In the last section, you added rotations to your set of transformations, along with reflections, translations, and dilations. In this section, you will compare two different classes of transformations: (1)congruence transformations that generate an image that is congruent to the preimage and (2) non-congruence transformations that generate an image that is not congruent to the preimage. Specifically, you will use side lengths and angle measures to distinguish between which types of transformations are congruence transformations, and which types are not.

Click the image below to open an interactive sketch in a new web browser tab or window. Use the interactive to investigate the relationships among side lengths and angle measures. Follow the detailed directions beneath the image to investigate each transformation.

Translations

• Select the translate option.
• Use the slider to translate Δ ABC, and use the check boxes to reveal the side lengths and angle measures for both Δ ABC and its translated image.
• Move the red Translate Point to adjust the translation.
• Use the interactive to answer the following questions.

Interactive popup. Assistance may be required.

Need additional directions for translations?

• What do you notice about the lengths of corresponding sides for both the preimage (Δ ABC) and the translated image (Δ A'''B'''C''')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The lengths of corresponding sides are the same, meaning that corresponding sides of a translated triangle are congruent.

• What do you notice about the measures of corresponding angles for both the preimage (Δ ABC) and the translated image (Δ A'''B'''C''')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The measures of corresponding angles are the same, meaning that corresponding angles of a translated triangle are congruent.

• Is Δ ABC, the preimage, congruent to Δ A'''B'''C''', the image? How do you know?

  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. Since all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent, Δ ABC is congruent to Δ A'''B'''C'''.

Rotations

• Click the double arrow in the top right corner of the sketch to reset the sketch.
• Select the Rotate option.
• Enter the desired angle of rotation.
• Use the slider to rotate Δ ABC, and use the check boxes to reveal the side lengths and angle measures for both Δ ABC and its rotated image.
• Move the red Rotation Center point to adjust the center of rotation.
• Repeat the process for several different angles of rotation.
• Use the interactive to answer the following questions.

Interactive popup. Assistance may be required.

Need additional directions for rotations?

• What do you notice about the lengths of corresponding sides for both the preimage (Δ ABC) and the rotated image (Δ A'B'C')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The lengths of corresponding sides are the same, meaning that corresponding sides of a translated triangle are congruent.

• What do you notice about the measures of corresponding angles for both the preimage (Δ ABC) and the rotated image (Δ A'B'C')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The measures of corresponding angles are the same, meaning that corresponding angles of a translated triangle are congruent.

• Is Δ ABC, the preimage, congruent to Δ A'B'C', the image? How do you know?

  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. Since all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent, Δ ABC is congruent to Δ A'B'C'.

Reflections

• Click the double arrow in the top right corner of the sketch to reset the sketch.
• Select the Reflect option.
• Use the slider to reflect Δ ABC fully, and use the check boxes to reveal the side lengths and angle measures for both Δ ABC and its rotated image.
• Move the red Mirror Line points to adjust the line of reflection.
• Use the interactive to answer the following questions.

Interactive popup. Assistance may be required.

Need additional directions for reflections?

• What do you notice about the lengths of corresponding sides for both the preimage (Δ ABC) and the reflected image (Δ A''B''C'')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The lengths of corresponding sides are the same, meaning that corresponding sides of a translated triangle are congruent.

• What do you notice about the measures of corresponding angles for both the preimage (Δ ABC) and the reflected image (Δ A''B''C'')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The measures of corresponding angles are the same, meaning that corresponding angles of a translated triangle are congruent.

• Is Δ ABC, the preimage, congruent to Δ A''B''C'', the image? How do you know?

  Interactive popup. Assistance may be required.

  Check Your Answer

  Yes. Since all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent, Δ ABC is congruent to Δ A''B''C''.

Dilations

Use the interactive below to investigate side lengths and angle relationships for dilations. Use the slider to adjust the scale factor of the dilation, and drag point D to adjust the center of dilation. Use the interactive to answer the questions that follow.

Interactive popup. Assistance may be required.

Need additional directions for dilations?

• What do you notice about the lengths of corresponding sides for both the preimage (Δ ABC) and the dilated image (Δ A'B'C')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The lengths of corresponding sides are not the same, but they are proportional.

• What do you notice about the measures of corresponding angles for both the preimage (Δ ABC) and the reflected image (Δ A'B'C')?

  Interactive popup. Assistance may be required.

  Check Your Answer

  The measures of corresponding angles are the same, meaning that corresponding angles of a translated triangle are congruent.

• Is Δ ABC, the preimage, congruent to Δ A'B'C', the image? How do you know?

  Interactive popup. Assistance may be required.

  Check Your Answer

  No. Since all pairs of corresponding sides are proportional and all pairs of corresponding angles are congruent, Δ ABC is similar to Δ A'B'C', but they are not congruent.

Pause and Reflect

A congruence transformation is a transformation in which the preimage and image are congruent figures. Which transformation(s) are congruence transformations?

Interactive popup. Assistance may be required.

Check Your Answer

Translations, reflections, and rotations are congruence transformations.

A non-congruence transformation is a transformation in which the preimage and image are not congruent figures. Which transformation(s) are non-congruence transformations?

Interactive popup. Assistance may be required.

Check Your Answer

Dilations are non-congruence transformations.

Which transformations preserve angle measures between the preimage and the image?

Interactive popup. Assistance may be required.

Check Your Answer

Translations, reflections, rotations, and dilations all preserve angle measures.

Which transformations preserve side lengths between the preimage and the image?

Interactive popup. Assistance may be required.

Check Your Answer

Translations, reflections, and rotations preserve side lengths.

Practice

1. Which of the following properties about geometric transformations are true?
   1. Dilations preserve congruence while reflections do not.
   2. Rotations and reflections both preserve a polygon’s side lengths.
   3. Dilations and translations both preserve a polygon’s angle measures.

   Interactive popup. Assistance may be required.

   Need a hint?

   Congruent figures have corresponding sides that are congruent and corresponding angles that are congruent. Congruence transformations preserve congruence between the preimage and image.
   Interactive popup. Assistance may be required.

   Check Your Answer

   II and III only
2. Trapezoid ABCD is transformed using the transformation (x, y) → (x − 4, y + 5). Which of the following statements about ABCD and A'B'C'D' are true?
   1. ABCD is congruent to A'B'C'D'.
   2. The corresponding angles of the two trapezoids are congruent, but the corresponding sides of the two trapezoids are not.
   3. A'B'C'D' is a translation of ABCD.

   Interactive popup. Assistance may be required.

   Need a hint?

   What type of transformation is indicated by (x, y) → (x − 4, y + 5)?
   Interactive popup. Assistance may be required.

   Check Your Answer

   I and III only
3. Which of the following transformations are congruence transformations?
   1. (x, y) → (−y, x)
   2. (x, y) → (x, −y)
   3. (x, y) → (1.5x, 1.5y)

   Interactive popup. Assistance may be required.

   Need a hint?

   Translations, rotations, and reflections are congruence transformations. Which algebraic rules indicate one of these types of transformations?
   Interactive popup. Assistance may be required.

   Check Your Answer

   I and II only