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9-6 Compositions of Reflections

9-6 Compositions of Reflections. Compositions of Reflections. If two figures are congruent, there is a transformation that maps one onto the other. If no reflection is involved, then the figures are either translation or rotation images of each other.

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9-6 Compositions of Reflections

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  1. 9-6 Compositions of Reflections

  2. Compositions of Reflections • If two figures are congruent, there is a transformation that maps one onto the other. • If no reflection is involved, then the figures are either translation or rotation images of each other. Example 1: The two figures are congruent. Is one figure a translation image of the other, a rotation image, or neither? Explain. a. b.

  3. Any translation or rotation can be expressed as the composition of two reflections. Theorem 9-1 A translation or rotation is a composition of two reflections. Theorem 9-2 A composition of reflections across two parallel lines is a translation. Theorem 9-3 A composition of reflections across two intersecting lines is a rotation.

  4. Example 2: Find the image of for the reflection across line followed by a reflection across line . Describe the resulting translation.

  5. Example 3: Lines intersect in point and form acute with measure 35. Find the image of for a reflection across line and then a reflection across line . Describe the resulting rotation.

  6. 9-4 Fundamental Theorem of Isometries • In a plane, one of two figures can be mapped onto the other by a composition of at most three reflections. • If two figures are congruent and have opposite orientations (but are not simply reflections of each other), then there is a slide and a reflection that will map one onto the other.

  7. Glide Reflection • The composition of a glide (translation) and a reflection across a line parallel to the direction of translation.

  8. Example 4: Find the image of for a glide reflection where the translation is and the reflection line is

  9. Theorem 9-5 IsometryClassification Theorem • There are only 4 isometries. They are the following:

  10. Example 5: Each figure is an isometry image of the figure. Tell whether their orientations are the same or opposite. Then classify the isometry.

  11. Example 5: Each figure is an isometry image of the figure. Tell whether their orientations are the same or opposite. Then classify the isometry.

  12. Practice • Pages 509-510 1-23 odd

  13. Homework

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