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

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

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

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

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

Theorem 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. B A E D C Proof for homework. 6

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

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

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

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

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

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

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

Assignment: 14.1