Transformational Proof: Informal and Formal

Kristin A. Camenga [email protected]

Houghton College November 12,2009

A General Approach to Solving Problems

Data Representation CLAIM Theorems Analysis

We used this approach to justify the claim that we could construct congruent angles.

Representation

 FBE, rays j & l

Data

compass length BE, circle radius BE

Analysis

compass length constant = circles congruent

Theorems

Congruent circles = congruent radii SSS, CPCTC How did we know these we’re true?

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”.

Toolbox Tools

• • • • • • Given Information (

data

) Definitions often w/ respect to the representation Postulates (axioms) Properties (could be from algebra) Previously proved theorems Logic (analysis)

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 equals DE.”

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

This is the idea of a transformation!

Builds intuition and understanding of meaning Generalizes to other geometries more easily

Key ideas of Informal Transformational Proofs We 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)

Let’s talk through an example!

link to…

http://www.youtube.com/watc h?v=O2XPy3ZLU7Y

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?

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?

To find the center of rotation, I connect two corresponding vertices. (∆A C D maps onto ∆D B A ) 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 .

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

Example: Parallelograms (Rigorous) • • • Given: Parallelogram ABDC Draw diagonal AD and let P be the midpoint of AD.

Rotate the figure 180 ⁰ 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

Today we are going to verify that isometries do preserve distance and angles.

We call this C orresponding P arts (of) C ongruent F igures (are) C ongruent