1. 6-3/6-4: Proving Triangles Similar
Geometry Chapter 6
6-3/6-4: Proving Triangles Similar

2. Warm-Up
For the given polygons,
a.) Determine whether the polygons are similar
b.) If the polygons are similar, write a similarity statement and find the scale factor. 𝑨 𝟐𝟏 𝑩 𝑪 𝑫 𝟏𝟓 𝟐𝟒 𝟏𝟐 𝑾 𝑿 𝒀 𝒁 𝟐𝟖 𝟐𝟎 𝟑𝟐 𝟏𝟔

3. Warm-Up In the diagram, ∆𝑫𝑬𝑭 ~ ∆𝑴𝑵𝑷. Find the value of x. 𝑴 𝑵 𝑷 𝟐𝟒 𝟏𝟖𝟑𝟎 𝑫 𝑬 𝑭 𝟔 𝟖 𝒙

4. Proving Triangles Similar
Objective: Students will be able to use theorems and postulates to prove that triangles are similar.
Agenda
Postulates/Theorems for Similarity
Practice

5. Proving Similar Polygons
To identify that two polygons are similar, then you must identify that:

6. Definition: Similar Polygons
To identify that two polygons are similar, then you must identify that:
1.) Corresponding angles are congruent

7. Definition: Similar Polygons
To identify that two polygons are similar, then you must identify that:
1.) Corresponding angles are congruent
2.) Corresponding sides are in proportion (i.e. Their side lengths have the same ratio.)

8. Definition: Similar Polygons
To identify that two polygons are similar, then you must identify that:
1.) Corresponding angles are congruent
2.) Corresponding sides are in proportion (i.e. Their side lengths have the same ratio.)
*For Triangles, however, we can prove that they are similar more quickly with the following:

9. A Postulate for Similar Triangles
Postulate 22 – AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 𝒀 𝑿 𝒁
If <𝑿 ≅ <𝑱
And <𝒀≅ <𝑲 𝑲 𝑱 𝑳
Then ∆𝑱𝑲𝑳 ~ ∆𝑿𝒀𝒁

10. Theorems for Similar Triangles
Theorem 6.2 – SSS Similarity Theorem: If the sides of two triangles are in proportion, then the triangles are similar.
If 𝑴𝑵 𝑻𝑼 = 𝑵𝑷 𝑼𝑽 = 𝑴𝑷 𝑻𝑽 𝑻 𝑼 𝑽 𝑴 𝑵 𝑷
Then ∆𝑴𝑵𝑷 ~ ∆𝑻𝑼𝑽

11. Theorems for Similar Triangles
Theorem 6.3 – SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. 𝑫 𝑬 𝑭
If <𝑨≅ <𝑫
𝑨𝑩 𝑫𝑬 = 𝑨𝑪 𝑫𝑭 𝑨 𝑩 𝑪
Then ∆𝑨𝑩𝑪 ~ ∆𝑫𝑬𝑭

12. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 6 9 12 𝑪 𝑫 𝑬

13. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵
𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬

14. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵
𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬
𝑲𝑵 𝑫𝑬 = 𝟐𝟏 𝟗 = 𝟕 𝟑

15. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵
𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬
𝑲𝑵 𝑫𝑬 = 𝟐𝟏 𝟗 = 𝟕 𝟑
𝑴𝑵 𝑪𝑬 = 𝟐𝟖 𝟏𝟐 = 𝟕 𝟑

16. Example 1
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 6 9 12 𝑪 𝑫 𝑬
𝑴𝑲 𝑪𝑫 = 𝑲𝑵 𝑫𝑬 = 𝑴𝑵 𝑪𝑬 = 𝟕 𝟑
∆𝑴𝑲𝑵 ~ ∆𝑪𝑫𝑬
By SSS Similarity

17. Example 2
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫
𝟐𝟔° 𝑯 𝑮 𝑲
𝟔𝟒°

18. Example 2
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
<𝑫 ≅ <𝑮 (Both Right Angles) 𝑬 𝑪 𝑫
𝟐𝟔° 𝑯 𝑮 𝑲
𝟔𝟒°

19. Example 2
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫
𝟐𝟔°
<𝑫 ≅ <𝑮 (Both Right Angles)
<𝑪≅ <𝑲 (Why?)
<𝑬≅ <𝑯 (Why?) 𝑯 𝑮 𝑲
𝟔𝟒°

20. Example 2
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫
𝟐𝟔°
∆𝑫𝑬𝑪 ~ ∆𝑮𝑯𝑲
By AA Similarity 𝑯 𝑮 𝑲
𝟔𝟒°

21. Example 3
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪 16 24 32 𝑴 𝑵 𝑷

22. Example 3
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪
𝑨𝑪 𝑵𝑷 = 𝟐𝟎 𝟑𝟐 = 𝟓 𝟖 16 24 32 𝑴 𝑵 𝑷
𝑩𝑪 𝑴𝑷 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑
𝑨𝑩 𝑴𝑵 = 𝟏𝟎 𝟏𝟔 = 𝟓 𝟖

23. Example 3
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪
𝑨𝑪 𝑵𝑷 = 𝟐𝟎 𝟑𝟐 = 𝟓 𝟖 16 24 32 𝑴 𝑵 𝑷
𝑩𝑪 𝑴𝑷 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑
𝑨𝑩 𝑴𝑵 = 𝟏𝟎 𝟏𝟔 = 𝟓 𝟖
The Triangles are NOT Similar

24. Example 4
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾

25. Example 4
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾
𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑

26. Example 4
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾
𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑
<𝑷𝑼𝑹 ≅ <𝑾𝑼𝑻 (Why?)

27. Example 4
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾
∆𝑷𝑼𝑹 ~ ∆𝑾𝑼𝑻
By SAS Similarity

28. Example 5
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁

29. Example 5
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁
𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑
<𝑷𝑼𝑹 ≅ <𝑾𝑼𝑻
However…

30. Example 5
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁
The congruent angles do not fall between the proportional sides.
Therefore, the Triangles are NOT Similar.

31. Example 6
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭

32. Example 6
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭
<𝑨𝑪𝑩 ≅ <𝑭𝑪𝑫 (Why?)
<𝑨 ≅ <𝑭 (Why?)

33. Example 6
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭
∆𝑨𝑪𝑩 ~ ∆𝑭𝑪𝑫
By AA Similarity

34. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱 𝑸 𝑹 𝑺

35. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱 𝑸 𝑹 𝑺
<𝑰 ≅ <𝑺 (Both 𝟕𝟎° Angles)
<𝑱≅ <𝑹 (Why?)
<𝑯≅ <𝑸 (Why?)

36. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱
<𝑰 ≅ <𝑺 (Both 𝟕𝟎° Angles)
<𝑱≅ <𝑹
<𝑯≅ <𝑸 𝑸 𝑹 𝑺
∆𝑯𝑰𝑱 ~ ∆𝑸𝑺𝑹
By AA Similarity

37. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬
𝟓𝟔° 𝑸 𝑹 𝑷
𝟑𝟑°

38. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬
𝟓𝟔°
Only 1 pair of angles are congruent. Not enough for AA. 𝑸 𝑹 𝑷
𝟑𝟑°
The Triangles are NOT Similar

39. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬
𝟓𝟔°
<𝑬≅ <𝑹 (Both 𝟗𝟎° Angles)
𝒎<𝑭=𝟓𝟔°; 𝒎<𝑮=𝟑𝟒°
𝒎<𝑸=𝟓𝟕°; 𝒎<𝑷=𝟑𝟑° 𝑸 𝑹 𝑷
𝟑𝟑°

40. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮

41. Final Practice 𝑮𝑰 𝑯𝑰 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑 𝑮𝑯 𝑯𝑲 = 𝟐𝟎 𝟑𝟎 = 𝟐 𝟑
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮
𝑮𝑰 𝑯𝑰 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑
𝑮𝑯 𝑯𝑲 = 𝟐𝟎 𝟑𝟎 = 𝟐 𝟑
𝑯𝑰 𝑰𝑲 = 𝟐𝟒 𝟑𝟔 = 𝟐 𝟑

42. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮
∆𝑯𝑰𝑮 ~ ∆𝑰𝑲𝑯
By SSS Similarity

43. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12
32° 𝑨 𝑪 𝑫 18
125° 22
23° 𝑶 𝑵 𝑷 33

44. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
𝑨𝑪 𝑶𝑵 = 𝟏𝟐 𝟐𝟐 = 𝟔 𝟏𝟏 12
32° 𝑨 𝑪 𝑫 18
125° 22
23° 𝑶 𝑵 𝑷 33
𝑪𝑫 𝑵𝑷 = 𝟏𝟖 𝟑𝟑 = 𝟔 𝟏𝟏
<𝑪 ≅ <𝑵
(Why?)

45. Final Practice
Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem.
∆𝑨𝑪𝑫 ~ ∆𝑶𝑵𝑷
By SAS Similarity 12
32° 𝑨 𝑪 𝑫 18
125° 22
23° 𝑶 𝑵 𝑷 33