Transformational Proof:Informal and Rigorous Kristin A. Camenga Kristin.email@example.com Houghton College November 12,2009
Note: all slides, resources, etc. from this talk will be posted on my home page under “Teacher Resources” within the next week or so.
Outline • What is transformational proof? • Why is transformational proof important? • How do you do a transformational proof? • What does transformational proof contribute to student learning? • Can I try?
What is transformational proof?
An Informal Example The Isosceles Triangle Theorem • Standard Method: Draw median and show triangles congruent • Transformational method: Draw angle bisector and reflect triangle over it to see that angles coincide
Key ideas • Uses transformations: reflections, rotations, translations, dilations • Depends on properties of the transformation: • Congruence is shown by showing one object is the image of the other under an isometry (preserves distance and angles) • Similarity is shown by showing one object is the image of the other under a similarity (preserves angle and ratio of distances)
Why is transformational proof important?
NYS Core Curriculum • G.RP.7 Construct a proof using a variety of methods (e.g., deductive, analytic, transformational) • G.G.57 Justify geometric relationships (perpendicularity, parallelism,congruence) using transformational techniques (translations,rotations, reflections)
Historical significance • Euclid’s Elements: • In the proof of SAS congruence, Euclid writes “If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE.” • This is the idea of a transformation! • Erlangen program: • 19th century program to unify geometry by looking at the transformations in different geometries and their invariants (tied to abstract algebra) • This approach to geometry is the current one in higher math
Advantages • More visual and intuitive; dynamic • Helpful in understanding geometry historically • Builds intuition and understanding of meaning • Generalizes to other geometries more easily
Informal Transformational Proofs • A visual, intuitive sense of how a transformation maps one shape to another • Builds on ideas of symmetry from elementary grades and could easily be done in middle school
Example: Arcs Cut by Parallel Lines (informal) Given: AB∥CD Prove: arc AC ≅ arc BD Idea: Reflect over the diameter perpendicular to CD.
Example: Parallelograms(Informal) Given: Parallelogram ABDC Idea: Rotate ABDC around the midpoint of the diagonal AD Results: • AC≅DB; AB≅DC • ∠B≅∠C • △ABD≅△DCA
Example: SAS(Informal) Given: AB≅A’B’, AC≅A’C’, ∠A≅∠A’ Prove: △ABC≅△A’B’C’ Idea: translate A to A’ and rotate △ABC until AB coincides with A’B’. Reflect over A’B’. Then the whole triangle coincides!
Example: Commutativity of Multiplication (Informal) … To show mxn=nxm, • Represent mxn as an array of dots with m rows and n columns • Rotate the array by 90 degrees and you have n rows and m columns, or nxm dots … … . . . … … … … … … … … . . . … … …
Background to formalize transformational proof • Experience that our vision can trick us • Transformations and their properties: • Isometries – reflections, rotations, translations • Preserve lengths • Preserve angles • Dilations • Preserve angles • Preserve ratios of lengths • Image lines are parallel to original lines • Symmetries of basic shapes (lines, circles) • Basic properties and axioms of geometry
Example: Isosceles Triangle Theorem (Rigorous) Given: △ABC, where AB≅AC • Draw the angle bisector AD. Therefore, ∠BAD≅∠CAD. • Reflect over AD. • AD reflects to itself. • ∠BAD reflects to ∠CAD since the angles are congruent and share side AD. • AB reflects to AC since they are corresponding rays of angles which coincide after reflection. • B reflects to C since A reflects to itself and AB≅AC. • BD reflects to CD since B reflects to C do and D reflects to itself and two points determine exactly one segment. • Since AB reflects to AC, B to C and BD to CD, ∠ABD reflects to ∠ACD. • Therefore ∠ABD≅∠ACD.
Example: Arcs Cut by Parallel Lines (Rigorous) Given: AB∥CD Prove: arc AC ≅ arc BD Draw diameter EF perpendicular to CD, intersecting CD at H and AB at G. • Since AB ∥CD, EH⊥ AB since it makes the same angle with both CD and AB. Also, since diameters bisect chords, CH≅HD and AG≅GB • Reflect over EF. • Since EF is a diameter, the circle reflects to itself • Lines CD and AB reflect to themselves since they are perpendicular to EF • Since CH≅HD and AG≅GB, A reflects to B and C reflects to D. • Since the circle reflects to itself and the endpoints of arc AC reflects to BD, arc AC reflects to BD. • Therefore arc AC ≅ arc BD
Example: Parallelograms(Rigorous) Given: Parallelogram ABDC • Draw diagonal AD and let P be the midpoint of AD. • Rotate the figure 180⁰ about point P. • Line AD rotates to itself. • Since P is the midpoint of AD, PA≅PD and A and D rotate to each other. • Since by definition of parallelogram, AB∥CD and AC∥BD, ∠BAD≅∠CDA and ∠CAD≅∠BDA. Therefore the two pairs of angles, ∠BAD and ∠CDA , and ∠CAD and ∠BDA, rotate to each other. • Since the angles ∠CAD and ∠BDA coincide, the rays AC and DB coincide. Similarly, rays AB and DC coincide because ∠BAD and ∠CDA coincide. • Since two lines intersect in only one point, C, the intersection of AC and DC, rotates to B, the intersection of DB and AB, and vice versa. • Therefore the image of parallelogram ABDC is parallelogram DCAB. • Based on what coincides, AC≅DB, AB≅DC, ∠B≅∠C, △ABD≅△DCA, and PC≅PB
Example: SAS(Formal) Given: AB≅A’B’, AC≅A’C’, ∠A≅∠A’ • Translate △ABC so that A coincides with A’. • Rotate △ABC so that ray AB coincides with ray A’B’. Since AB≅A’B’, B coincides with B’. • If C and C’ are on different sides of line AB, reflect △ABC over line AB. • Since ∠A≅∠A’ and the rays AB and A’B’ coincide and are on the same side of the angle, ∠A coincides with∠A’. • Since the angles coincide, the other rays AC and A’C’ coincide. • Since AC≅A’C’, C coincides with C’. • Since B coincides with B’ and C with C’ and two points determine exactly one line, BC coincides with B’C’. Since all the sides coincide, the angles do, as well. • Since all sides and angles coincide, △ABC≅△A’B’C’
Example: June Geometry Regents #38 Given △ABC and △EDC. C is the midpoint of BD and AE. Prove: AB∥DE • Rotate the figure around the point C 180⁰ • Lines AE and BD rotate to themselves by symmetries of a line • Since C is the midpoint, BC≅CD and AC≅CE, so B rotates to D and A rotates to E. • Since two points determine a unique line, AB rotates to ED. • Since ∠A has sides of AB and AE, it rotates to the angle with sides ED and EA, or ∠E. Therefore ∠A≅∠E • Since , alternate interior angles are congruent and AB ∥DE.
Student Benefits • Builds on students’ intuitive ideas so they can participate in proof from the beginning • Encourages visual and spatial thinking, helping students consider the same ideas in multiple ways • Serves as a guide for students to remember theorems and figure out problems • Promotes understanding by offering an alternate explanation
Student Benefits • Reinforces properties of transformations • Gives application for a number of different axioms or theorems we don’t use frequently • Another method to analyze geometric figures, looking at them piece by piece to get the whole
Here are a few for you to try! • Vertical angles are congruent. • If the base angles of a triangle are congruent, then the sides opposite those angles are congruent. • If a quadrilateral has diagonals that are perpendicular bisectors of each other, then it is a rhombus. • HL: If two right triangles have congruent hypotenuses and one pair of legs congruent, then the triangles are congruent.
Resources • Wallace, Edward C., and West, Stephen F., Roads to Geometry: section on transformational proof • Henderson, David W., and Taimina, Daina, Experiencing Geometry • The eyeballing game http://woodgears.ca/eyeball/