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6-4 Proving Triangles Congruent - ASA, AAS PROOF Write the specified type of proof. 1. two-column proof Given: bisects ABD and ACD. Prove: SOLUTION: Proof: Statements (Reasons) 1. bisects ABD and ACD. (Given) 2. ABC DBC (Definition of angular bisector) 3. (Reflexive Property) 4. ACB DCB (Definition of angular bisector) 5. (ASA) ANSWER: Proof: Statements (Reasons) 1. bisects ABD and ACD. (Given) 2. ABC DBC (Def. of bisector) 3. (Refl. Prop.) 4. ACB DCB (Def. of bisector) 5. (ASA) 2. flow proof Given: Prove: SOLUTION: eSolutions Manual - Powered by Cognero Page 1 2. ABC DBC (Def. of bisector) 3. (Refl. Prop.) 4. ACB DCB (Def. of bisector) 6-4 Proving Triangles (ASA) Congruent - ASA, AAS 5. 2. flow proof Given: Prove: SOLUTION: ANSWER: 3. paragraph proof Given: Prove: bisects SOLUTION: eSolutions Manual - Powered by Cognero Proof: We are given MLJ. So, K M, and bisects KLM. Since bisects by the Angle-Angle-Side Congruence Theorem. KLM, we know Page 2 KLJ 6-4 Proving Triangles Congruent - ASA, AAS 3. paragraph proof Given: Prove: bisects SOLUTION: Proof: We are given MLJ. So, K M, and bisects KLM. Since bisects by the Angle-Angle-Side Congruence Theorem. KLM, we know KLJ K M, and bisects KLM. Since bisects by the Angle-Angle-Side Congruence Theorem. KLM, we know KLJ ANSWER: Proof: We are given MLJ. So, 4. two-column proof Given: m G = m J = 90 Prove: SOLUTION: Proof: Statements (Reasons) 1. m G = m J = 90 (Given) 2. G J (Definition of congruent angles) 3. GHF JFH (Alternate Interior angles are congruent.) 4. (Reflexive Prop.) 5. (AAS) ANSWER: Proof: Statements (Reasons) 1. m G = m J = 90 (Given) 2. G J (Def. of s.) 3. GHF JFH (Alt. Int. s are .) 4. (Ref. Prop.) 5. (AAS) 5. BRIDGE BUILDING A surveyor needs to find the distance from point A to point B across a canyon. She places a stake at A, and a coworker places a stake at B on the other side of the canyon. The surveyor then locates C on the same side of the canyon as A such that A fourth stake is placed at E, the midpoint of Finally, a stake . is placed D such by that and D, E, and B are sited as lying along the same line eSolutions Manualat- Powered Cognero Page 3 a. Explain how the surveyor can use the triangles formed to find AB. b. If AC = 1300 meters, DC = 550 meters, and DE = 851.5 meters, what is AB? Explain your reasoning. 2. G J (Def. of s.) 3. GHF JFH (Alt. Int. s are .) 4. (Ref. Prop.) 6-4 Proving Triangles Congruent - ASA, AAS 5. (AAS) 5. BRIDGE BUILDING A surveyor needs to find the distance from point A to point B across a canyon. She places a stake at A, and a coworker places a stake at B on the other side of the canyon. The surveyor then locates C on the same side of the canyon as A such that A fourth stake is placed at E, the midpoint of Finally, a stake is placed at D such that and D, E, and B are sited as lying along the same line. a. Explain how the surveyor can use the triangles formed to find AB. b. If AC = 1300 meters, DC = 550 meters, and DE = 851.5 meters, what is AB? Explain your reasoning. SOLUTION: a. We know BAE and DCE are congruent because they are both right angles. is congruent to by the Midpoint Theorem. From the Vertical Angles Theorem, DEC BEA. By ASA, the surveyor knows that By CPCTC, so the surveyor can measure and know the distance between A and B. b. Since DC = 550 m and then by the definition of congruence, AB = 550 m. ANSWER: a. We know BAE and DCE are congruent because they are both right angles. is congruent to by the Midpoint Theorem. From the Vertical Angles Theorem, DEC BEA. By ASA, the surveyor knows that By CPCTC, so the surveyor can measure and know the distance between A and B. b. 550 m; Since DC = 550 m and then by the definition of congruence, AB = 550 m. PROOF Write a paragraph proof. 6. Given: bisects BED; BCE and Prove: ECD are right angles. SOLUTION: Proof: We are given that bisects BED and BCE and ECD are right angles. Since all right angles are congruent, BCE ECD. By the definition of angle bisector, Reflexive Property tells us that By Angle-Side-Angle Congruence Postulate, BEC DEC. The ANSWER: Proof: We are given that bisects BED and BCE and ECD are right angles. Since all right angles are congruent, BCE by Cognero ECD. By the definition of angle bisector, BEC DEC. The Reflexive Property tells eSolutions Manual - Powered Page 4 us that By Angle-Side-Angle Congruence Postulate, Midpoint Theorem. From the Vertical Angles Theorem, DEC BEA. By ASA, the surveyor knows that By CPCTC, so the surveyor can measure and know the distance between A and B. 6-4 Proving Triangles Congruent - ASA, AAS b. 550 m; Since DC = 550 m and then by the definition of congruence, AB = 550 m. PROOF Write a paragraph proof. 6. Given: bisects BED; BCE and Prove: ECD are right angles. SOLUTION: Proof: We are given that bisects BED and BCE and ECD are right angles. Since all right angles are congruent, BCE ECD. By the definition of angle bisector, Reflexive Property tells us that By Angle-Side-Angle Congruence Postulate, BEC DEC. The ANSWER: Proof: We are given that bisects BED and BCE and ECD are right angles. Since all right angles are congruent, BCE ECD. By the definition of angle bisector, BEC DEC. The Reflexive Property tells us that By Angle-Side-Angle Congruence Postulate, 7. Given: Prove: bisects SOLUTION: Proof: It is given that W Y, and bisects YZX. The Angle-Side-Angle Congruence Postulate tells us that WZY. By the definition of angle bisector, WZX Proof: It is given that W Y, and bisects WZY. By the definition of angle bisector, YZX. The Angle-Side-Angle Congruence Postulate tells us that WZX ANSWER: 8. TOYS The object of the toy shown is to make the two spheres meet and strike each other repeatedly on one side of the wand and then again on the other side. If JKL MLK and JLK MKL, prove that SOLUTION: Proof: Statements (Reasons) eSolutions Manual - Powered by Cognero 1. JKL MLK, JLK 2. (Reflexive Prop.) Page 5 MKL (Given) The Angle-Side-Angle Congruence Postulate tells us that ANSWER: Proof: It Triangles is given thatCongruent W Y, bisects WZY. By the definition of angle bisector, 6-4 Proving - ASA, AAS and YZX. The Angle-Side-Angle Congruence Postulate tells us that WZX 8. TOYS The object of the toy shown is to make the two spheres meet and strike each other repeatedly on one side of the wand and then again on the other side. If JKL MLK and JLK MKL, prove that SOLUTION: Proof: Statements (Reasons) 1. JKL MLK, JLK 2. (Reflexive Prop.) 3. (ASA) 4. (CPCTC) MKL (Given) ANSWER: Proof: Statements (Reasons) 1. JKL MLK, JLK 2. (Refl. Prop.) 3. (ASA) 4. (CPCTC) MKL (Given) PROOF Write a two-column proof. 9. Given: V is the midpoint of Prove: SOLUTION: Proof: Statements (Reasons) 1. V is the midpoint of 2. 3. 4. 5. (Given) (Midpoint Theorem) VWX VYU (Alternate Interior Angles Theorem) VUY VXW (Alternate Interior Angles Theorem) (AAS) ANSWER: Proof: Statements (Reasons) 1. V is the midpoint of 2. 3. 4. 5. (Given) (Midpoint Theorem) VWX VYU (Alt. Int. Thm.) VUY VXW (Alt. Int. Thm.) (AAS) eSolutions Manual - Powered by Cognero Page 6 1. JKL MLK, JLK MKL (Given) 2. (Refl. Prop.) 3. 6-4 Proving Triangles (ASA) Congruent - ASA, AAS 4. (CPCTC) PROOF Write a two-column proof. 9. Given: V is the midpoint of Prove: SOLUTION: Proof: Statements (Reasons) 1. V is the midpoint of 2. 3. 4. 5. VWX VUY (Given) (Midpoint Theorem) VYU (Alternate Interior Angles Theorem) VXW (Alternate Interior Angles Theorem) (AAS) ANSWER: Proof: Statements (Reasons) 1. V is the midpoint of 2. 3. 4. 5. (Given) (Midpoint Theorem) VWX VYU (Alt. Int. Thm.) VUY VXW (Alt. Int. Thm.) (AAS) 10. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. (Given) 2. SPM QPR (Vertical angles are congruent.) 3. SMP QRP (Alternate Interior Angle Theorem) 4. (AAS) ANSWER: Proof: Statements (Reasons) eSolutions Manual - Powered by Cognero 1. (Given) 2. SPM QPR (Vert. s are .) 3. SMP QRP (Alt. Int. Thm.) Page 7 2. (Midpoint Theorem) 3. VWX VYU (Alt. Int. Thm.) 4. VUY VXW (Alt. Int. Thm.) 6-4 Proving Triangles (AAS) Congruent - ASA, AAS 5. 10. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. (Given) 2. SPM QPR (Vertical angles are congruent.) 3. SMP QRP (Alternate Interior Angle Theorem) 4. (AAS) ANSWER: Proof: Statements (Reasons) 1. (Given) 2. SPM QPR (Vert. s are .) 3. SMP QRP (Alt. Int. Thm.) 4. (AAS) 11. CCSS ARGUMENTS Write a flow proof. Given: A and C are right angles. ABE CBD, Prove: SOLUTION: Proof: eSolutions Manual - Powered by Cognero ANSWER: Page 8 1. (Given) 2. SPM QPR (Vert. s are .) 3. SMP QRP (Alt. Int. - ASA, Thm.)AAS 6-4 Proving Triangles Congruent 4. (AAS) 11. CCSS ARGUMENTS Write a flow proof. Given: A and C are right angles. ABE CBD, Prove: SOLUTION: Proof: ANSWER: Proof: 12. PROOF Write a flow proof. Given: bisects JML; Prove: eSolutions Manual - Powered by Cognero SOLUTION: J L. Page 9 6-4 Proving Triangles Congruent - ASA, AAS 12. PROOF Write a flow proof. Given: bisects JML; Prove: J L. SOLUTION: Proof: ANSWER: Proof: 13. CCSS MODELING A high school wants to hold a 1500-meter regatta on Lake Powell but is unsure if the lake is long enough. To measure the distance across the lake, the crew members locate the vertices of the triangles below and find the measures of the lengths of as shown below. eSolutions Manual - Powered by Cognero Page 10 6-4 Proving Triangles Congruent - ASA, AAS 13. CCSS MODELING A high school wants to hold a 1500-meter regatta on Lake Powell but is unsure if the lake is long enough. To measure the distance across the lake, the crew members locate the vertices of the triangles below and find the measures of the lengths of as shown below. a. Explain how the crew team can use the triangles formed to estimate the distance FG across the lake. b. Using the measures given, is the lake long enough for the team to use as the location for their regatta? Explain your reasoning. SOLUTION: a. HJK GFK since all right angles are congruent. We are given that HKJ and FKG are vertical angles, so HKJ Theorem. By ASA, so by CPCTC. b. Since and HJ = 1350, FG = 1350. If the regatta is to be 1500 m, the lake is not long enough, since 1350 < 1500. FKG by the Vertical Angles ANSWER: a. HJK GFK since all right angles are congruent. We are given that HKJ and FKG are vertical angles, so HKJ FKG by the Vertical Angles Theorem. By ASA, so by CPCTC. b. No; HJ = 1350 m, so FG = 1350 m. If the regatta is to be 1500 m, the lake is not long enough, since 1350 < 1500. ALGEBRA Find the value of the variable that yields congruent triangles. 14. SOLUTION: Since , the corresponding sides are congruent. Therefore, congruence, BC = WX. By the definition of ANSWER: x =3 15. eSolutions Manual - Powered by Cognero Page 11 ANSWER: 6-4 Proving x = 3 Triangles Congruent - ASA, AAS 15. SOLUTION: Since , the corresponding sides are congruent. Therefore, congruence, HJ = QJ. By the definition of ANSWER: y =5 16. THEATER DESIGN The trusses of the roof of the outdoor theater shown below appear to be several different pairs of congruent triangles. Assume that trusses that appear to lie on the same line actually lie on the same line. Refer to the figure on page 280. a. If b. If c. If bisects CBD and CAD, prove that and FCA EDA, prove that BHG BEA, HGJ EAD, and JGB DAB, prove that SOLUTION: a. Given: Prove: bisects CBD and CAD. Proof: Statements (Reasons) 1. bisects CBD and CAD. (Given) 2. ABC ABD, CAB DAB (Definition of bisect) 3. (Reflexive Property) 4. (ASA) b. Given: FCA EDA Prove: eSolutions Manual - Powered by Cognero Page 12 2. ABC ABD, CAB DAB (Definition of bisect) 3. (Reflexive Property) 4. (ASA) 6-4 Proving Triangles Congruent - ASA, AAS b. Given: FCA EDA Prove: Proof: Statements (Reasons) 1. FCA EDA (Given) 2. (CPCTC) 3. CAF DAE (Vertically opposite angles are congruent.) 4. (ASA) c. Given: BHG BEA, HGJ EAD, JGB Prove: DAB Proof: Statements (Reasons) 1. BHG BEA, HGJ EAD, JGB DAB (Given) 2. m HGJ = m EAD, m JGB = m DAB (Definition of Congruence) 3. m HGJ + m JGB = m HGB, m EAD + m DAB = m EAB (Addition Property of Equality) 4. m EAD + m DAB = m HGB, m EAD + m DAB = m EAB (Angle Addition Postulate) 5. m HGB = m EAB (Substitution) 6. HGB EAB (Definition of congruence) 7. (AAS) ANSWER: a. Given: Prove: bisects CBD and eSolutions Manual - Powered by Cognero CAD. Page 13 7. (AAS) ANSWER: 6-4 Proving Triangles Congruent - ASA, AAS a. Given: bisects CBD and CAD. Prove: Proof: Statements (Reasons) 1. bisects CBD and CAD. (Given) 2. ABC ABD, CAB DAB (Def. of bisect) 3. (Refl. Prop.) 4. (ASA) b. Given: FCA EDA Prove: Proof: Statements (Reasons) 1. FCA 2. (CPCTC) 3. CAF DAE (Vert. 4. (ASA) c. Given: BHG Prove: EDA (Given) s are .) BEA, HGJ EAD, JGB DAB Proof: eSolutions Manual - Powered by Cognero Statements (Reasons) 1. BHG BEA, Page 14 HGJ EAD, JGB DAB (Given) 6-4 Proving Triangles Congruent - ASA, AAS Proof: Statements (Reasons) 1. BHG BEA, HGJ EAD, JGB DAB (Given) 2. m HGJ = m EAD, m JGB = m DAB (Def. of ) 3. m HGJ + m JGB = m HGB, m EAD + m DAB = m EAB (Add Prop of =) 4. m EAD + m DAB = m HGB, m EAD + m DAB = m EAB (Angle Add. Post.) 5. m HGB = m EAB (Subst.) 6. HGB EAB (Def. of congruence) 7. (AAS) PROOF Write a paragraph proof. 17. Given: C is the midpoint of Prove: SOLUTION: Proof: We are given that Since is perpendicular to is perpendicular to m is perpendicular to and C is the midpoint of is perpendicular to CED = 90. Since m BAC because all right angles are congruent. from the Midpoint Theorem. because they are vertical angles. Angle-Side-Angle postulate gives us that corresponding parts of congruent triangles are congruent. BAC = 90. ECD . CED ACB because ANSWER: Proof: We are given that Since midpoint of is perpendicular to is perpendicular to is perpendicular to m CED = 90. Since 90. CED BAC because all right angles are congruent. because they are vertical angles. Angle-Side-Angle gives us that parts of congruent triangles are congruent. 18. Given: F and C is the is perpendicular to m BAC = from the Midpt. Thm. ECD ACB because corresponding J, Prove: SOLUTION: Proof: F J and because it is given. FHG JGH because they are alternate interior angles. By the Reflexive Property, eSolutions Cognero So Manual - Powered by by the Angle-Angle-Side postulate. Then triangles are congruent. Page 15 since corresponding parts of congruent Since midpoint of is perpendicular to m CED = 90. Since 90. CED BAC because all right angles are congruent. because they are vertical angles. Angle-Side-Angle gives us that 6-4 Proving Trianglestriangles Congruent - ASA, AAS parts of congruent are congruent. 18. Given: F is perpendicular to m BAC = from the Midpt. Thm. ECD ACB because corresponding J, Prove: SOLUTION: Proof: F J and because it is given. FHG JGH because they are alternate interior angles. By the Reflexive Property, So by the Angle-Angle-Side postulate. Then triangles are congruent. since corresponding parts of congruent ANSWER: Proof: F J and because it is given. FHG JGH because they are alternate interior angles. By the Reflexive Property, So by the Angle-Angle-Side postulate. Then since corresponding parts of congruent triangles are congruent. PROOF Write a two-column proof. 19. Given: K M , Prove: KPL MRL SOLUTION: Proof: Statements (Reasons) 1. K M , (Given) 2. KPR and MRP are both right angles. (Definition of Perpendicular lines.) 3. KPR MRP (All right angles are congruent.) 4. (Reflexive Prop.) 5. (AAS) 6. (CPCTC) 7. (Vertical angles are congruent.) 8. (AAS) 9. (CPCTC) ANSWER: Proof: Statements (Reasons) 1. K M , (Given) eSolutions Manual - Powered by Cognero 2. KPR and MRP are both right angles. (Def. of ⊥) 3. KPR MRP (All rt. s are congruent.) 4. (Refl. Prop.) Page 16 Proof: F J and because it is given. FHG JGH because they are alternate interior angles. By the Reflexive Property, So by the Angle-Angle-Side postulate. Then 6-4 Proving Triangles Congruent - ASA, AAS since corresponding parts of congruent triangles are congruent. PROOF Write a two-column proof. 19. Given: K M , Prove: KPL MRL SOLUTION: Proof: Statements (Reasons) 1. K M , (Given) 2. KPR and MRP are both right angles. (Definition of Perpendicular lines.) 3. KPR MRP (All right angles are congruent.) 4. (Reflexive Prop.) 5. (AAS) 6. (CPCTC) 7. (Vertical angles are congruent.) 8. (AAS) 9. (CPCTC) ANSWER: Proof: Statements (Reasons) 1. K M , (Given) 2. KPR and MRP are both right angles. (Def. of ⊥) 3. KPR MRP (All rt. s are congruent.) 4. (Refl. Prop.) 5. (AAS) 6. (CPCTC) 7. (Vertical angles are .) 8. (AAS) 9. (CPCTC) 20. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. (Given) 2. Manual QRV - Powered (Vertical angles are congruent.) SRW eSolutions by Cognero 3. (SAS) 4. VQR SWR (CPCTC) Page 17 6. (CPCTC) 7. (Vertical angles are .) 8. (AAS) 6-4 Proving Triangles (CPCTC) Congruent - ASA, AAS 9. 20. Given: Prove: SOLUTION: Proof: Statements (Reasons) 1. (Given) 2. QRV SRW (Vertical angles are congruent.) 3. (SAS) 4. VQR SWR (CPCTC) 5. QRT URW (Vertical angles are congruent.) 6. (ASA) 7. (CPCTC) ANSWER: Proof: Statements (Reasons) 1. (Given) 2. QRV SRW (Vert. s are 3. (SAS) 4. VQR SWR (CPCTC) 5. QRT URW (Vert. s are 6. (ASA) 7. .) .) (CPCTC) 21. FITNESS The seat tube of a bicycle forms a triangle with each seat and chain stay as shown. If each seat stay makes a 44° angle with its corresponding chain stay and each chain stay makes a 68° angle with the seat tube, show that the two seat stays are the same length. SOLUTION: Proof: Statements (Reasons) 1. m ACB = 44, m ADB = 44, m CBA = 68, m DBA = 68 (Given) 2. m ACB = m ADB, m CBA = m DBA (Substitution) eSolutions by Cognero 3. Manual ACB - Powered CBA DBA (Definition of congruence) ADB, 4. (Reflexive Property) 5. (AAS) Page 18 4. VQR SWR (CPCTC) 5. QRT URW (Vert. s are .) 6. (ASA) 6-4 Proving Triangles Congruent - ASA, AAS 7. (CPCTC) 21. FITNESS The seat tube of a bicycle forms a triangle with each seat and chain stay as shown. If each seat stay makes a 44° angle with its corresponding chain stay and each chain stay makes a 68° angle with the seat tube, show that the two seat stays are the same length. SOLUTION: Proof: Statements (Reasons) 1. m ACB = 44, m ADB = 44, m CBA = 68, m DBA = 68 (Given) 2. m ACB = m ADB, m CBA = m DBA (Substitution) 3. ACB ADB, CBA DBA (Definition of congruence) 4. (Reflexive Property) 5. (AAS) 6. (CPCTC) ANSWER: Proof: Statements (Reasons) 1. m ACB = 44, m ADB = 44, m CBA = 68, m DBA = 68 (Given) 2. m ACB = m ADB, m CBA = m DBA (Subst.) 3. ACB ADB, CBA DBA (Def. of ) 4. (Refl. Prop.) 5. (AAS) 6. (CPCTC) 22. OPEN ENDED Draw and label two triangles that could be proved congruent by ASA. SOLUTION: Sample answer: ANSWER: Sample answer: eSolutions Manual - Powered by Cognero Page 19 3. ACB ADB, CBA DBA (Def. of 4. (Refl. Prop.) 5. 6-4 Proving Triangles (AAS) Congruent - ASA, AAS 6. (CPCTC) ) 22. OPEN ENDED Draw and label two triangles that could be proved congruent by ASA. SOLUTION: Sample answer: ANSWER: Sample answer: 23. CCSS CRITIQUE Tyrone says it is not possible to show that since ADE ACB, and A A by the Reflexive Property, Explain. Lorenzo disagrees, explaining that Is either of them correct? SOLUTION: Tyrone is correct. Lorenzo showed that all three corresponding angles were congruent, but AAA is not a proof of triangle congruence. ANSWER: Tyrone; Lorenzo showed that all three corresponding angles were congruent, but AAA is not a proof of triangle congruence. 24. REASONING Find a counterexample to show why SSA (Side-Side-Angle) cannot be used to prove the congruence of two triangles. SOLUTION: Sample answer: To find a counterexample, show a set of triangles with corresponding SSA congruence and then show that at least one pair of the other 2 corresponding angles are not congruent. If SSA was a valid congruence theorem, then each pair of corresponding angles would be congruent. Consider triangles ABC and XYZ. . is obtuse while C Z. However, is acute so eSolutions Manual - Powered by Cognero Page 20 Lorenzo showed that all three corresponding angles were congruent, but AAA is not a proof of triangle congruence. ANSWER: Tyrone; Lorenzo showed that all -three angles were congruent, but AAA is not a proof of triangle 6-4 Proving Triangles Congruent ASA,corresponding AAS congruence. 24. REASONING Find a counterexample to show why SSA (Side-Side-Angle) cannot be used to prove the congruence of two triangles. SOLUTION: Sample answer: To find a counterexample, show a set of triangles with corresponding SSA congruence and then show that at least one pair of the other 2 corresponding angles are not congruent. If SSA was a valid congruence theorem, then each pair of corresponding angles would be congruent. Consider triangles ABC and XYZ. . is obtuse while C Z. However, is acute so ANSWER: Sample answer: C Z. 25. CHALLENGE Using the information given in the diagram, write a flow proof to show that SOLUTION: Proof: eSolutions Manual - Powered by Cognero Page 21 6-4 Proving Triangles Congruent - ASA, AAS SOLUTION: Proof: ANSWER: Proof: eSolutions Manual - Powered by Cognero Page 22 6-4 Proving Triangles Congruent - ASA, AAS ANSWER: Proof: 26. WRITING IN MATH Summarize the methods described in Lessons 4-3, 4-4, and 4-5 for proving triangle congruence into a chart that explains when to use each method. SOLUTION: eSolutions Manual - Powered by Cognero Page 23 26. WRITING IN MATH Summarize the methods described in Lessons 4-3, 4-4, and 4-5 for proving triangle 6-4 Proving Triangles Congruent - ASA, AAS congruence into a chart that explains when to use each method. SOLUTION: ANSWER: 27. Given: is perpendicular to eSolutions Manual - Powered by Cognero 1 2. Page 24 6-4 Proving Triangles Congruent - ASA, AAS 27. Given: is perpendicular to 1 2. Which theorem or postulate could be used to prove A AAS B ASA C SAS D SSS SOLUTION: Given: By the definition of perpendicular lines, And by the Reflexive property. Therefore, by ASA postulate, The correct choice is B. That is, ANSWER: B 28. SHORT RESPONSE Write an expression that can be used to find the values of s(n) in the table. SOLUTION: Is the expression linear? To check, confirm if there is a constant rate of increase (slope). As the input value n increases from -1 to 0, the output value s(n) increases from 2.75 to 3.00. As the input value n increases from 0 to , the output value s(n) increases from 3.00 to 3.25. We see a constant increase of 0.25 for every increase in input by 1. The same constant can be found when testing other values of n. This constant increase of 0.25 is equivalent to a slope m of . We now know the expression is linear. The yintercept is at (0, 3.00), so the value of b in mx + b is 3 Therefore, the expression for s(n) could be ANSWER: 29. ALGEBRA If –7 is multiplied by a number greater than 1, which of the following describes the result? F a number greater than 7 G a number between –7 and 7 H a number greater than –7 J a number less than –7 SOLUTION: eSolutions Manual - Powered by Cognero When a number greater than 1 is multiplied with –7, we get a negative number less than –7. Page 25 ANSWER: 6-4 Proving Triangles Congruent - ASA, AAS 29. ALGEBRA If –7 is multiplied by a number greater than 1, which of the following describes the result? F a number greater than 7 G a number between –7 and 7 H a number greater than –7 J a number less than –7 SOLUTION: When a number greater than 1 is multiplied with –7, we get a negative number less than –7. For example: 4 × –7 = –28 –28 < –7 2 × –7 = –14 –14 < –7 1.1 × –7 = –7.7 –7.7 < –7 The correct choice is J. ANSWER: J 30. SAT/ACT A 15 B 21 C 25 D 125 E 225 SOLUTION: The correct choice is A. ANSWER: A Determine whether Explain. 31. A(6, 4), B(1, –6), C(–9, 5), X(0, 7), Y(5, –3), Z(15, 8) SOLUTION: Use the distance formula to find the length of each side of the triangles. The side lengths of the triangle ABC are: Manual - Powered by Cognero eSolutions The side lengths of the triangle XYZ are: Page 26 The correct choice is A. ANSWER: 6-4 Proving Triangles Congruent - ASA, AAS A Determine whether Explain. 31. A(6, 4), B(1, –6), C(–9, 5), X(0, 7), Y(5, –3), Z(15, 8) SOLUTION: Use the distance formula to find the length of each side of the triangles. The side lengths of the triangle ABC are: The side lengths of the triangle XYZ are: The corresponding sides have the same measure and are congruent. by SSS. ANSWER: AB = BC = AC = same measure and are congruent. XY = YZ = by SSS. XZ = The corresponding sides have the 32. A(0, 5), B(0, 0), C(–2, 0), X(4, 8), Y(4, 3), Z(6, 3) SOLUTION: Use the distance formula to find the length of each side of the triangles. The side lengths of the triangle ABC are: The side lengths of the triangle XYZ are: The corresponding sides have the same measure and are congruent. by SSS. ANSWER: AB = 5, BC = 2, AC = congruent. XY = 5, YZ = 2, XZ = by SSS. eSolutions Manual - Powered by Cognero the corresponding sides have the same measure and are Page 27 33. ALGEBRA If RS = 7, ST = 5, RT = 9 + x, JL = 2x – 10, and JK = 4y – 5, draw and label a figure to represent the congruent triangles. Then find x and y. ANSWER: AB = BC = AC = XY = 6-4 Proving Triangles Congruent same measure and are congruent.- ASA, AAS YZ = by SSS. XZ = The corresponding sides have the 32. A(0, 5), B(0, 0), C(–2, 0), X(4, 8), Y(4, 3), Z(6, 3) SOLUTION: Use the distance formula to find the length of each side of the triangles. The side lengths of the triangle ABC are: The side lengths of the triangle XYZ are: The corresponding sides have the same measure and are congruent. by SSS. ANSWER: AB = 5, BC = 2, AC = congruent. XY = 5, YZ = 2, XZ = by SSS. the corresponding sides have the same measure and are 33. ALGEBRA If RS = 7, ST = 5, RT = 9 + x, JL = 2x – 10, and JK = 4y – 5, draw and label a figure to represent the congruent triangles. Then find x and y. SOLUTION: Corresponding sides of triangles RST and JKL are congruent. Since Solve the equation for y. . Since RT = JL. eSolutions Manual Solve for x.- Powered by Cognero Page 28 The corresponding sides have the same measure and are congruent. by SSS. ANSWER: AB = 5, BC = 2, AC Congruent = 5, YZ =AAS 2, XZ = XY = 6-4 Proving Triangles - ASA, congruent. by SSS. the corresponding sides have the same measure and are 33. ALGEBRA If RS = 7, ST = 5, RT = 9 + x, JL = 2x – 10, and JK = 4y – 5, draw and label a figure to represent the congruent triangles. Then find x and y. SOLUTION: Corresponding sides of triangles RST and JKL are congruent. Since Solve the equation for y. Since Solve for x. . RT = JL. ANSWER: x = 19; y = 3 34. BUSINESS Maxine charges $5 to paint a mailbox and $4 per hour to mow a lawn. Write an equation to represent the amount of money Maxine can earn from a homeowner who has his or her mailbox painted and lawn mowed. SOLUTION: Assume that x represents the time Maxine spends mowing a lawn and let y be the amount of money Maxine canPage 29 earn. The expression that represents the amount of money Maxine can earn could be: eSolutions Manual - Powered by Cognero 6-4 Proving Triangles Congruent - ASA, AAS 34. BUSINESS Maxine charges $5 to paint a mailbox and $4 per hour to mow a lawn. Write an equation to represent the amount of money Maxine can earn from a homeowner who has his or her mailbox painted and lawn mowed. SOLUTION: Assume that x represents the time Maxine spends mowing a lawn and let y be the amount of money Maxine can earn. The expression that represents the amount of money Maxine can earn could be: ANSWER: y = 4x + 5 Copy and complete each truth table. 35. SOLUTION: ANSWER: 36. SOLUTION: ANSWER: eSolutions Manual - Powered by Cognero Page 30 6-4 Proving Triangles Congruent - ASA, AAS 36. SOLUTION: ANSWER: PROOF Write a two-column proof for each of the following. 37. Given: 2 1 1 3 Prove: SOLUTION: Proof: Statements (Reasons) 1. 2 1, 1 3 (Given) 2. 2 3 (Transitive Property) 3. (If alternative interior angles are congruent, then the lines are parallel.) ANSWER: Proof: Statements (Reasons) 1. 2 1, 1 3 (Given) 2. 2 3 (Trans. Prop.) 3. (If alt. int. s are lines are .) 38. Given: MJK KLM LMJ and KLM are supplementary. Prove: eSolutions Manual - Powered by Cognero Page 31 Statements (Reasons) 1. 2 1, 1 3 (Given) 2. 2 3 (Trans. Prop.) 6-4 Proving Triangles Congruent AAS .) 3. s are - ASA, (If alt. int. lines are 38. Given: MJK KLM LMJ and KLM are supplementary. Prove: SOLUTION: Proof: Statements (Reasons) 1. MJK KLM, LMJ and KLM are supplementary. (Given) 2. m MJK = m KLM (Definition of congruent angles.) 3. m LMJ + m KLM = 180 (Definition of supplementary angles) 4. m LMJ + m MJK = 180 (Substitution) 5. LMJ and MJK are supplementary. (Definition of supplementary angles) 6. (If consecutive interior angles are supplementary, then the lines are parallel.) ANSWER: Proof: Statements (Reasons) 1. MJK KLM, LMJ and KLM are suppl. (Given) 2. m MJK = m KLM (Def. of s ) 3. m LMJ + m KLM = 180 (Def. of suppl. s) 4. m LMJ + m MJK = 180 (Subst.) 5. LMJ and MJK are suppl. (Def. of suppl. s) 6. (If cons. int. s are suppl., lines are ||.) eSolutions Manual - Powered by Cognero Page 32