Math 416

1. Math 416 Geometry Isometries

2. Topics Covered • 1) Congruent Orientation – Parallel Path • 2) Isometry • 3) Congruent Relation • 4) Geometric Characteristic of Isometry • 5) Composite • 6)Geometry Properties • 7) Pythagoras – 30 - 60

3. Congruent Figures • Any two figures that are equal in every aspect are said to be congruent • Equal is every aspect means… • All corresponding angles • All corresponding side lengths • Areas • Perimeters

4. Congruent Figures • We also note that we are talking about any figures in the plane not just triangles • However, it seems in most geometry settings, we deal with triangles • We hope this section will allow you to look at all shapes but…

5. Orientation • One of the most important characteristics of shapes in the plane is its orientation • How the shape is oriented means the order that corresponding point appear

6. Orientation • Consider A A’ #1 #2 B C B’ C’

7. Orientation • To establish the order of the points, we need two things; • #1) A starting point – that is a corresponding point • #2) A direction – to establish order • “Consistency is the core of mathematics” • I will choose A and A’ as my starting points

8. Orientation • I will choose counterclockwise as my direction • Hence in triangle #1 we have A -> B > C. • In Triangle #2 we have A’ -> B’ -> C’

9. Orientation • Since the corresponding points match, we say the two figures have the same orientation. • Consider A A’ #1 #2 B C C’ B’

10. Orientation Vocabulary • These figures do not have the same orientation • Same orientation can be phrased as follows; – orientation is preserved - orientation is unchanged - orientation is constant

11. Orientation Vocabulary • Different Orientation can be stated - orientation is not preserved - orientation is changed - orientation is not constant

12. Parallel Path • We are interested how one congruent figure gets to the other • We are interested how one congruent figure is transformed into another • We call the line joining corresponding points its path • i.e. A A’ is the path • If we look at all the paths between corresponding points, we can determine if all the paths are parallel.

13. Examples A’ A B’ C’ B C These are a parallel path

14. Examples A’ A C’ B’ B C These are not parallel paths It is called Intersecting Paths

15. Types of Isometries • There are 4 Isometries 1) Translation 2) Rotation 3) Reflection 4) Glide Reflection

16. Translation • Translation – moving points of a figure represented by the letter t. • As you may recall t (-2,4) (x – 2, y + 4) You move on the x axis minus 2 and on the y axis you move plus 4.

17. Rotations • Rotations: Rotations can be either 90, 180, 270 or 360 degrees. • Rotations can be clockwise or counter-clockwise • Represented by the letter r

18. Reflection • You can have reflections of x • You can have a reflections of y

19. Glide Reflection • Glide reflection occurs when the orientation is not preserved AND does not have a parallel path. • Can be best seen with examples…

20. Tree Diagram • We can define the four isometries by the way of these two characteristics Orientation Same?Parallel Path? TRANSLATION YES YES No ROTATION YES REFLECTION No GLIDE REFLECTION No

21. Table Representation

22. Notes • The biggest problem is establishing corresponding points. • It is easy when they tell you AA’, BB’ but it is usually not the case  • Let’s try two examples… what kind of isometric figures are these… • You may choose to cut up the figure on a piece of paper which can help locate the points…

23. Example #1 • Consider (we assume they are congruent) • We need to establish the points. Look for clues (bigger, 90 and smaller angle). Bigger Angle Bigger Angle 90° Smaller Angle 90° Smaller Angle

24. Which Isometric Figure? ORIENTATION? PARALLEL PATH? • Hence orientation ABC A’C’B’ are NOT the same… • Parallel paths… No! A A’ B’ B C C’ GLIDE REFLECTION

25. Example #2 B’ A C’ B C A’ ORIENTATION? PARALLEL PATHS? ABC and A’B’C’ – Orientation the same Not Parallel Paths ROTATION

26. Other Figures • When the figure is NOT a triangle, you can usually get away with just checking three points. The hard part is finding them. Let’s take a look at two more examples

27. Example with a Square A ° Glide Reflection B ° C A’ C’ B’ Orientation / Parallel Paths? Orientation Changed, Not Parallel

28. Practice 90o counter clockwise rotation ° ° Orientation? Parallel? Orientation Same; Not Parallel  Rotation

29. The Congruency Relation • When we know two shapes are congruent (equal), we use the symbol. CongruentSymbol

30. Congruency Relation • Hence if we say HGIJ KLMN • We note • H corresponds to K • G corresponds to L • I corresponds to M • J corresponds to N

31. Congruency Relation • From this we state the following equalities. • Line length • HG = KL (1st two) • GI = LM (second two) • IJ = MN (last two) • HJ = KN (outside two)

32. Congruency Relation • Angles • < HGI = < KLM (1st two) • < GIJ = < LMN (second two) • < IJH = < MNK (last two first) • < JHG = < NKL (last one 1st two) • We have established all this without seeing the figure!

33. Exam Question • State the single isometry. State the congruency relation and the resulting equalities. M A B K L N D C Hence BACD KMNL

34. Exam Question • We can also can note that… • B  K • D  L • C  N • A  M Clockwise • Orientation / Parallel Path?

35. Exam Solution • Orientation Changed • Parallel Path • Reflection

36. Line Length BA = KM AC = MN CD = NL DB = LK Angles < BAC = < KMN < ACD = < MNL < CDB = < NLK < DBA = < LKM Other Findings

37. Test Question Given ABCDE FGHIJ True or False? A E B D C • You should draw a diagram to clarify… F J G I H < ABC = HIJ False < ABC = HGF True False BC = HI

38. Two Isometries – Double the fun! • At certain points, we may impose more than one isometry. • Consider 3 1 2 Math #1 Math #1 Math #1 We say 1  2 is a reflection of s 2  3 is a rotation r

39. Notes • We would say that the composite is r ° s after We can say there is a rotation after a reflection. So you should read from right to left

40. Notes • We also note that 1 – 3 is a glide reflection (gr) • Hence r ° s = gr

41. Practice • Consider 2 3 1 Math is fun Math is fun Math is fun 1  2 t 2  r Thus r ° t = r

42. Geometry RemindersComplimentary Angles • Here are some reminders of things you should know. Complimentary angles add up to 90o. Thus <a + <b = 90o b a

43. Supplementary Angles Supplementary angles add up to 180o. All straight lines form an angles of 180o. Thus <a + < b = 180o a b

44. Vertically Opposite Angles Vertically opposite angles are equal. Thus <a = <c and <b = <d a d b c

45. Isoscelles Triangles The angles opposite the equal sides are equal or vice versa x x

46. Angles in a Triangle a Angles in a triangle add up to 180o. Thus <a + <b + <c = 180o. b c

47. Parallel Lines Transversal Line When a line (transversal) crosses two parallel lines, four angles are created at each line a b c d w x y z

48. Parallel Lines • The following relationship between each group is created. • Alternate Angles - both inside (between lines) & the opposite side of tranversal are EQUAL. Thus, < c = < x < d = < w a b c d w x y z

49. Corresponding Angles • Both same side of tranversal one between parallel lines the other outside parallel lines are EQUAL • <a = <w • <c <y • <b = < x • <d = <z a b c e d w x y z <b & <e are called alternate interior angle

50. Supplemental Angles • Both same side of transversal • Both between parallel lines • Add up to 180° • Therefore, <c + <w = 180° • <d + <x = 180°