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# 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