Lesson 4.4 Congruence and Transformations Learning
Description: Lesson 4.4 Congruence and Transformations Learning Target: Understand congruence transformations. Success Criteria: I can identify congruent figures. I can describe congruence transformations. I can identify the types of symmetry in a
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slide1. Lesson 4.4 Congruence and Transformations<br>
slide2. Learning Target:
Understand congruence transformations. Success Criteria:
I can identify congruent figures.
I can describe congruence transformations.
I can identify the types of symmetry in a tessellation.
I can use congruence transformations to solve problems.<br>
slide3. Vocabulary
congruent figures, p. 194
congruence transformation,
p. 195
tessellation, p. 196
translational symmetry, p. 196<br>
slide4. Example 1 Identifying Congruent Figures Identify any congruent figures in the
coordinate plane. Explain. Square NPQR is a translation of square ABCD
2 units left and 6 units down. So, square ABCD
and square NPQR are congruent. KLM is a reflection of EFG in the x-axis.
So, EFG and KLM are congruent. SOLUTION<br>
slide5. Congruence Transformations
Another name for a rigid motion or a combination of rigid motions is a congruence transformation because the preimage and image are congruent. The terms rigid motion and congruence transformation are interchangeable.<br>
slide6. Example 2 Describing a Congruence Transformation Two sides of ABCD rise from left to right, and the corresponding sides of EFGH fall from left to right. If you reflect ABCD in the y-axis, as shown, then the image, A′B′C′D′, will have the same orientation
as EFGH. So, a congruence transformation that maps ABCD to EFGH is a reflection in the y-axis, followed by a translation of 4 units down. SOLUTION<br>
slide7. Example 2 Describing a Congruence Transformation Check Verify that the corresponding sides and the corresponding angles are congruent. Using a protractor to estimate the angle measures, you can see that the
a
corresponding angles appear to be congruent. ✓<br>
slide8. Tessellations
A tessellation is the covering of a plane with one or more congruent figures so that there are no gaps or overlaps. A tessellation has translational symmetry when the tessellation can be mapped onto itself by a translation.<br>
slide9. Example 3 Identifying Transitional Symmetry Determine whether the tessellation has translational symmetry. If so, identify two translations that map the tessellation onto itself. SOLUTION Because a translation can map the tessellation onto itself, it has translational symmetry. Two such translations are shown below.<br>
slide10. Using Theorems about Congruence Transformations
Compositions of two reflections result in either a translation or a rotation. A composition of two reflections in parallel lines results in a translation, as described in the following theorem.<br>
slide12. Example 4 Using the Reflections in Parallel Lines Theorem a. Name any segments congruent to
each segment: GH, HB, and GA. c. What is the length of GG″? SOLUTION<br>
slide13. A composition of two reflections in intersecting lines results in a rotation, as described in the following theorem.<br>
slide14. Example 5 Using the Reflections in Intersecting Lines Theorem In the diagram, the figure is reflected in line k. The image is then reflected in line m.
Describe a single transformation that maps F to F″. SOLUTION<br>
slide2. Learning Target:
Understand congruence transformations. Success Criteria:
I can identify congruent figures.
I can describe congruence transformations.
I can identify the types of symmetry in a tessellation.
I can use congruence transformations to solve problems.<br>
slide3. Vocabulary
congruent figures, p. 194
congruence transformation,
p. 195
tessellation, p. 196
translational symmetry, p. 196<br>
slide4. Example 1 Identifying Congruent Figures Identify any congruent figures in the
coordinate plane. Explain. Square NPQR is a translation of square ABCD
2 units left and 6 units down. So, square ABCD
and square NPQR are congruent. KLM is a reflection of EFG in the x-axis.
So, EFG and KLM are congruent. SOLUTION<br>
slide5. Congruence Transformations
Another name for a rigid motion or a combination of rigid motions is a congruence transformation because the preimage and image are congruent. The terms rigid motion and congruence transformation are interchangeable.<br>
slide6. Example 2 Describing a Congruence Transformation Two sides of ABCD rise from left to right, and the corresponding sides of EFGH fall from left to right. If you reflect ABCD in the y-axis, as shown, then the image, A′B′C′D′, will have the same orientation
as EFGH. So, a congruence transformation that maps ABCD to EFGH is a reflection in the y-axis, followed by a translation of 4 units down. SOLUTION<br>
slide7. Example 2 Describing a Congruence Transformation Check Verify that the corresponding sides and the corresponding angles are congruent. Using a protractor to estimate the angle measures, you can see that the
a
corresponding angles appear to be congruent. ✓<br>
slide8. Tessellations
A tessellation is the covering of a plane with one or more congruent figures so that there are no gaps or overlaps. A tessellation has translational symmetry when the tessellation can be mapped onto itself by a translation.<br>
slide9. Example 3 Identifying Transitional Symmetry Determine whether the tessellation has translational symmetry. If so, identify two translations that map the tessellation onto itself. SOLUTION Because a translation can map the tessellation onto itself, it has translational symmetry. Two such translations are shown below.<br>
slide10. Using Theorems about Congruence Transformations
Compositions of two reflections result in either a translation or a rotation. A composition of two reflections in parallel lines results in a translation, as described in the following theorem.<br>
slide12. Example 4 Using the Reflections in Parallel Lines Theorem a. Name any segments congruent to
each segment: GH, HB, and GA. c. What is the length of GG″? SOLUTION<br>
slide13. A composition of two reflections in intersecting lines results in a rotation, as described in the following theorem.<br>
slide14. Example 5 Using the Reflections in Intersecting Lines Theorem In the diagram, the figure is reflected in line k. The image is then reflected in line m.
Describe a single transformation that maps F to F″. SOLUTION<br>