Obj. 35 Triangle Similarity
- 1. Obj. 35 Triangle Similarity The student is able to (I can): • Identify similar polygons • Prove certain triangles are similar by using AA~, SSS~, and SAS~ • Use triangle similarity to solve problems.
- 2. similar polygons Two polygons are similar if and only if their corresponding angles are congruent and their corresponding side lengths are proportional. Example: 6 N 5 3 M O 4 L 12 X 8 10 E ∠N ≅ ∠X ∠L ≅ ∠S ∠E ≅ ∠A ∠O ≅ ∠M S 6 A 3 4 5 6 = = = 6 8 10 12 NOEL ~ XMAS
- 3. Note: A similarity statement describes two similar polygons by listing their corresponding vertices. Example: NOEL ~ XMAS Note: To check whether two ratios are equal, cross-multiply them–the products should be equal. Example: 3 4 = 6 8 24 = 24
- 4. Example Determine whether the rectangles are similar. If so, write the similarity ratio and a similarity statement. Q 15 U 6 D A R 25 E 10 T C All of the angles are right angles, so all the angles are congruent. QUAD ~ RECT 6 15 = ? sim. ratio: 3 10 25 5 150 = 150
- 5. Angle-Angle Similarity (AA~) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. P M A D C O ∠M ≅ ∠P ∠A ≅ ∠O Therefore, ∆MAC ~ ∆POD by AA~
- 6. Side-Side-Side Similarity (SSS~) If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. N 18 W 12 30 24 H 40 O 16 Y T WH HY WY = = NO OT NT Therefore, ∆WHY ~ ∆NOT by SSS~
- 7. Side-Angle-Side Similarity (SAS~) If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar. E U 5 2.5 L 2 V LU LV = TE TX T 4 X ∠L ≅ ∠T Therefore, ∆LUV ~ ∆TEX by SAS~
- 8. Example Explain why the triangles are similar and write a similarity statement. X 34º L E 56º U V T 90 — 56 = 34º Therefore m∠V = m∠X, thus ∠V ≅ ∠X. Since m∠U = m∠E = 90º, ∠U ≅ ∠E Therefore, ∆LUV ~ ∆TEX by AA~
- 9. Example Verify that ∆SAT ~ ∆ORT R 20 S 12 15 T 16 O A ∠ATS ≅ ∠RTO (Vertical angles ≅) 12 15 = ? 16 20 240 = 240 Therefore, ∆SAT ~ ∆ORT by SAS~