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Triangles: Ratios and Similarity Theorems

In this review sheet, learn how to find proportions and solve using cross-multiplication, apply Triangle Similarity Theorems, and use similar triangle proofs. Also, discover tips for problem-solving and test-taking.

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Triangles: Ratios and Similarity Theorems

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  1. Review Sheet Chapter Seven Ratios: To get a proportion, we must set up a ratio that has the corresponding parts (parts from the same triangle have to be in either the top or the bottom). Solve using cross-multiplication. Triangle Similarity Theorems: All similar triangles must have their corresponding angles congruent !! All sides must have the same scaling factor with their corresponding side AA – (includes ASA and AAS) – if two angles are congruent in a triangle then the third angle must be congruent SAS – sides must have the same scaling factor (be in the same ratio); included angle SSS – all sides must have the same scaling factor Proofs: Use similar steps to congruent triangle proofs. Need to show angles congruent (parallel lines, vertical angles, etc) and sides having the same ratio (scaling factor) Similar triangles (or figures) problem solving: 1) Draw a picture of triangles, if you are not given one or if the picture given is too complex 2) Find corresponding parts (angles must be congruent and order still rules!) 3) Set up a proportion; make sure the tops (and bottoms) come from the same triangles! 4) Solve using cross multiplication 5) Check answer to make sure it makes sense Test Taking Tips: Check your answer and make sure that it makes sense in the picture If the figure is smaller, then the corresponding part must be smaller than the given piece of the larger a b --- = --- ad = cb a and b must come from same triangle (c and d from other) ! c d

  2. Similar triangles: 30 36 ----- = ----- 15 x 30 x = 540 x = 18 • SSM: • Scaling factor is 1/2 • ED matches to 36 (order rules) • ED = ½(36) = 18

  3. And any other triangle option

  4. Triangles and Logic • SSM: • Read the equations and see which look right Which angles match up? A  T, C  C and B  R look for ratios that match corresponding sides and are consistent – all one triangle on top and the other triangle on the bottom

  5. Coordinate Relations and Transformations • SSM: • measure the 30 side • measure the 105 side • Not to scale! • Answer closest # to 105 since 28 is close to 30 set up similar triangles: 28 ST ---- = ----- 30 ST = 2940 ST = 98 30 105

  6. Ch 7 Coordinate Relations and Transformations • SSM: • isosceles • Eliminate C Isosceles triangle with legs bigger than the base. Only triangles A and B satisfy that. Scaling factor of ¾ only fits triangle A completely.

  7. Ch 7 Coordinate Relations and Transformations • SSM: • draw figure • draw lines of symmetry Similar triangle proofs (AA, SAS and SSS) CBD and ABE have right angles and a 2:1 ratio between long and short legs of the right triangle

  8. Ch 7 Coordinate Relations and Transformations • SSM: • order rules • shared angle R Similarity theorems (AA, SAS, SSS) with shared angle R, either another corresponding angle or the sides on both sides of angle R. Options A and B do not fit order rules. Option D has the correct sides.

  9. Ch 7 Triangles • SSM: • no help A  Q C  S Only AA similarity has two relationships for similarity. Order rules!

  10. Ch 7 Triangles • SSM: • no help Triangle similarity theorems are AA, SAS and SSS. With no side information given, the later two are tougher to prove. Hidden feature of “bowtie” gives us vertical angles and a second angle in choice A.

  11. Ch 7 Triangles • SSM: • look for scaling factor Need to find a scaling factor (3) that each of the sides of the given triangle is multiplied and we get one of the answers Answer C is another Pythagorean triple, but not in right proportion.

  12. Ch 7 Polygons, Circles, and Three-Dimensional Figures • SSM: • slightly less than 1 to 1 • less than 55 Need to set up a proportion to solve for model’s width 555.5 55 --------- = ------- 505(55) = 555.5x 27775 = 555.5x 505 x 50 = x

  13. AB // CD Given A  D Alt Interior Angles B  C Alt Interior Angles AFB  DFC Vertical Angles ABF ~ DCF AA Similarity

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