8 3similar Triangles

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8 3similar Triangles

  1. 1. Similar Triangles Lesson 8-3 Focus – Apply the properties of similar triangles and tomorrow, prove that triangles are similar. WA State Standards: G.3.A and G.3.B
  2. 2. Congruent Triangles A 100° 115 f t ft 0 10 45° 35° B 150 ft C …have matching angles that are congruent. …have matching sides that are congruent. F 150 ft 45° E 35° t 0f 10 115 ft 100° D
  3. 3. Congruent Triangles A m∠C = m∠E 100° 115 f t m∠A = m∠D ft 0 10 45° 35° m∠B = m∠F B 150 ft C …have matching angles that are congruent. …have matching sides that are congruent. F AC ≈ DE 150 ft 45° E 35° t 0f CB ≈ EF 10 115 BA ≈ FD ft 100° D
  4. 4. Similar Triangles IF and ONLY IF − Vertices match up so corresponding angles are congruent. − Corresponding sides are in proportion. 4 Ratios of each side are 3 12 30° 12 30° 16 16 75° 75° 6 75° 75° 8
  5. 5. Triangle Similarity Postulate If two angles of one triangle are equal in measure to two angles of another triangle, then the two triangles are similar. AA (angle/angle) similarity
  6. 6. AA? • You will also see: • SAS • Side/Angle/Side • ASA • Angle/Side/Angle • SSS • Side/Side/Side Knowing these letters will help with proofs later.
  7. 7. Are they similar? Only one angle is given as congruent. Two must be given to use Angle/Angle or AA Similarity.
  8. 8. Are they similar? Use Angle/Angle or AA Similarity. Two congruent angles show triangles are similar.
  9. 9. Similar? 30° Find the missing side of each 30° triangle to find two 30° angles and a 120° angle 120° for each of these similar triangles. 30°
  10. 10. Is VABC similar to VAEF? A E F Sometimes it helps to separate the two triangles and look at each angle separately. B C
  11. 11. Find the missing side 30° We previously 21 ft determined that these triangles are similar. 7 ft 30° We can set up ratios to find the missing 28 ft side. n ft 120° Start with a label on top and bottom. 30° short 21 7 = = long 28 n
  12. 12. In today’s lesson… • We found that congruent triangles have both congruent angles and sides. • Similar triangles have congruent angles. • We can use the AA similarity to determine if triangles are similar. • We can use ratios to determine a missing side’s length when similar triangles are used. WA State Standards: G.3.A and G.3.B
  13. 13. Assign: 453: 4-8; 12-13 457: 1-4 This statue can be seen in downtown Seattle in the Pacific Place Mall on the main level.
  14. 14. Day Two Yesterday, we found that…. • We found that congruent triangles have both congruent angles and sides. • Similar triangles have congruent angles. • We can use the AA similarity to determine if triangles are similar. Today’s Focus-Prove that triangles are similar.
  15. 15. Overlapping Triangles  It is sometimes useful to redraw as separate triangles to name the congruent sides and angles of those triangles. AF : AC A A AE : AB EF : BC E F B C
  16. 16. Is V ABC similar to V AEF? If so, what Is the missing side? E B y 12 15 x 12 A C F 16
  17. 17. It often helps to separate the two attached triangles. 15 x y 12 12 16
  18. 18. Prove: A line drawn from a point on one side of a triangle parallel to another side forms a triangle similar to the original triangle.
  19. 19. Did you notice that the words corresponding occur with parallel lines and triangles? Corresponding Angles in triangles are different than when working with parallel lines, but in both cases are congruent.
  20. 20. A D E Given: VABC ; DE || BC B Prove: VADE : VABC C 1. DE || BC 1. Given 2. m∠ADE = m∠ABC 3. If two || lines are m∠AED = m∠ACB intersected by a transversal, then corresponding angles are = in measure 3. VADE : VABC 4. AA Similarity WA State Standards: G.3.A and G.3.B
  21. 21. B 9. Given: AB || DE C E Prove: VABC : VEDC A 1. AB || DE 1.Given D 2. m∠A = m∠E 2.Alternate Interior Angles from m∠B = m∠D Transversal/Parallel Lines Theorem 3.AA Similarity Theorem 3. VABC : VEDC WA State Standards: G.3.A and G.3.B
  22. 22. Overlapping Similar Triangles Theorem If a line is drawn from a point on one side of a triangle parallel to another side, …then it forms a triangle similar to the original triangle.
  23. 23. Solve by using proportions 4 6 = 5 x 6 ft 30 1 =7 4 ft 4 2 5 ft x
  24. 24. Assign: 453: 9; 14, 16, 21a, 23, 26 457: 5-7

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