8 3 Similar Triangles

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  • 1. Similar Triangles Focus – Apply the properties of similar triangles and tomorrow, prove that triangles are similar. Lesson 8-3 WA State Standards: G.3.A and G.3.B
  • 2. Congruent Triangles …have matching angles that are congruent. …have matching sides that are congruent. A B C D F E 100° 35° 45° 115 ft 150 ft 100 ft 150 ft 115 ft 100 ft 35° 100° 45°
  • 3. Congruent Triangles …have matching angles that are congruent. …have matching sides that are congruent. A B C D F E 100° 35° 45° 115 ft 150 ft 100 ft 150 ft 115 ft 100 ft 35° 100° 45°
  • 4. Similar Triangles
    • IF and ONLY IF
      • Vertices match up so corresponding angles are congruent.
      • Corresponding sides are in proportion .
    30° 30° 75° 75° 75° 75° 16 12 12 16 6 8 Ratios of each side are
  • 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. AA?
    • You will also see:
    • SAS
    • ASA
    • SSS
    • Knowing these letters will help with proofs later .
    • Side/Angle/Side
    • Angle/Side/Angle
    • Side/Side/Side
  • 7. Are they similar?
    • Only one angle is given as congruent. Two must be given to use Angle/Angle or AA Similarity .
  • 8. Are they similar? Use Angle/Angle or AA Similarity . Two congruent angles show triangles are similar.
  • 9. Similar?
    • Find the missing side of each triangle to find two 30° angles and a 120° angle for each of these similar triangles.
    120° 30° 30° 30°
  • 10. Is ABC similar to AEF? A E F C B Sometimes it helps to separate the two triangles and look at each angle separately.
  • 11. Find the missing side
    • We previously determined that these triangles are similar . We can set up ratios to find the missing side.
    • Start with a label on top and bottom.
    120° 30° 30° 30° 7 ft 21 ft 28 ft n ft
  • 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. 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. 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. Overlapping Triangles
    • It is sometimes useful to redraw as separate triangles to name the congruent sides and angles of those triangles.
    A C B A F E
  • 16. Is ABC similar to AEF? If so, what Is the missing side? y 15 16 x 12 12 A B C E F
  • 17. It often helps to separate the two attached triangles. y 15 16 x 12 12
  • 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.
    • 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.
    WARNING
  • 20. Given: ; Prove:
    • 1.
    • 2.
    • 3.
    • Given
    • If two || lines are intersected by a transversal, then corresponding angles are = in measure
    • AA Similarity
  • 21. 9. Given: Prove:
    • 1.
    • 2.
    • 3.
    • 1.Given
    • 2.Alternate Interior Angles from Transversal/Parallel Lines Theorem
    • 3.AA Similarity Postulate
    A E D B C
  • 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. Solve by using proportions 4 ft 6 ft 5 ft x
  • 24. Assign: 453: 9; 14, 16, 21a, 23, 26 457: 5-7