[G.CO.6] Congruence #6

Objective

Common Core Text:

  • Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Said Differently:

    • Perform transformations

    • Determine if two figures are congruent to each other

Example

http://en.wikipedia.org/wiki/Congruence_(geometry)#mediaviewer/File:Congruence_en.gif

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Explanation

Performing transformations is discussed in [G.CO.5]

Congruence:

  • According to Wikipedia,

So two figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.

  • Said differently, shapes are congruent if they are the same size and shape. If the shapes are reflections, that's also congruence.

  • Also, since rigid motions are transformations that don't change the size or shape of an object, we can also say that two figures are congruent if you can make one from the other using rigid motions

    • Remember, rigid motions are translation, rotation, and reflection.

The symbol for congruence is ≅

Example 1

Are these 2 shapes congruent? Give your answer in terms of rigid motions.

Answer: Yes.

  • We can create A'B'C'D' by translating ABCD down 3 and right 1. Translation is a rigid motion, so the shapes are congruent. ABCD ≅ A'B'C'D

Example 2

Are these 2 shapes congruent? Give your answer in terms of rigid motions.

Answer: Yes.

  • We can create "other" by

    • First, rotating ABCD 90° clockwise

    • Then, reflecting it across the y-axis.

  • Rotation and reflection are rigid motions, so the shapes are congruent. ABCD ≅ Other

Example 3

Are these 2 shapes congruent? Give your answer in terms of rigid motions.

Answer: Yes.

  • We can create A'B'C'D' by rotating ABCD 90° clockwise. Rotation is a rigid motion, so the two shapes are congruent. ABCD ≅ A'B'C'D

Example 1

Are these 2 shapes congruent? Give your answer in terms of rigid motions.

Answer: No

    • The shape has been stretched out horizontally, so it is no longer the same shape. Horizontal stretch is not a rigid motion. A'B'C'D' can not be created from ABCD by translation, rotation, or reflection, so the two shapes are not congruent