 1: Determine congruent triangles using SSS, SAS, ASA, AAS. Aligns with CO-B.7.

MathBitsNotebook: Geometry:

• Transformations includes lessons, then practice problems on transformations as functions, symmetry, reflections, translations, rotations, rigid transformations, and compositions (sequences) of transformations.  Use with CO-A.2, CO-A.3, CO-A.4, CO-A.5, and CO-B.6.
• Congruent Triangles includes lessons, then practice problems on rigid motions and congruence, numerical practice with congruence, recognizing congruent triangles, completing congruent triangle proofs (this latter is constructed response), congruent triangles (multiple choice), and congruent triangle proofs (this latter is constructed response with proofs - rigid motion, direct, double, indirect, flow).  Use with CO-B.7 and CO-B.8.

AlgebraLab.org: Congruent Triangles: Lesson and fill in the blank practice problems, including word problems.  Be sure to show the related AlgebraLab documents for additional practice problems.

Khan Academy: HSG-G-CO includes HSG-CO.B.6 through HSG-CO.B.8 Common Core aligned problems.

Also see the following practice questions with videos.

• Rigid transformations overview and quizzes: Aligns with CO-B.6.
• Explore rigid transformations and congruence: Aligns with CO-B.6, CO-B.7.
• Congruent triangles 2: Find angles in congruent triangles. Aligns with CO-B.6, CO-B.7.

Math.com: Congruent Figures: Lesson and fill-in the blank matching problems.

Illustrative Mathematics: Geometry:

• Reflections and Isosceles Triangles: Aligns with CO-B.
• Properties of Congruent Triangles: Aligns with CO-B.7.

Mathematics Assessment Project: High School: Geometry: Task 452: Analyzing Congruence Proofs

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks.  Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5. Task 5: Symmetries of Quadrilaterals; Task 6: Symmetries of Regular Polygons; and Task 7: Quadrilaterals-Beyond Definition align with standard CO-B.6.  Task 9: Congruent Triangles aligns with standards CO-B.6, CO-B.7, CO-B.8.

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use, and student activity sheet.  Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

New York City School System: High School Geometry: Company Logo: Unit Overview: “This 3-4 week unit uses an investigation of rigid motions and geometric theorems to teach students how to verify congruence of plane figures and use the implications of congruence to solve problems and create proofs about geometric relationships. Students will demonstrate mastery of the content by making sense of the Company Logo Performance Task and persevering in solving the task.” Geometry standards addressed include CO-B.6, CO-B.7, CO-C.9, CO-C.10, and SRT-B.5.  Mathematical Practices standards include MP.1, MP.3, and MP.6.

Imagine Learning Classroom (formerly LearnZillion) subscription needed: Lesson set: Prove theorems concerning triangles and parallelograms: Six video lessons: Use triangle congruence (SAS), use triangle similarity to prove Triangle Midsection Theorem and Concurrency of Medians Theorem, use parallel lines and triangle congruence theorems to prove properties of parallelograms, including properties of diagonals of parallelograms, use properties of parallelograms and triangles to solve a real-world problem.  Aligns with CO-C.10 and CO-C.11.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary. The Broken Stick Experiment: Triangles, Random Numbers and Probability: "In this video lesson, during in-class sessions between video segments, students will learn among other things: 1) how to generate random numbers; 2) how to deal with probability; and 3) how to construct and draw portions of the X-Y plane that satisfy linear inequalities."  Aligns with CO-C.10.

Saltire Software: Geometry Applet Gallery contains some excellent applets, which are easy to use and suitable for exploring several geometry theorems. Select from the following:

• Basic Geometry: Nine applets for exploration of angles on a chord, angles on a chord and center, similar triangles, angles between parallel lines, right kites (2 applets), perpendicular bisectors of a right triangle, the incircle, and Pythagorean triples.  Students can drag points to explore concepts.
• Triangle Calculators: Four triangle calculators are presented.  Students can enter SSS, SAS, or ASA, and see the unique triangle that is formed with the applet.  If they enter SSA, they see that the resulting triangle is not unique (two are possible) or no triangle is possibly formed with given information.

Shodor Interactivate: Angles: This lesson addresses acute, obtuse, and right angles as well as relationships between angles formed by parallel lines crossed by a transversal, including vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 1 manipulative for CO-C.9, 15 manipulatives for CO-C.10, 2 manipulatives for CO-C-11. Among those:

• Parallel lines: Angles with a transversal: Aligns with CO-C.9.
• Interior angles of a triangle sum to Pi (radians): a straight line is formed.  Aligns with CO-C.10.
• The Sum of the Interior Angles of a Triangle Equals 180 Degrees by Paper Folding: Aligns with CO-C.10.
• Halving a Triangle: The median divides the triangle into two halves of equal area.  Aligns with CO-C.10.
• The Medians of a Triangle Divide It into Three Smaller Triangles of Equal Area: Aligns with CO-C.10, CO-C.11.
• The Medial Division of Triangles: The medians of a triangle divide it into six smaller triangles of equal area.  Aligns with CO-C.10.
• The Medians of a Triangle Are Concurrent: A Visual Proof: Aligns with CO-C.10.
• The Centroid of a Triangle Divides Each Median in the Ratio 1:2: Aligns with CO-C.10.
• Triangle Inequality: The sum of lengths of two sides of a triangle is at least as large as the length of the third side.  Aligns with CO-C.10.
• Euclid, Book 3, Proposition 22: in a cyclic quadrilateral, opposite angles sum to 180 degrees: Aligns with C-A.2, C-A.3, and CO-C.10.
• Angles in a quadrilateral total 360 degrees: Aligns with CO-C.10.
• Classifying quadrilaterals: Aligns with CO-A.3, CO-C.11.

YouTube Videos:

• Theorems for parallel lines and transversals: Use with CO-C.9.
• Triangle Application theorems:  Use with CO-C.10.
• Using Properties of Parallelograms: Use with CO-C.11.

Khan Academy: Practice questions with videos.

• Triangle inequality theorem: Aligns with CO-C.10.

Shodor Interactivate: Angles: This applet with drop-down choices for responses provides an interactive quiz for learners to practice their knowledge of acute, obtuse, and alternate angles. Also, practice relationships between vertical, adjacent, alternate, same-side, and corresponding angles.  Learners can use either one or two transversals crossing parallel lines.

National Library of Virtual Manipulatives: walk through a visual proof of the Pythagorean theorem.

AlgebraLab.org:  Lesson and fill in the blank practice problems.  Be sure to show the related AlgebraLab documents for additional practice problems on each topic.

• Parallel Lines and Transversals
• Slope of Parallel and Perpendicular Lines

Khan Academy: HSG-G-CO includes HSG-CO.C.9 through HSG-CO.C.11 Common Core aligned problems.

Also see the following practice questions with videos.

• Line and angle proofs: Aligns with CO-C.9.

Math.com: Angles and Intersecting Lines: Lesson and fill-in the blank matching problems.

MathBitsNotebook: Geometry:

• Segments, Angles and Triangles includes lessons, then practice problems on segments, segments in triangles, midsegments, angles, angles in triangles, isosceles triangles, triangle inequality, pre-proof reasoning (this latter is constructed response), and practice with proofs (this latter is constructed response).  Use with CO-A.1, and CO-C.9, CO-C.10.
• Parallels and Perpendiculars includes lessons, then practice problems on angles and parallel lines, perpendicular lines, and proofs involving parallel and perpendicular lines (this latter is constructed response).  Use with CO-A.1 and CO-C.9.
• Quadrilaterals includes lessons, then practice problems on properties of quadrilaterals (fill-in chart, then a quiz), numerical problems and algebraic problems about quadrilaterals, applications of quadrilaterals, and proofs dealing with quadrilaterals (this latter includes both multiple choice and constructed response).  Use with CO-C.11.

Illustrative Mathematics: Geometry:

• Tangent lines and the radius of a circle: Aligns with CO-C.9.
• Points equidistant from two points in a plane: Aligns with CO-C.9.
• Is this a parallelogram?: Aligns with CO-C.11.
• Parallelograms and Translations: Aligns with CO-C.11.
• Midpoints of the sides of a parallelogram: Aligns with CO-C.11.
• Congruence of parallelograms: Aligns with CO-C.11 and SRT-B.5.

Inside Mathematics: MARS Tasks:

• Quadrilaterals: Students use geometric properties to solve a problem.  This task aligns with standards CO-C.9 and CO-C.11.
• Circles in Triangles: Students use geometric properties of circles and triangles to prove that two triangles are congruent.  This task aligns with geometry congruence standards CO-C.9 and CO-C.10 and geometry circle standards C-A.2 and C-A.3.

Mathematics Assessment Project: High School: Geometry:

• Task 429: Solving Geometry Problems: Floodlights
• Task 212: Evaluating Statements About Length and Area
• Task 419: Proofs of the Pythagorean Theorem

Mathematics Vision Project, Secondary 2 Student Edition:

• Module 5: Geometric Figures: This module contains 8 classroom tasks addressing geometry congruence standards CO-C.9, CO-C.10 and CO-C.11.

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use, and student activity sheet.  Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

New York City School System: High School Geometry: Company Logo: Unit Overview: “This 3-4 week unit uses an investigation of rigid motions and geometric theorems to teach students how to verify congruence of plane figures and use the implications of congruence to solve problems and create proofs about geometric relationships. Students will demonstrate mastery of the content by making sense of the Company Logo Performance Task and persevering in solving the task.” Geometry standards addressed include CO-B.6, CO-B.7, CO-C.9, CO-C.10, and SRT-B.5.  Mathematical Practices standards include MP.1, MP.3, and MP.6.

Desmos: Geometry Tool

GeoGebra: free open-source software

Khan Academy: Geometric Constructions unit includes tutorial videos on the following:

• Constructing bisectors of lines and angles: Aligns with CO-D.12.
• Constructing a line tangent to a circle: Aligns with CO-D.12.
• Constructing regular polygons inscribed in circles: Square, equilateral triangle, hexaxon.  Aligns with CO-D.13.

MathsIsFun.com: Constructions: This is an excellent series of 17 animated geometric constructions, which learners would do with a compass, pencil, and straightedge only.  They demonstrate how to construct a perpendicular bisector, angle bisector, 30-45-60 degree angles and equilateral triangle, perpendicular line to a line from a point on or off the line, parallel lines, congruent angles, a line cut into "n" pieces.  Circle constructions include to find a center, tangent lines from an external point, inscribed and circumscribed circles for triangles, circle touching three points, and a pentagon construction.  Aligns with CO-D.12 and CO-D.13.

Math Open Reference: Constructions: The site provides step-by-step animations of constructions with compass and straightedge.  The Table of Contents for the Constructions is comprehensive for lines, line segments, angles, triangles, circles, arcs, ellipses, polygons, and non-Euclidean constructions.  Aligns with CO-D.12 and CO-D.13.

Robo Compass is an online app for geometric constructions, transformations, patterns and tesselations.  It works "exactly as you do using a physical straightedge, compass and protractor."  Documentation is included.

Whistler Alley Mathematics: Geometry Construction Reference: This includes illustrations and steps for geometric constructions.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 20 manipulatives for CO-D.12, 3 manipulatives for CO-D.13. Among those:

• Distance of a point to a segment: Aligns with CO-D.12.
• Triangle side length inequality: Construct a triangle with three given sides.  Aligns with CO-D.12.
• Angle bisectors of two intersecting lines: Aligns with CO-D.12.
• Angle bisectors in a triangle: Aligns with CO-D.12.
• The perpendicular bisectors of a triangle: Aligns with CO-D.12.
• Triangle: scalene, isosceles, equilateral: Aligns with CO-D.12.
• Mascheroni's construction of the center of a circle: Aligns with CO-D.12.
• Constructing a regular pentagon with a ruler and compass: Aligns with CO-D.12.
• Indestructible perpendicular lines: Aligns with CO-D.12.
• Bisection of line segment: Aligns with CO-D.12.
• Largest triangle inscribed in a circle: the equilateral triangle.  Aligns with CO-D.13.
• Largest isosceles triangle inscribed in a circle: the equilateral triangle (via exploration).  Aligns with CO-D.13.
• Maximum rectangle inscribed in a circle: the square.  Aligns with CO-D.13.

Khan Academy: HSG-G-CO includes HSG-CO.D.12 Common Core aligned problems.

MathBitsNotebook: Geometry: Constructions and Concurrencies includes lessons on how to construct with compass and straightedge and other tools (e.g., a reflective edge, string, paper-folding, and dynamic geometric software), points of concurrency, and locus theorems.  Practice problem are on basic and more constructions, using constructions, constructions and reflections, points on concurrency, and locus.  Use with CO-C.10, CO-D.12 and CO-D.13.

Illustrative Mathematics: Geometry:

• Angle bisection and midpoints of line segments: Aligns with CO-D.12.
• Construction of perpendicular bisector: Aligns with CO-D.12.
• Reflected Triangles: Aligns with CO-A.5 and CO-D.12.
• Inscribing a square in a circle: This task aligns with standard CO-D.13.

Mathematics Assessment Project: High School: Geometry: Task 403: Inscribing and Circumscribing Right Triangles

Mathematics Vision Project, Secondary 1 Student Edition: Module 6: Congruence, Construction, and Proof: This module contains 14 classroom tasks.  Geometry standards addressed in Module 5 include CO-A.1, CO-A.2, CO-A.3, CO-A.4, CO-A.5 and CO-B.6, CO-B.7, CO-B.8, and CO-D.12, CO-D.13, and GPE-B.5. Task 11: Under Construction; Task 12: More Under Construction; Task 13: Justifying Construction; and Task 14: Construction Blueprints align with standards CO-D.12 and CO-D.13.