Pandit Geometry Unit 3-lesson 06 and 09
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This geometry lesson covers similarity and transformations. It begins with a warm-up activity identifying errors in a dilation. Students then sort transformation cards and write sequences of transformations. They practice identifying similar triangles using the Angle-Angle Similarity Theorem. The lesson emphasizes that similar figures have a sequence of rigid motions and dilations that map one onto the other. It introduces the definition of similarity and provides examples proving triangles similar based on angle and side relationships.
Pandit Geometry Unit 3-lesson 06 and 09
- 1. GEOMETRY Lesson 6 Connecting Similarity and Transformations Unit 3 Similarity
- 2. Learning Goal Geometry Let’s identify similar figures. Unit 3 ● Lesson 6
- 3. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up What’s wrong with this dilation? Why is GHFE not a dilation of ADCB? Dilation Miscalculation Unit 3 ● Lesson 6 ● Activity 1
- 4. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up What’s wrong with this dilation? Why is GHFE not a dilation of ADCB? ● noticed the figures are not the same shape ● noticed the sides are not scaled by the same scale factor ● noticed that and are not on rays and ● Dilation Miscalculation Unit 3 ● Lesson 6 ● Activity 1
- 5. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. 1. Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories. 2. Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other. 3. For all the cards that could include a dilation, what scale factor is used to go from Figure F to Figure G? What scale factor is used to go from Figure G to Figure F? Suggestion ● figures that are similar ● figures that are congruent ● figures that are neither congruent nor similar Card Sort: Not-So-Rigid Transformations Unit 3 ● Lesson 6 ● Activity 2
- 6. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. ● Put this in you notebook
- 7. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Are the triangles similar? 1. Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other. 2. Write a similarity statement about the 2 figures, and explain how you know they are similar. 3. Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement? Alphabet Soup Unit 3 ● Lesson 6 ● Activity 2
- 8. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Are the triangles similar? 1. Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other. Translate figure ABE by the directed line segment from A to Q . Dilate figure A’B’E’ using center A and scale factor 0.4. Alphabet Soup Unit 3 ● Lesson 6 ● Activity 2
- 9. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Are the triangles similar? 1. Write a similarity statement about the 2 figures, and explain how you know they are similar. ABE ~ QRT or BEA ~ RTQ. or . The figures are similar because a sequence of rigid motions and dilations takes one onto the other. Alphabet Soup Unit 3 ● Lesson 6 ● Activity 2
- 10. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Lesson Synthesis Determine if each statement must be true, could possibly be true, or definitely can’t be true. Explain or show your reasoning. 1. Congruent figures are similar. 2. Similar figures are congruent. Forward and Backwards? Unit 3 ● Lesson 6
- 11. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Glossary One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second. Triangle A’B’C’ is similar to triangle ABC because a rotation with center B followed by a dilation with center P takes ABC to A’B’C’. Unit 3 ● Lesson 6 similar
- 12. Learning Targets Geometry Unit 3 ● Lesson 6 ● I can write similarity statements. ● I know the definition of similarity.
- 13. GEOMETRY Lesson 9 Conditions for Triangle Similarity Unit 3 Similarity
- 14. Learning Goal Geometry Let’s prove some triangles similar. Unit 3 ● Lesson 9
- 15. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up: Math Talk How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1
- 16. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up: Math Talk How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1 By translating and rotating to take triangle P’Q’R’ onto triangle STU. By the Angle-Side-Angle Triangle Congruence Theorem. By the Side-Angle-Side Triangle Congruence Theorem.
- 17. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up: Math Talk How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1 By dilating triangle PQR by scale factor 1/2 , and then translating and rotating to take triangle P’Q’R’ onto triangle STU . By dilating triangle PQR by scale factor 1/2, and then knowing QR is the same length as UT, so P’Q’R’ is congruent to STU by the Angle-Side-Angle Triangle Congruence Theorem. Congruent figures can be taken onto each other using rigid motions, so there is a dilation and sequence of rigid motions that takes PQR onto STU.
- 18. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up: Math Talk How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1 By translating and rotating to take triangle G’H’I’ onto triangle MNO. By the Angle-Side-Angle Triangle Congruence Theorem.
- 19. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Warm-up: Math Talk How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1 Dilate triangle GHI by scale factor 𝑀𝑁 𝐺𝐻 . Then G’H’ is the same length as MN, so G’H’I’ is congruent to MNO by the Angle-Side-Angle Triangle Congruence Theorem. G’H’I’ is a dilation of GHI, so there is a dilation and sequence of rigid motions that takes GHI onto MNO, and therefore, GHI is similar to MNO .
- 20. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Put this into your notebook
- 21. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle. 1. Explain how you know the triangles are similar. 2. How long are the sides labeled x and y? Any Two Angles? Unit 3 ● Lesson 9 ● Activity 3
- 22. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle. Explain how you know the triangles are similar. I know the angles of a triangle sum to 180 degrees, so I figured out the missing angle in each triangle. Both triangles have angles of 40, 60, and 80 degrees, so they must be similar by the Angle-Angle Triangle Similarity Theorem. Any Two Angles? Unit 3 ● Lesson 9 ● Activity 3
- 23. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Here are 2 triangles. One triangle has a 60 degree angle and a 40 degree angle. The other triangle has a 40 degree angle and an 80 degree angle. How long are the sides labeled x and y? x is 4.8 units long and y is 10 units long Any Two Angles? Unit 3 ● Lesson 9 ● Activity 3
- 24. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Lesson Synthesis The main ideas to draw out of this lesson are: ● the Angle-Angle Triangle Similarity Theorem: if two pairs of corresponding angles in triangles are congruent, then the triangles are similar ● Knowing the measures of any two angles from one triangle, and any two angles of the other triangle, is enough information to determine if the Angle-Angle Triangle Similarity Theorem can be used. The given angles do not have to be corresponding. This uses the Triangle Sum Theorem (angle A + angle B + angle C = 180 degrees). Conditions for Triangle Similarity Unit 3 ● Lesson 9
- 25. Learning Targets Geometry Unit 3 ● Lesson 9 I can explain why the Angle- Angle Triangle Similarity Theorem works.
- 26. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Lesson Synthesis Cool-down Priya noticed in the last activity that between the 2 triangles, you only need to know 4 angles to show that they are similar. She wondered which fourth angle would work to prove that triangle RST is similar to triangle EFG. Draw a sketch of the triangles. Then pick one angle measurement that would prove the triangles are similar. Explain to Priya why knowing that angle would be enough. Any Four Angles? Unit 3 ● Lesson 9 ● Activity 4 In triangle RST: ● Angle R is 90° ● Angle S is 25° ● Angle T is x° In triangle EFG: ● Angle E is 90° ● Angle F is y° ● Angle G is z°
- 27. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Lesson Synthesis Cool-down Priya noticed in the last activity that between the 2 triangles, you only need to know 4 angles to show that they are similar. She wondered which fourth angle would work to prove that triangle RST is similar to triangle EFG. If you knew , you would know the triangles were similar by the Angle-Angle Triangle Similarity Theorem. If you knew , you would know the triangles were similar by the Triangle Angle Sum Theorem and the Angle-Angle Triangle Similarity Theorem. Any Four Angles? Unit 3 ● Lesson 9 ● Activity 4 In triangle RST: ● Angle R is 90° ● Angle S is 25° ● Angle T is x° In triangle EFG: ● Angle E is 90° ● Angle F is y° ● Angle G is z°
- 28. Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Illustrative Mathematics. Glossary One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second. Triangle A’B’C’ is similar to triangle ABC because a rotation with center B followed by a dilation with center P takes ABC to A’B’C’. Unit 3 ● Lesson 9 similar
- 29. This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by- nc/4.0/.This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/. All curriculum excerpts are under the following licenses: IM 9–12 Math is copyright 2019 by Illustrative Mathematics. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). This material includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information. The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
- 30. This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by- nc/4.0/.This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/. All curriculum excerpts are under the following licenses: IM 9–12 Math is copyright 2019 by Illustrative Mathematics. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). This material includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information. The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
Editor's Notes
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- #27: If you knew , you would know the triangles were similar by the Angle-Angle Triangle Similarity Theorem. If you knew , you would know the triangles were similar by the Triangle Angle Sum Theorem and the Angle-Angle Triangle Similarity Theorem.
- #28: If you knew , you would know the triangles were similar by the Angle-Angle Triangle Similarity Theorem. If you knew , you would know the triangles were similar by the Triangle Angle Sum Theorem and the Angle-Angle Triangle Similarity Theorem.