Transformational Proof: Informal and Formal

1. Transformational Proof:Informal and Formal Kristin A. Camenga Kristin.camenga@houghton.edu Houghton College November 12,2009

2. A General Approach to Solving Problems Data Representation CLAIM Theorems Analysis

3. We used this approach to justify the claim that we could construct congruent angles. RepresentationFBE, rays j & lDatacompass length BE,circle radius BEAnalysiscompass length constant = circles congruentTheoremsCongruent circles = congruent radiiSSS, CPCTC How did we know these we’re true?

4. What is a “proof”? In geometry, a proof is the justification of a statement or claim through deductive reasoning. Statements that are proven are called theorems. To complete the reasoning process, we use a variety of “tools” from our “toolbox”.

5. Toolbox Tools • Given Information (data) • Definitions often w/ respect to the representation • Postulates (axioms) • Properties (could be from algebra) • Previously proved theorems • Logic (analysis)

6. In this unit, we will be doing informal transformational proofs. In the next unit we will be doing formal proofs. Why use this approach? • More visual and intuitive; dynamic • Helpful in understanding geometry historically • 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! • Builds intuition and understanding of meaning • Generalizes to other geometries more easily

7. Key ideas of Informal Transformational ProofsWe already know these! • Uses transformations: reflections, rotations, translations and compositions of these. • 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)

8. Let’s talk through an example! link to… http://www.youtube.com/watch?v=O2XPy3ZLU7Y

9. Example: Show informally that the two triangles of a parallelogram formed by a diagonal are congruent. What do I have to write to “prove” this using transformations?

10. Using patty paper, I can see that ∆ACD maps onto ∆DBA through a rotation. (Do you see that it’s not a reflection?) What is the nature of the rotation?

11. To find the center of rotation, I connect two corresponding vertices. (∆ACD maps onto ∆DBA) I see that the center of rotation is point P. Is there anything special about P? What is the degree measure of the rotation? Use patty paper and a protractor.

12. Informal Proof(what you have to write) ∆ACD maps onto ∆DBA by R(180◦ , P) where P is the midpoint of the diagonal. So ∆ACD∆DBA

13. 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

14. Today we are going to verify that isometries do preserve distance and angles. We call this Corresponding Parts (of) Congruent Figures (are) Congruent