1. Similarity of Triangles
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2. In geometry, two polygons are similar when one is a replica (scale model) of the other.
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3. Consider Dr. Evil and Mini Me from Mike Meyers’ hit movie Austin Powers. Mini Me is supposed to be an exact replica of Dr. Evil.
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5. The following are similar figures.II
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6. The following are non-similar figures.II
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7. Feefee the mother cat, lost her daughters, would you please help her to find her daughters. Her daughters have the similar footprint with their mother.
Feefee’s footprint
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8. Which of the following is similar to the above triangle?1.
Which of the following is similar to the above triangle? B A C
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9. Note: One triangle is a scale model of the other triangle.
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10. How do we know if two triangles are similar or proportional?
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11. Triangles are similar (~) if corresponding angles are equal and the ratios of the lengths of corresponding sides are equal.
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12. The sum of the measure of the angles of a triangle is 1800.
Interior Angles of Triangles A B C
The sum of the measure of the angles of a triangle is 1800.
Ð A + Ð B + ÐC =1800
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13. Determine whether the pair of triangles is similar. Justify your answer.
Answer: Since the corresponding angles have equal measures, the triangles are similar.
Example 6-1b

14. If the product of the extremes equals the product of the means then a proportion exists.
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15. This tells us that  ABC and  XYZ are similar and proportional.
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16. Q: Can these triangles be similar?
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17. Answer—Yes, right triangles can also be similar but use the criteria.
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19. Do we have equality?
This tells us our triangles are not similar. You can’t have two different scaling factors!
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20. If we are given that two triangles are similar or proportional what can we determine about the triangles?
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21. The two triangles below are known to be similar, determine the missing value X.
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23. In the figure, the two triangles are similar. What are c and d ?B C P Q R 10 6 c 5 4 d
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24. In the figure, the two triangles are similar. What are c and d ?B C P Q R 10 6 c 5 4 d
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25. Sometimes we need to measure a distance indirectly
Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles. h 6 17
102
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26. MODELING A REAL-LIFE PROBLEM
Error Analysis
GEOMETRY CONNECTION Two students are visiting the mysterious statues on Easter Island in the South Pacific. To find the heights of two statues that are too tall to measure, they tried a technique involving proportions. They measured the shadow lengths of the statues at 2:00 P.M. and again at 3:00 P.M.
3:00
2:00

27. a b a b 2:00 3:00 a = b a = b a = b a = b a = b a = b
Error Analysis
SOLUTION
They let a and b represent the heights of the two statues. Because the ratios of corresponding sides of similar triangles are equal, the students wrote the following two equations. 27 a 18 b 30 a 20 b
2:00
3:00 a 27 = b 18 a 30 = b 20 a = 27 18 b a = 30 20 b
30 ft
27 ft
18 ft
20 ft a = 3 2 b a = 3 2 b

28. Draw Similar Rectangles ABCD and EFGH whose lengths and widths are 16 and 12 and 12 and 9 respectively.

29. 12 16 9 12

30. Two triangles are called “similar”
if their corresponding angles have
the same measure.      

31. a A b B c C Two triangles are called “similar”
if their corresponding angles have
the same measure. 
Ratios of corresponding
sides are equal. C A  a c     b B a A b B c C = =

32. Mary is 5 ft 6 inches tall.
She casts a 2 foot shadow.
The tree casts a 7 foot shadow.
How tall is the tree?

33. Mary is 5 ft 6 inches tall.
She casts a 2 foot shadow.
The tree casts a 7 foot shadow.
How tall is the tree?
Mary’s height
Tree’s height
Mary’s shadow
Tree’s shadow = x
5.5 2 7

34. Mary is 5 ft 6 inches tall.
She casts a 2 foot shadow.
The tree casts a 7 foot shadow.
How tall is the tree?
5.5 x 2 7 =
Mary’s height
Tree’s height
Mary’s shadow
Tree’s shadow = x
5.5 2 7

35. 5.5 x 2 7 =
7 ( 5.5 ) = 2 x
= 2 x
x =
The height of the tree is feet

36. Find the missing measures if the pair of triangles is similar.
Corresponding sides of similar triangles are proportional.
and
Example 6-2b

37. Find the cross products.
Divide each side by 4.
Answer: The missing measure is 7.5.
Example 6-2b

38. Find the missing measures if each pair of triangles is similar. a.
Answer: The missing measures are 18 and 42.
Example 6-2c

39. Find the missing measures if each pair of triangles is similar. b.
Answer: The missing measure is 5.25.
Example 6-2c

40. Shadows Richard is standing next to the General Sherman Giant Sequoia three in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree?
Since the length of the shadow of the tree and Richard’s height are given in meters, convert the length of Richard’s shadow to meters.
Example 6-3a

41. Let the height of the tree.
Simplify.
Let the height of the tree.
Richard’s shadow
Tree’s shadow
Richard’s height
Tree’s height
Cross products
Answer: The tree is about 84 meters tall.
Example 6-3a

42. Answer: The length of Trudie’s shadow is about 0.98 meter.
Tourism Trudie is standing next to the Eiffel Tower in France. The height of the Eiffel Tower is 317 meters and casts a shadow of 155 meters. If Trudie’s height is 2 meters, how long is her shadow?
Answer: The length of Trudie’s shadow is about meter.
Example 6-3b

44. Similarity of Triangles
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