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14.1 Ratio & Proportion

14.1 Ratio & Proportion. The student will learn about:. similar triangles. 1. 1. D. E. F. Triangle Similarity. Definition . If the corresponding angles in two triangles are congruent, and the sides are proportional, then the triangles are similar. A. B. C. D. E. F. AAA Similarity.

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14.1 Ratio & Proportion

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  1. 14.1 Ratio & Proportion The student will learn about: similar triangles. 1 1

  2. D E F Triangle Similarity Definition. If the corresponding angles in two triangles are congruent, and the sides are proportional, then the triangles are similar. A B C

  3. D E F AAA Similarity Theorem. If the corresponding angles in two triangles are congruent, then the triangles are similar. A Since the angles are congruent we need to show the corresponding sides are in proportion. B C

  4. If the corresponding angles in two triangles are congruent, then the triangles are similar. Prove: A B C Given: A=D, B=E, C=F What will we prove? What is given? Construction Why? (1) E’ so that AE’ = DE (2) F’ so that AF’ = DF Construction Why? Why? SAS. (3) ∆AE’F’  ∆DEF (4) AE’F =E =  B Why? CPCTE & Given (5) E’F’ ∥ BC Why? Corresponding angles Why? Prop Thm (6) AB/AE’ = AC /AF’ E’ F’ (7) AB/DE = AC /DF Why? Substitute (8) AC/DF = BC/EF is proven in the same way. QED 4

  5. AA Similarity A D E B F C Theorem. If two corresponding angles in two triangles are congruent, then the triangles are similar. In Euclidean geometry if you know two angles you know the third angle. 5

  6. Theorem A D E B If a line parallel to one side of a triangle intersects the other two sides, then it cuts off a similar triangle. Don’t confuse this theorem with If a line intersects two sides of a triangle , and cuts off segments proportional to these two sides, then it is parallel to the third side. C 6 Proof for homework.

  7. SAS Similarity A D E F B C Theorem. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. 7

  8. If the two pairs of corresponding sides are proportional, and the included angles are congruent, then the triangles are similar. A E’ F’ B D C E F What will we prove? What is given? Given: AB/DE =AC/DF, A=D Prove: ∆ABC ~ ∆DEF Why? Construction (1) AE’ = DE, AF’ = DF Why? SAS (2) ∆AE’F’  ∆DEF Why? Given & substitution (1) (3) AB/AE’ = AC/AF’ Basic Proportion Thm Why? (4) E’F’∥ BC (5) B =  AE’F’ Why? Corresponding angles (6) A =  A Reflexive Why? AA Why? (7) ∆ABC  ∆AE’F’ (8) ∆ABC  ∆DEF Why? Substitute 2 & 7 QED 8

  9. SSS Similarity Theorem. If the corresponding sides are proportional, then the triangles are similar. A D E F B C 9 Proof for homework.

  10. Right Triangle Similarity C b a h x c - x c A B Theorem. The altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle. 10 Proof for homework.

  11. Pythagoras Revisited C b a h x c - x c B A From the warm up: And of course then, a 2 + b 2 = cx + c(c – x) = cx + c 2 – cx = c2 11

  12. Geometric Mean. It is easy to show that b = √(ac) or 6 = √(4 ·9) Construction of the geometric mean. 12

  13. Summary. • We learned about AAA similarity. • We learned about SAS similarity. • We learned about SSS similarity. • We learned about similarity in right triangles. 13

  14. Assignment: 14.1

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