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## PowerPoint Slideshow about 'GBK Geometry' - giulio

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### GBK Geometry

Jordan Johnson

Today’s plan

- Greeting
- Hand in Problem Write-up
- Lesson: Isometries, Congruence, and Symmetry
- Study Guide
- Clean-up

Isometries / Rigid Motions

- A transformation is an isometry (or rigid motion) iff it preserves distances and angles.

Example Isometry Proof:Rotation Preserves Distance

Example Isometry Proof:Rotation Preserves Distance

Example Isometry Proof:Rotation Preserves Distance

Definitions

- Transformations definition of congruence:
- Two figures are congruent iff there is an isometry in which one is the image of the other.

Definition

- A glide reflection is the composite of a translation and a reflection in a line parallel to the direction of the translation.
- In other words: Reflect, then translate, or vice versa.

2-fold Rotational Symmetries

3-fold Rotational Symmetry

Flag of the Isle of Man

Translational Symmetry

Other neat examples: autologlyphs and ambigrams.

Translational Symmetry

Summary

- Each isometry has an associated symmetry:
- Reflection symmetry with respect to a line:
- Figure coincides with its reflection image through the line

- Rotation symmetry:
- Figure coincides with its image when rotated less than 360°
- n-fold symmetry, if the smallest such rotation is 360°⁄n

- Translation symmetry:
- Figure coincides with its translation image

- Reflection symmetry with respect to a line:

Homework

- Problem for final
- Study guide
- Any make-up portfolio work
- Recommended review for Unit 5:
- Proof problems:
- Two reflections across parallel lines= translation perpendicular to the lines of reflection.
- Prove that the 3 transformations are isometries.

- Asgs #37-38.

- Proof problems:

Clean-up / Reminders

- Pick up all trash / items.
- Push in chairs (at front and back tables).
- See you tomorrow!

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