1. 7-3 Triangle Similarity I CAN -Use the triangle similarity theorems to
determine if two triangles are similar.
Use proportions in similar triangles to solve
for missing sides.

2. Recall in 7-2, to prove that two polygons are similar you had to:
Show all corresponding angles are congruent
AND
Show all corresponding sides are proportional
Triangle Similarity Theorems are “shortcuts”
for showing two triangles are similar.

4. Because A  E and C  D justification
Example D 9 B E 12 10 A C 5 6 18 F
Similarity Statement Reason
ABC ~
EFD
by AA
Because A  E and C  D
justification

5. Check It Out! Example
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C  F.
B  E by the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.

7. Example: Verifying Triangle Similarity
Verify that the triangles are similar.
∆PQR and ∆STU
Therefore ∆PQR ~ ∆STU by SSS ~.

8. Example Longest side QS = 6 = 2 Legs QP = PS = 4 = 2 QS SR 6 39 R
Longest side QS = 6 = 2 QR 6 4 6 S P 4
Legs QP = PS = 4 = 2
QS SR
∆QPS ~ ∆QSR by SSS

10. Example : Verifying Triangle Similarity
Verify that the triangles are similar.
∆DEF and ∆HJK
D  H by the Definition of Congruent Angles.
Therefore ∆DEF ~ ∆HJK by SAS ~.

11. Check It Out! Example
Verify that ∆TXU ~ ∆VXW.
TXU  VXW by the Vertical Angles Theorem.
Therefore ∆TXU ~ ∆VXW by SAS ~.

12. Find the value of x such that ∆ACE ~ ∆BCD
∆ACE ~ ∆BCD by AA C 3 3 12 D B D B x 12 28 E C A = x
x + 3 E A
12(x + 3) = 84 28
12x + 36 = 84
– 36 – 36
12x = 48
x = 4

13. Example : Finding Lengths in Similar Triangles
Explain why ∆ABE ~ ∆ACD, and then find CD.
Step 1 Prove triangles are similar.
A  A by Reflexive Property of , and B  C since they are both right angles.
Therefore ∆ABE ~ ∆ACD by AA ~.

14. Example Continued
Step 2 Find CD.
x(9) = 5(3 + 9)
9x = 60

15. Check It Out! Example
Explain why ∆RSV ~ ∆RTU and then find RT.
Step 1 Prove triangles are similar.
It is given that S  T.
R  R by Reflexive Property of .
Therefore ∆RSV ~ ∆RTU by AA ~.

16. Check It Out! Example 3 Continued
Step 2 Find RT.
RT(8) = 10(12)
8RT = 120
RT = 15