1. Pearson Unit 3 Topic 9: Similarity 9-3: Proving Triangles Similar Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(7)(B) Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.
(1)(E) Create and use representations to organize, record, and communicate mathematical ideas.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(8)(A) Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.

3. Figures that are similar (~) have the same shape but not necessarily the same size.

4. There are several ways to prove certain triangles are similar
There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

6. This was accidently left off of the notes packet. Copy the words.

8. Example 1: Yes; vertical angles are congruent.B)
Yes; vertical angles are congruent.
The triangles are similar by AA~.
The third angle in ΔJKL = 80.
The third angle in ΔPQR = 25.
The triangles are not similar.

9. Example 2: Yes; ratios of corresponding sides are
6/8, 6/8, and 9/12. Those simplify to
3/4, 3/4, and 3/4. The triangles are
similar by SSS~.
Yes; ratios of corresponding sides are
6/8 and 12/16. Those simplify to
3/4 and 3/4. The right angle is the
Included angle in both triangles.
The triangles are similar by SAS~.

10. A similarity ratio is the ratio of the lengths of
the corresponding sides of two similar polygons.
The similarity ratio of ∆ABC to ∆DEF is , or .
The similarity ratio of ∆DEF to ∆ABC is , or 2.

11. Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.
Writing Math

12. Example: 3 ∆PQR and ∆STW P and R = 72. W = 62 and T = 56.
Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement.
∆PQR and ∆STW
P and R = 72. W = 62 and T = 56.
Corresponding angles are not congruent.
The triangles are not similar.

13. Example: 4 Determine if ∆JLM ~ ∆NPS.
If so, write the similarity ratio and a similarity statement.
Step 1 Identify pairs of congruent angles.
Step 2 Compare corresponding sides.
∡N  ∡M, ∡L  ∡P (both given),
∡S  ∡J by Third ∡s Theorem.
Thus the similarity ratio is , and ∆LMJ ~ ∆PNS.

14. When you work with proportions, be sure the ratios compare corresponding measures.
Helpful Hint

15. Example: 5 A→ A→ Explain why the triangles are similar and write a
similarity statement.
Statements
Reasons A→
1. ∡B  ∡E
1. Right Angles Theorem
2. m∡C = 47⁰
2. Triangle Sum Theorem A→
3. ∡C  ∡F
3. def. of  angles
4. ∆ABC ~ ∆DEF
4. AA~

16. Example: 6

17. Example: 7
X ft
5.5 ft
6 ft
34 ft

18. Example: 8 S→ S→ S→ Therefore ∆PQR ~ ∆STU by SSS ~.
Verify that the triangles are similar.
∆PQR and ∆STU S→ S→ S→
Therefore ∆PQR ~ ∆STU by SSS ~.

19. Example: 9 S→ A→ S→ Therefore ∆DEF ~ ∆HJK by SAS ~.
Verify that the triangles are similar.
∆DEF and ∆HJK S→ A→
∡D  ∡H by the Definition of Congruent Angles. S→
Therefore ∆DEF ~ ∆HJK by SAS ~.

20. Example: 10 S→ A→ S→ Therefore ∆TXU ~ ∆VXW by SAS ~.
Verify that ∆TXU ~ ∆VXW. S→ A→
∡TXU  ∡VXW by the Vertical Angles Theorem. S→
Therefore ∆TXU ~ ∆VXW by SAS ~.

21. Example: 11 A→ A→ Therefore ∆ABE ~ ∆ACD by AA ~.
Explain why ∆ABE ~ ∆ACD, and then find CD.
Step 1 Prove triangles are similar. A→
∡EBA  ∡C since they are both right angles. A→
∡A  ∡A by Reflexive Property of 
Therefore ∆ABE ~ ∆ACD by AA ~.

22. Example: 11: continued Step 2 Find CD.
Corr. sides are proportional. Seg. Add. Postulate.
Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA.
x(9) = 5(3 + 9)
Cross Products Prop.
9x = 60
Simplify.
Divide both sides by 9.

23. Example: 12 A→ A→ Therefore ∆RSV ~ ∆RTU by AA ~.
Explain why ∆RSV ~ ∆RTU and then find RT.
Step 1 Prove triangles are similar. A→
given that ∡RSV  ∡T A→
∡R  ∡R by Reflexive Property of 
Therefore ∆RSV ~ ∆RTU by AA ~.

24. Example: 12 continued Step 2 Find RT. Corr. sides are proportional.
Substitute RS for 10, 12 for TU, 8 for SV.
RT(8) = 10(12)
Cross Products Prop.
8RT = 120
Simplify.
RT = 15
Divide both sides by 8.