1. SIMILAR
TRIANGLES

2. Similar triangles are triangles that have the same shape but not necessarily the same size.D F E
ABC  DEF
When we say that triangles are similar we know:
A  D AB DE BC EF AC DF
= =
B  E
C  F

3. Six of those statements are true as a result of the similarity of the two triangles. However, if we need to prove that a pair of triangles are similar how many of those statements do we need? Because we are working with triangles and the measure of the angles and sides are dependent on each other. We do not need all six. There are three special combinations that we can use to prove similarity of triangles.
1. AA Similarity Postulate
 2 pairs of congruent angles
2. SSS Similarity Theorem
 3 pairs of proportional sides
3. SAS Similarity Theorem
 2 pairs of proportional sides and congruent angles between them

4. 8.4 – Similar Triangles (AA)

5. MNO  QRP mN = mR mO = mP
AA Similarity Postulate - If two angles of one triangle are congruent to two angles of another triangle, then the to triangles are similar. Q P R M N O
70
50
50
70
mN = mR
MNO  QRP
mO = mP

7. Example:
Given: ABC and DEF are isosceles right triangles
Prove:

8. Example: J O
Given:
Prove: M H S

9. 8.5 – Proving that Triangles are Similar (SSS and SAS)

10. ABC  DFE SSS Similarity Theorem
If corresponding sides of two triangles are proportional, then the two triangles
are similar. E F D
9.6
10.4 A B C 5 13 12 4
ABC  DFE

11. Proof of SSS Similarity Theorem
Given:
Prove:
Draw so that and
Then because ____________.
So by _______. So
Since PS = ML, then SQ = MN and PQ = LN
Then by ________.
Use CPCTC and AA sim postulate to show P R Q M T L N

12. GHI  LKJ mH = mK SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of another triangle, and then lengths of the sides including these angles are proportional, then the triangle are similar. L J K
7.5 G H I 5
70
70 7
10.5
mH = mK
GHI  LKJ

14. The SAS Similarity Theorem does not work unless the congruent angles fall between the proportional sides. For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. We do not have the information that we need. L J K
7.5 G H I 5
50 7
50
10.5
Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

15. Are the following triangles similar
Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. F G H K J
Yes, FGH  KJH because of the AA~ Postulate

16. Are the following triangles similar
Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. M O R G H I 6 10 3 4
No, these are not similar because

17. Are the following triangles similar
Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. A 20 25 X Y 25 30 B C
No, these are not similar because

18. Are the following triangles similar
Are the following triangles similar? If so, what similarity statement can be made. Name the postulate or theorem you used. A P J B C 3 5 2 8
Yes, APJ  ABC because of the SSS~ Postulate.

19. Explain why these triangles are similar. Then find the value of x.
4.5 3 5 x
These 2 triangles are similar because of the AA~ Postulate.
x=7.5

20. Explain why these triangles are similar. Then find the value of x.5 x
70
110 3 3
These 2 triangles are similar because of the AA~ Postulate.
x=2.5

21. Explain why these triangles are similar. Then find the value of x.24 14 22
These 2 triangles are similar because of the AA~ Postulate.
x=12

22. Explain why these triangles are similar. Then find the value of x.6 9 2 x
These 2 triangles are similar because of the AA~ Postulate.
x= 12

23. Explain why these triangles are similar. Then find the value of x.4 5 x 15
These 2 triangles are similar because of the AA~ Postulate.
x=8

24. Explain why these triangles are similar. Then find the value of x.
7.5 12 18
These 2 triangles are similar because of the AA~ Postulate.
x= 15