Download Transformational Proof: Informal and Rigorous

Transformational Proof:
Informal and Formal
Kristin A. Camenga
[email protected]
Houghton College
November 12,2009
A General Approach to
Solving Problems
Data
Representation
CLAIM
Analysis
Theorems
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 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
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.
(∆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
.
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 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
Today we are going to verify that
isometries do preserve distance
and angles. We call this
Corresponding
Parts (of)
Congruent
Figures (are)
Congruent