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Understanding Congruence and Similarity in Polygons: A Comprehensive Guide

This educational resource explores the concepts of congruence and similarity in polygons. It includes exercises to compare ratios using greater than, less than, or equal to signs. Additionally, it covers the essential properties of congruent polygons, angles, and segments, explaining their relationships through theorems and examples. Students will learn how to solve proportions and identify congruence and similarity in various figures through practical exercises. Perfect for enhancing understanding of geometric shapes and their properties.

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Understanding Congruence and Similarity in Polygons: A Comprehensive Guide

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  1. Exercise Compare by using >, <, or =. 912 1116 >

  2. Exercise Compare by using >, <, or =. 1218 812 =

  3. Exercise Compare by using >, <, or =. 1628 1321 <

  4. Exercise Solve the proportion. x15 1612 = x = 20

  5. 145 45 d = = 2 = 2.8 Exercise Solve the proportion. 57 2d =

  6. Congruent Polygons • Congruent polygons are polygons with the same size and shape.

  7. C • F • A • B • D • E

  8. same place in different figures • corresponding angles • corresponding sides

  9. Congruent Angles • Congruent angles are angles with the same measure.

  10. Congruent Segments • Congruent segments are segments with the same length.

  11. congruence symbol

  12. AD • BE • CF • Corresponding Angles • Corresponding angles are congruent (have the same measure).

  13. C • F • A • B • D • E

  14. ACDF • ABDE • BCEF • Corresponding Sides • Corresponding sides are congruent (have the same length).

  15. X Example 1 RSTXYZ. Complete each statement. S Y R T Z X R

  16. Y Example 1 RSTXYZ. Complete each statement. S Y R T Z X S

  17. Z Example 1 RSTXYZ. Complete each statement. S Y R T Z X T

  18. XZ Example 1 RSTXYZ. Complete each statement. S Y R T Z X RT

  19. XY Example 1 RSTXYZ. Complete each statement. S Y R T Z X RS

  20. YZ Example 1 RSTXYZ. Complete each statement. S Y R T Z X ST

  21. Similar Polygons • Similar polygons are polygons that have the same shape but not necessarily the same size. The symbol ~ means “is similar to.”

  22. Theorem • If two polygons are similar, then the corresponding angles are congruent and the lengths of the corresponding sides are proportional.

  23. AD • BE • CF B • Corresponding Angles 9 6 12 A C E 6 4 D 8 F

  24. ABDE • ACDF • BCEF B • Corresponding Sides 9 6 12 A C E 6 4 D 8 F

  25. ABDE • 6 4 • 3 2 = = • ACDF • 12 8 • 3 2 = = • BCEF • 9 6 • 3 2 = =

  26. scale factor—ratio of corresponding dimensions in similar figures

  27. Example 2 RST ~ XYZ. Use a proportion to find XY. Y S 10 15 9 X 12 Z 18 R T

  28. XZRT • XYRS = • XY9 • 2 3 = • 3 • 3 • 3(XY) = 18 XY = 6

  29. FD • ABFE = Example ABC ~ FED. Complete the ratio. D C 8 6 A F B E AC

  30. Example ABC ~ FED. If BC = 9, what is ED? D C 8 6 A F B E 12

  31. Example ABC ~ FED. If the perimeter of ABC is 30, what is the perimeter of FED?

  32. D C 8 6 A F B E 40

  33. Example ABC ~ FED. If mA = 85° and m E = 30°, what is the mC? D C 8 6 A F B E 65°

  34. Example Are PQR and JKL similar? L Q 8 6 18 J 12 P 12 8 K no R

  35. Example What length of PQ would make them similar? L Q 8 6 18 J 12 P 12 8 K 9 R

  36. Example Assume the two parallelograms are similar. 12 B C F G 9 6 A D E FG = 8

  37. Example Assume the two parallelograms are similar. 12 B C F G 9 6 A D E AE = 4

  38. Example If the diagonal AC = 15, what is the length of EG? 12 B C F G 9 6 A D E 10

  39. Example What is the perimeter of EFGD? 12 B C F G 9 6 A D E 28

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