IntroductionRigid motions can also be called congruency transformations. A congruency transformation moves a geometric figure but keeps the same size and shape. Preimages and images that are congruent are also said to be isometries. If a figure has undergone a rigid motion or a set of rigid motions, the preimage and image are congruent. When two figures are congruent, they have the same shape and size. Remember that rigid motions are translations, reflections, and rotations. Non-rigid motions are dilations, stretches, and compressions. Non-rigid motions are transformations done to a figure that change the figure’s shape and/or size.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Key Concepts• To decide if two figures are congruent, determine if

the original figure has undergone a rigid motion or set of rigid motions.

• If the figure has undergone only rigid motions (translations, reflections, or rotations), then the figures are congruent.

• If the figure has undergone any non-rigid motions (dilations, stretches, or compressions), then the figures are not congruent. A dilation uses a center point and a scale factor to either enlarge or reduce the figure. A dilation in which the figure becomes smaller can also be called a compression. 2

1.4.2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued• A scale factor is a multiple of the lengths of the sides

from one figure to the dilated figure. The scale factor remains constant in a dilation.

• If the scale factor is larger than 1, then the figure is enlarged.

• If the scale factor is between 0 and 1, then the figure is reduced.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued• To calculate the scale factor, divide the length of the

sides of the image by the lengths of the sides of the preimage.

• A vertical stretch or compression preserves the horizontal distance of a figure, but changes the vertical distance.

• A horizontal stretch or compression preserves the vertical distance of a figure, but changes the horizontal distance.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued• To verify if a figure has undergone a non-rigid motion,

compare the lengths of the sides of the figure. If the sides remain congruent, only rigid motions have been performed.

• If the side lengths of a figure have changed, non-rigid motions have occurred.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued

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1.4.2: Defining Congruence in Terms of Rigid Motions

Non-Rigid Motions: Dilations

Enlargement/reduction Compare with . The size of each side changes by a constant scale factor. The angle measures have stayed the same.

Key Concepts, continued

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1.4.2: Defining Congruence in Terms of Rigid Motions

Non-Rigid Motions: Vertical Transformations

Stretch/compression Compare with . The vertical distance changes by a scale factor. The horizontal distance remains the same. Two of the angles have changedmeasures.

Key Concepts, continued

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1.4.2: Defining Congruence in Terms of Rigid Motions

Non-Rigid Motions: Horizontal Transformations

Stretch/compression Compare with . The horizontal distance changes by a scale factor. The vertical distance remains the same. Two of the angles have changedmeasures.

Guided Practice

Example 1Determine if the two figures to the right are congruent by identifying the transformations that havetaken place.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued

2. Identify the transformations that have occurred.The orientation has changed, indicating a rotation or a reflection.

The second triangle is a mirror image of the first, but translated to the right 4 units.

The triangle has undergone rigid motions: reflection and translation (shown on the next slide).

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued

3. State the conclusion.The triangle has undergone two rigid motions: reflection and translation. Rigid motions preserve size and shape. The triangles are congruent.

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1.4.2: Defining Congruence in Terms of Rigid Motions

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Guided Practice

Example 2Determine if the two figuresto the right are congruent by identifying the transformations that havetaken place.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued

2. Identify the transformations that have occurred.The orientation has stayed the same, indicating translation, dilation, stretching, or compression. The vertical and horizontal distances have changed. This could indicate a dilation.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued

3. Calculate the scale factor of the changes in the side lengths.Divide the image side lengths by the preimage side lengths.

The scale factor is constant between each pair of sides in the preimage and image. The scale factor is 2, indicating a dilation. Since the scale factor is greater than 1, this is an enlargement.

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued

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1.4.2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued

4. State the conclusion.The triangle has undergone at least one non-rigid motion: a dilation. Specifically, the dilation is an enlargement with a scale factor of 2. The triangles are not congruent because dilation does not preserve the size of the original triangle.

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1.4.2: Defining Congruence in Terms of Rigid Motions

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