Pandit Geometry Unit 3-lesson 06 and 09

Lesson 9: Conditions for Triangle Similarity
Lesson 6: Connecting Similarity and Transformations

1. Lesson 6 Connecting Similarity and Transformations Unit 3 Similarity

2. Let’s identify similar figures. Unit 3 ● Lesson 6

3. What’s wrong with this dilation? Why is GHFE not a dilation of ADCB ? Dilation Miscalculation Unit 3 ● Lesson 6 ● Activity 1

4. 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. 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. Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other. 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. Put this in you notebook

7. Are the triangles similar ? Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other. Write a similarity statement about the 2 figures, and explain how you know they are similar. 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. Are the triangles similar ? 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. Are the triangles similar ? 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. Determine if each statement must be true, could possibly be true, or definitely can’t be true. Explain or show your reasoning. Congruent figures are similar. Similar figures are congruent. Forward and Backwards? Unit 3 ● Lesson 6

11. 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. Unit 3 ● Lesson 6 I can write similarity statements. I know the definition of similarity.

13. Lesson 9 Conditions for Triangle Similarity Unit 3 Similarity

14. Let’s prove some triangles similar. Unit 3 ● Lesson 9

15. How could you justify each statement? Angle-Side-Angle As A Helpful Tool Unit 3 ● Lesson 9 ● Activity 1

16. 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. 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. 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. 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  . 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. Put this into your notebook

21. 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. How long are the sides labeled x and y ? Any Two Angles? Unit 3 ● Lesson 9 ● Activity 3

22. 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. 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. 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. Unit 3 ● Lesson 9 I can explain why the Angle-Angle Triangle Similarity Theorem works.

26. 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. 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. 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.