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# 8 3 Similar Triangles

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### 8 3 Similar Triangles

1. 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. 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. 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. 4. Similar Triangles <ul><li>IF and ONLY IF </li></ul><ul><ul><li>Vertices match up so corresponding angles are congruent. </li></ul></ul><ul><ul><li>Corresponding sides are in proportion . </li></ul></ul>30° 30° 75° 75° 75° 75° 16 12 12 16 6 8 Ratios of each side are
5. 5. Triangle Similarity Postulate <ul><li>If two angles of one triangle are equal in measure to two angles of another triangle, </li></ul><ul><li>then the two triangles are similar . </li></ul><ul><li>AA (angle/angle) similarity </li></ul>
6. 6. AA? <ul><li>You will also see: </li></ul><ul><li>SAS </li></ul><ul><li>ASA </li></ul><ul><li>SSS </li></ul><ul><li>Knowing these letters will help with proofs later . </li></ul><ul><li>Side/Angle/Side </li></ul><ul><li>Angle/Side/Angle </li></ul><ul><li>Side/Side/Side </li></ul>
7. 7. Are they similar? <ul><li>Only one angle is given as congruent. Two must be given to use Angle/Angle or AA Similarity . </li></ul>
8. 8. Are they similar? Use Angle/Angle or AA Similarity . Two congruent angles show triangles are similar.
9. 9. Similar? <ul><li>Find the missing side of each triangle to find two 30° angles and a 120° angle for each of these similar triangles. </li></ul>120° 30° 30° 30°
10. 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. 11. Find the missing side <ul><li>We previously determined that these triangles are similar . We can set up ratios to find the missing side. </li></ul><ul><li>Start with a label on top and bottom. </li></ul>120° 30° 30° 30° 7 ft 21 ft 28 ft n ft
12. 12. In today’s lesson… <ul><li>We found that congruent triangles have both congruent angles and sides. </li></ul><ul><li>Similar triangles have congruent angles. </li></ul><ul><li>We can use the AA similarity to determine if triangles are similar. </li></ul><ul><li>We can use ratios to determine a missing side’s length when similar triangles are used. </li></ul>WA State Standards: G.3.A and G.3.B
13. 13. Assign: 453: 4-8; 12-13 457: 1-4 <ul><li>This statue can be seen in downtown Seattle in the Pacific Place Mall on the main level. </li></ul>
14. 14. Day Two <ul><li>Yesterday, we found that…. </li></ul><ul><li>We found that congruent triangles have both congruent angles and sides. </li></ul><ul><li>Similar triangles have congruent angles. </li></ul><ul><li>We can use the AA similarity to determine if triangles are similar. </li></ul><ul><li>Today’s Focus- Prove that triangles are similar. </li></ul>
15. 15. Overlapping Triangles <ul><li>It is sometimes useful to redraw as separate triangles to name the congruent sides and angles of those triangles. </li></ul>A C B A F E
16. 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. 17. It often helps to separate the two attached triangles. y 15 16 x 12 12
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. <ul><li>Did you notice that the words corresponding occur with parallel lines and triangles? </li></ul><ul><li>Corresponding Angles in triangles are different than when working with parallel lines, but in both cases are congruent. </li></ul>WARNING
20. 20. Given: ; Prove: <ul><li>1. </li></ul><ul><li>2. </li></ul><ul><li>3. </li></ul><ul><li>Given </li></ul><ul><li>If two || lines are intersected by a transversal, then corresponding angles are = in measure </li></ul><ul><li>AA Similarity </li></ul>
21. 21. 9. Given: Prove: <ul><li>1. </li></ul><ul><li>2. </li></ul><ul><li>3. </li></ul><ul><li>1.Given </li></ul><ul><li>2.Alternate Interior Angles from Transversal/Parallel Lines Theorem </li></ul><ul><li>3.AA Similarity Postulate </li></ul>A E D B C
22. 22. Overlapping Similar Triangles Theorem <ul><li>If a line is drawn from a point on one side of a triangle parallel to another side, </li></ul><ul><li>…then it forms a triangle similar to the original triangle. </li></ul>
23. 23. Solve by using proportions 4 ft 6 ft 5 ft x
24. 24. Assign: 453: 9; 14, 16, 21a, 23, 26 457: 5-7
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