Lesson 21: Correspondence and Transformations
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Lesson 21 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 21: Correspondence and Transformations Student Outcomes ο§ Students practice applying a sequence of rigid motions from one figure onto another figure in order to demonstrate that the figures are congruent. Lesson Notes In Lesson 21, students consolidate their understanding of congruence in terms of rigid motions with knowledge of corresponding vertices and sides of triangles. They identify specific sides and angles in the pre-image of a triangle that map onto specific angles and sides of the image. If a rigid motion results in every side and every angle of the pre-image mapping onto every corresponding side and angle of the image, they assert that the triangles are congruent. Classwork Opening Exercise (6 minutes) Opening Exercise The figure to the right represents a rotation of β³ π¨π©πͺ ππ° around vertex πͺ. Name the triangle formed by the image of β³ π¨π©πͺ. Write the rotation in function notation, and name all corresponding angles and sides. Corresponding Angles Corresponding Sides ∠π¨ → ∠π¬ ∠π© → ∠π ∠πͺ → ∠πͺ Μ Μ Μ Μ → π¬π Μ Μ Μ Μ π¨π© Μ Μ Μ Μ → ππͺ Μ Μ Μ Μ π©πͺ Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ π¬πͺ β³ π¬ππͺ πΉπ«,ππ° (β³ π¬ππͺ) Discussion (5 minutes) Discussion In the Opening Exercise, we explicitly showed a single rigid motion, which mapped every side and every angle of β³ π¨π©πͺ onto β³ π¬ππͺ. Each corresponding pair of sides and each corresponding pair of angles was congruent. When each side and each angle on the pre-image maps onto its corresponding side or angle on the image, the two triangles are congruent. Conversely, if two triangles are congruent, then each side and angle on the pre-image is congruent to its corresponding side or angle on the image. Lesson 21: Correspondence and Transformations This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 170 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 21 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Example (8 minutes) Example Μ Μ Μ Μ is one diagonal of the square. β³ π¨π©πͺ is a reflection of β³ π¨π«πͺ across segment π¨πͺ. Complete π¨π©πͺπ« is a square, and π¨πͺ the table below, identifying the missing corresponding angles and sides. A B D a. Corresponding Angles Corresponding Sides ∠π©π¨πͺ → ∠π«π¨πͺ ∠π¨π©πͺ → ∠π¨π«πͺ Μ Μ Μ Μ → π¨π« Μ Μ Μ Μ π¨π© Μ Μ Μ Μ π©πͺ → Μ Μ Μ Μ π«πͺ ∠π©πͺπ¨ → ∠π«πͺπ¨ Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ π¨πͺ C Are the corresponding sides and angles congruent? Justify your response. Since a reflection is a rigid transformation, all angles and sides maintain their size. b. Is β³ π¨π©πͺ ≅ β³ π¨π«πͺ? Justify your response. Yes, since β³ π¨π«πͺ is a reflection of β³ π¨π©πͺ, they must be congruent. Exercises (20 minutes) Exercises Each exercise below shows a sequence of rigid motions that map a pre-image onto a final image. Identify each rigid motion in the sequence, writing the composition using function notation. Trace the congruence of each set of corresponding sides and angles through all steps in the sequence, proving that the pre-image is congruent to the final image by showing that every side and every angle in the pre-image maps onto its corresponding side and angle in the image. Finally, make a statement about the congruence of the pre-image and final image. 1. Sequence of Rigid Motions (2) rotation, translation x A’’ Composition in Function Notation π»ββββββββββββ (πΉπ,ππ° (β³ π¨π©πͺ)) π©′ π©′′ Sequence of Corresponding Sides Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ → π¨′′π©′′ π¨π© Μ Μ Μ Μ → Μ Μ Μ Μ Μ Μ Μ π©πͺ π©′′πͺ′′ Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ Μ Μ Μ π¨′′πͺ′′ Sequence of Corresponding Angles π¨ → π¨′′ π© → π©′′ πͺ → πͺ′′ Triangle Congruence Statement β³ π¨π©πͺ ≅β³ π¨′′π©′′πͺ′′ A’ Lesson 21: Correspondence and Transformations This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 171 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 21 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 2. Sequence of Rigid Motions (3) reflection, translation, rotation Composition in Function Notation πΉπ¨′′,πππ° (π»ββββββββββββ (πΜ Μ Μ Μ π«π¬ (β³ π¨π©πͺ))) π©′ π©′′ Sequence of Corresponding Sides Μ Μ Μ Μ π¨π© → Μ Μ Μ Μ Μ Μ Μ Μ Μ π¨′′′π©′′′ Μ Μ Μ Μ → Μ Μ Μ Μ Μ Μ Μ Μ Μ π©πͺ π©′′′πͺ′′′ Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ Μ Μ Μ Μ Μ π¨′′′πͺ′′′ Sequence of Corresponding Angles π¨ → π¨′′′ π© → π©′′′ πͺ → πͺ′′′ Triangle Congruence Statement β³ π¨π©πͺ ≅ β³ π¨′′′π©′′′πͺ′′′ Sequence of Rigid Motions (3) reflections Composition in Function Notation ππΏπ Μ Μ Μ Μ ( πΜ Μ Μ Μ Μ Μ Μ Μ Μ Μ (β³ π¨π©πͺ))) π©π¨′ ( ππ©πͺ Sequence of Corresponding Sides Μ Μ Μ Μ → ππΏ Μ Μ Μ Μ π¨π© Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ ππ Μ Μ Μ Μ → πΏπ Μ Μ Μ Μ π©πͺ Sequence of Corresponding Angles π¨→π π©→πΏ πͺ→π Triangle Congruence Statement β³ π¨π©πͺ ≅β³ ππΏπ 3. Closing (1 minute) ο§ When each side and each angle of a triangle on the pre-image maps onto its corresponding side or angle on the image, the two triangles are congruent. Conversely, if two triangles are congruent, then each side and angle on the pre-image is congruent to its corresponding side or angle on the image. Exit Ticket (5 minutes) Lesson 21: Correspondence and Transformations This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 172 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 21 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name ___________________________________________________ Date____________________ Lesson 21: Correspondence and Transformations Exit Ticket Complete the table based on the series of rigid motions performed on β³ π΄π΅πΆ below. A Sequence of Rigid Motions (2) X C' B A' Composition in Function Notation C'' B' B'' Y A'' Sequence of Corresponding Sides Sequence of Corresponding Angles Triangle Congruence Statement Lesson 21: Correspondence and Transformations This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 173 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 21 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions Complete the table based on the series of rigid motions performed on β³ π¨π©πͺ below. A Sequence of Rigid Motions (2) rotation, reflection Composition in Function Notation (ππΏπ Μ Μ Μ Μ (πΉπͺ,ππ°Λ )) Sequence of Corresponding Sides Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ → π¨′′π©′′ π¨π© Μ Μ Μ Μ → Μ Μ Μ Μ Μ Μ Μ π©πͺ π©′′πͺ′′ Μ Μ Μ Μ π¨πͺ → Μ Μ Μ Μ Μ Μ Μ π¨′′πͺ′′ Sequence of Corresponding Angles ∠π¨ → ∠π¨′ ∠π© → ∠π©′ ∠πͺ → ∠πͺ′ Triangle Congruence Statement β³ π¨π©πͺ ≅β³ π¨′′π©′′πͺ′′ X C' B A' C'' B' B'' Y A'' Problem Set Sample Solutions 1. Exercise 3 mapped β³ π¨π©πͺ onto β³ ππΏπ in three steps. Construct a fourth step that would map β³ ππΏπ back onto β³ π¨π©πͺ. Construct an angle bisector for the ∠π¨π©π, and reflect β³ ππΏπ over that line. 2. Explain triangle congruence in terms of rigid motions. Use the terms corresponding sides and corresponding angles in your explanation. Triangle congruence can be found using a series of rigid motions in which you map an original or pre-image of a figure onto itself. By doing so, all the corresponding sides and angles of the figure map onto their matching corresponding sides or angles, which proves the figures are congruent. Lesson 21: Correspondence and Transformations This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 174 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
