has an inscribed rhombus ARST. If A square with sides of length 2 in. rests as shown on a

The proof of this theorem is left to the student; see Exercise 33. Note that this proof also requires the use of CSSTP. General Rule: First prove that two triangles are similar. Then form a proportion involv- ing the lengths of corresponding sides. Finally, apply the Means-Extremes Property. Illustration: See the following proof and Example 3 an alternative form of the proof. STRATEGY FOR PROOF 왘 Proving Products of Lengths Equal The lengths of the corresponding altitudes of similar triangles have the same ratio as the lengths of any pair of corresponding sides. THEOREM 5.3.2 PROOF Statements Reasons

1. 2.

3. 4. DE BC = AE AC 䉭ADE ~ 䉭ABC ∠A ⬵ ∠A ∠ADE ⬵ ∠B 1. Given 2. Identity 3. AA 4. CSSTP 쮿 NOTE: In this proof, DE appears above BC because the sides with these names lie opposite in the two similar triangles. AE and AC are the lengths of the sides opposite the congruent and corresponding angles and . That is, corresponding sides of similar triangles always lie opposite corresponding angles. ∠B ∠ADE ∠A The paragraph style of proof is generally used in upper-level mathematics classes. These paragraph proofs are no more than modified two-column proofs. Compare the following two-column proof to the paragraph proof found in Example 3. GIVEN: in Figure 5.14 PROVE: NP QR = RP MN ∠M ⬵ ∠Q PROOF Statements Reasons

1. 2.

3. 4.

5. NP QR = RP MN NP RP = MN QR 䉭MPN 䉭QPR ∠1 ⬵ ∠2 ∠M ⬵ ∠Q 1. Given hypothesis 2. Vertical angles are 3. AA 4. CSSTP 5. Means-Extremes Property ⬵ EXAMPLE 3 Use a paragraph proof to complete this problem. GIVEN: in Figure 5.14 PROVE: NP QR = RP MN ∠M ⬵ ∠Q P M R N Q 1 2 Figure 5.14 In addition to AA, there are other methods that can be used to establish similar tri- angles. To distinguish the following techniques for showing triangles similar from methods for proving triangles congruent, we use SAS~ and SSS~ to identify the simi- larity theorems. We prove SAS~ in Example 6 and prove SSS~ at our website. F D E G H Figure 5.15 PROOF: By hypothesis, . Also, by the fact that vertical angles are congruent. Now by AA. Using CSSTP, Then by the Means-Extremes Property. NOTE: In the proof, the sides selected for the proportion were carefully chosen. The statement to be proved suggested that we include NP, QR, RP, and MN in the proportion. NP QR = RP MN NP RP = MN QR . 䉭MPN 䉭QPR ∠1 ⬵ ∠2 ∠M ⬵ ∠Q 쮿 Exs. 5–7 If an angle of one triangle is congruent to an angle of a second triangle and the pairs of sides including the angles are proportional, then the triangles are similar. THEOREM 5.3.3 SAS~ If the three sides of one triangle are proportional to the three corresponding sides of a second triangle, then the triangles are similar. THEOREM 5.3.4 SSS~ EXAMPLE 4 In Figure 5.15, . Also, , , and . Find the value of x and the measure of each angle. Solution With Identity and Given, by SAS~. By CASTC, , so . The sum of angles in is , so . Then and . In turn, , , and . m ∠D = 78° m ∠F = m∠DHG = 46° m ∠E = ∠DGH = 56° x = 56 3x = 168 3x + 12 = 180 x + x + 22 + x - 10 = 180 䉭DEF m ∠F = x - 10 ∠F ⬵ ∠DHG 䉭DGH ~ 䉭DEF DG DE = DH DF ∠D ⬵ ∠D m ∠DHG = x - 10 m ∠D = x + 22 m ∠E = x DG DE = DH DF Consider this application of Theorem 5.3.3. Warning SSS and SAS prove that triangles are congruent. SSS~ and SAS~ prove that triangles are similar. 쮿 Along with AA and SAS , Theorem 5.3.4 SSS provides the third and final method of establishing that triangles are similar. EXAMPLE 5 Which method AA, SAS~, or SSS~ establishes that ? See Figure 5.16. a , , , , and b , , , , , and Solution a SAS ; b SSS ; AB XT = AC XN = BC TN AC XN = AB XT TN = 12 XN = 6 XT = 9 BC = 8 AC = 4 AB = 6 XT = 12 AB = 8 XN = 9 AC = 6 ∠A ⬵ ∠X 䉭ABC 䉭XTN B C A Figure 5.16 T N X We close this section by proving Theorem 5.3.3 SAS . To achieve this goal, we prove a helping theorem by the indirect method. In Figure 5.17, we say that sides and are divided proportionally by if . DA CD = EB CE DE CB CA If a line segment divides two sides of a triangle proportionally, then this line segment is parallel to the third side of the triangle. LEMMA 5.3.5 쮿 Exs. 8–10 B A C D 1 E Figure 5.17 B A C D E F GIVEN: with PROVE: PROOF: in . Applying Property 3 of Section 5.1, we have so . Now suppose that is not parallel to Through D, we draw It follows that . With , it follows that by the rea- son AA. By CSSTP, . Using the starred statements and substitution, CA CD = CB CF 12 䉭CDF 䉭CAB ∠C ⬵ ∠C ∠CDF ⬵ ∠A DF 7 AB. AB . DE CA CD = CB CE CD + DA CD = CE + EB CE , 䉭ABC DA CD = EB CE DE 7 AB DA CD = EB CE 䉭ABC EXAMPLE 6 GIVEN: a nd ; PROVE: 䉭ABC 䉭DEC CA CD = CB CE 䉭DEC 䉭ABC B A C D 1 E Applying the Means-Extremes Property, . Dividing each side of the last equation by CB, we find that . That is, F must coincide with E; it follows that . In Example 6, we use Lemma 5.3.5 to prove the SAS theorem. DE 7 AB CF = CE CB CE CB CF = CB CE = CB CF A both ratios are equal to CA CD B . 쮿 Exercises 5–8 U V W E D H G Exercises 9, 10 X V W S R Statements Reasons 1. and ; 2. 3. 4. 5. 6. 7. 䉭ABC ~ 䉭DEC ∠C ⬵ ∠C ∠1 ⬵ ∠A ‹ DE 7 AB DA CD = EB CE CA - CD CD = CB - CE CE CA CD = CB CE 䉭DEC 䉭ABC 1. Given 2. Property 3 of Section 5.1 3. Substitution 4. Lemma 5.3.5 5. If 2 lines are cut by a trans., corr. are 6. Identity 7. AA ⬵ ∠s 7 쮿 Exs. 11, 12 Exercises 5.3

1. What is the acronym that is used to represent the

statement “Corresponding angles of similar triangles are congruent?”

2. What is the acronym that is used to represent the

statement “Corresponding sides of similar triangles are proportional?”

3. Classify as true or false:

a If the vertex angles of two isosceles triangles are congruent, the triangles are similar. b Any two equilateral triangles are similar.

4. Classify as true or false:

a If the midpoints of two sides of a triangle are joined, the triangle formed is similar to the original triangle. b Any two isosceles triangles are similar. In Exercises 5 to 8, name the method AA, SSS , or SAS that is used to show that the triangles are similar.

5. , ,

and UV = 3 2 RS WV = 3 2 TS WU = 3 2 TR

10. and DF = 3

DH DE = 3 DG

6. and

7. and

8. In Exercises 9 and 10, name the method that explains why . 9. DG DE = DH DF 䉭DGH 䉭DEF TR WU = TS WV = RS UV TR WU = TS WV ∠T ⬵ ∠W ∠R ⬵ ∠U ∠T ⬵ ∠W In Exercises 11 to 14, provide the missing reasons.

11. Given: ; ;

Prove: 䉭VWR 䉭VXT VX ⬜ TS VW ⬜ RS ⵥRSTV PROOF Statements Reasons 1. ; ; 1. ? 2. and 2. ? are rt. 3. 3. ? 4. 4. ? 5. 5. ? 䉭VWR 䉭VXT ∠R ⬵ ∠T ∠VWR ⬵ ∠VXT ∠s ∠VXT ∠VWR VX ⬜ TS VW ⬜ RS ⵥRSTV C D E A B N P M Q R Exercises 15, 16 F H G K Exercises 17, 18

12. Given: and

Prove: 䉭ABE 䉭CTB ⵥABCD 䉭DET PROOF Statements Reasons 1. and 1. ? 2. 2. Opposite sides of a are 3. 3. ? 4. 4. ? 5. 5. ? 6. 6. ?

13. Given: ; M and N are

midpoints of and respectively Prove: 䉭AMN 䉭ABC AC , AB 䉭ABC 䉭ABE 䉭CTB ∠E ⬵ ∠CBT ED 7 CB ∠EBA ⬵ ∠T 7 ⵥ AB 7 DT ⵥABCD 䉭DET B C A M N PROOF Statements Reasons 1. ; M and N are the 1. ? midpoints of and respectively 2. and 2. ? 3. 3. ? 4. 4. ? and 5. 5. ? 6. 6. ?

14. Given:

with trisected at P and Q and trisected at R and S Prove: 䉭XYZ 䉭PYR YZ XY 䉭XYZ 䉭AMN 䉭ABC AM AB = AN AC = MN BC MN BC = 1 2 AN AC = 1 2 , AM AB = 1 2 , MN = 1 2 BC AN = 1 2 AC AM = 1 2 AB AC , AB 䉭ABC X Y Z R P S Q PROOF Statements Reasons 1. ; trisected at 1. ? P and Q; trisected at R and S 2. and 2. Definition of trisect 3. 3. ? 4. 4. ? 5. 5. ? In Exercises 15 to 22, complete each proof.

15. Given:

, Prove: 䉭MNP 䉭QRP QR ⬜ RP MN ⬜ NP 䉭XYZ 䉭PYR ∠Y ⬵ ∠Y YR YZ = YP YX YP YX = 1 3 YR YZ = 1 3 YZ XY 䉭XYZ PROOF Statements Reasons 1. ? 1. Given 2. N and QRP are 2. ? right 3. ? 3. All right are 4. 4. ? 5. ? 5. ?

16. Given:

See figure for Exercise 15. Prove: PROOF Statements Reasons 1. ? 1. Given 2. 2. ? 3. ? 3. If two lines are cut by a transversal, the cor- responding are 4. ? 4. ?

17. Given:

Prove: 䉭HJK 䉭FGK ∠H ⬵ ∠F ⬵ ∠s 7 ∠M ⬵ ∠RQP 䉭MNP 䉭QRP MN 7 QR ∠P ⬵ ∠P ⬵ ∠s ∠s ∠s PROOF Statements Reasons 1. ? 1. Given 2. 2. ? 3. ? 3. ?

18. Given:

, See figure for Exercise 17. Prove: PROOF Statements Reasons 1. ? 1. Given 2. G and J are right 2. ? 3. 3. ? 4. 4. ? 5. ? 5. ?

19. Given:

Prove: ∠N ⬵ ∠R RQ NM = RS NP = QS MP ∠HKJ ⬵ ∠GKF ∠G ⬵ ∠J ∠s ∠s 䉭HJK 䉭FGK HG ⬜ FG HJ ⬜ JF ∠HKJ ⬵ ∠FKG PROOF Statements Reasons 1. ? 1. ? 2. and 2. ? 3. ? 3. AA 4. ? 4. ?

22. Given:

, Prove: AB DC = BC CE AC 7 DE AB 7 DC ∠S ⬵ ∠U ∠R ⬵ ∠V R P M S Q N E F D H G U R V T S C E D A B B A C D E Exercises 27, 28 E C D B A Exercises 23–26 PROOF Statements Reasons 1. ? 1. Given 2. ? 2. SSS 3. ? 3. CASTC

20. Given:

Prove: ∠DGH ⬵ ∠E DG DE = DH DF PROOF Statements Reasons 1. ? 1. ? 2. 2. ? 3. 3. ? 4. ? 4. ?

21. Given:

Prove: RT VT = RS VU RS 7 UV 䉭DGH 䉭DEF ∠D ⬵ ∠D PROOF Statements Reasons 1. 1. ? 2. ? 2. If 2 lines are cut by a trans. corr. are 3. ? 3. Given 4. 4. ? 5. 5. ? 6. ? 6. ? In Exercises 23 to 26, . 䉭ABC 䉭DBE 䉭ACB 䉭DEC ∠ACB ⬵ ∠E ⬵ ∠s 7 AB 7 DC

23. Given: , ,

Find: EB HINT: Let , and solve an equation.

24. Given: ,

E is the midpoint of Find: DE

25. Given: , ,

Find: DB

26. Given: , ,

Find: DB

27. with .

If , , and , find EB. CE = 6 DA = 8 CD = 10 ∠CDE ⬵ ∠B 䉭CDE 䉭CBA AD = 5 CE = 4 CB = 12 AD = 4 DE = 8 AC = 10 CB CB = 12 AC = 10 EB = x CB = 6 DE = 6 AC = 8