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TEKS Focus: (7)(B) Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems. (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (8)(A) Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems. Figures that are similar (~) have the same shape but not necessarily the same size. There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent. This was accidently left off of the notes packet. Copy the words. A) B) Yes; vertical angles are congruent. The triangles are similar by AA~. The third angle in ΔJKL = 80. The third angle in ΔPQR = 25. The triangles are not similar. Example 2: Yes; ratios of corresponding sides are 6/8, 6/8, and 9/12. Those simplify to 3/4, 3/4, and 3/4. The triangles are similar by SSS~. Yes; ratios of corresponding sides are 6/12 and 8/16. Those simplify to 1/2 and 1/2. The right angle is the Included angle in both triangles. The triangles are similar by SAS~. A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio of ∆ABC to ∆DEF is , or The similarity ratio of ∆DEF to ∆ABC is , or 2. . Writing Math Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Example: 3 Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. ∆PQR and ∆STW P and R = 72. W = 62 and T = 56. Corresponding angles are not congruent. The triangles are not similar. Example: 4 Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio and a similarity statement. Step 1 Identify pairs of congruent angles. Step 2 Compare corresponding sides. ∡N ∡M, ∡L ∡P (both given), ∡S ∡J by Third ∡s Theorem. Thus the similarity ratio is , and ∆LMJ ~ ∆PNS. Helpful Hint When you work with proportions, be sure the ratios compare corresponding measures. Example: 5 Explain why the triangles are similar and write a similarity statement. Statements A→ 1. ∡B ∡E Reasons 1. Right Angles Theorem 2. m∡C = 47⁰, m∡C= 47⁰ 2. Triangle Sum Theorem A→ 3. ∡C ∡F 3. Transitive Prop. of 4. ∆ABC ~ ∆DEF 4. AA~ Example: 6 Example: 7 X ft 5.5 ft 6 ft 34 ft Example: 8 Verify that the triangles are similar. ∆PQR and ∆STU S→ S→ S→ Therefore ∆PQR ~ ∆STU by SSS ~. Example: 9 Verify that the triangles are similar. ∆DEF and ∆HJK S→ A→ ∡D ∡H by the Definition of Congruent Angles. S→ Therefore ∆DEF ~ ∆HJK by SAS ~. Example: 10 Verify that ∆TXU ~ ∆VXW. S→ A→ ∡TXU ∡VXW by the Vertical Angles Theorem. S→ Therefore ∆TXU ~ ∆VXW by SAS ~. Example: 11 Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A→ A→ ∡EBA ∡C since they are both right angles. ∡A ∡A by Reflexive Property of Therefore ∆ABE ~ ∆ACD by AA ~. Example: 11: continued Step 2 Find CD. Corr. sides are proportional. Seg. Add. Postulate. Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. x(9) = 5(3 + 9) 9x = 60 Cross Products Prop. Simplify. Divide both sides by 9. Example: 12 Explain why ∆RSV ~ ∆RTU and then find RT. Step 1 Prove triangles are similar. A→ given that ∡RSV ∡T A→ ∡R ∡R by Reflexive Property of Therefore ∆RSV ~ ∆RTU by AA ~. Example: 12 continued Step 2 Find RT. Corr. sides are proportional. Substitute RS for 10, 12 for TU, 8 for SV. RT(8) = 10(12) 8RT = 120 RT = 15 Cross Products Prop. Simplify. Divide both sides by 8.