6-4 Proving Triangles Congruent - ASA, AAS. PROOF Write the specified type of proof. 1. two-column proof Given: bisects ABD and ACD.
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1 PROOF Write the specified type of proof. 1. two-column proof Given: bisects ABD and ACD. 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) 1. bisects ABD and ACD. (Given) 2. ABC DBC (Def. of bisector) 3. (Refl. Prop.) 4. ACB DCB (Def. of bisector) 5. (ASA) esolutions Manual - Powered by Cognero Page 1
2 2. flow proof Given: esolutions Manual - Powered by Cognero Page 2
3 3. paragraph proof Given: bisects We are given K M, and bisects KLM. Since bisects KLM, we know KLJ MLJ. So, by the Angle-Angle-Side Congruence Theorem. We are given K M, and bisects KLM. Since bisects KLM, we know KLJ MLJ. So, by the Angle-Angle-Side Congruence Theorem. 4. two-column proof Given: m G = m J = 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) 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) esolutions Manual - Powered by Cognero Page 3
4 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 = meters, what is AB? Explain your reasoning. 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. 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. esolutions Manual - Powered by Cognero Page 4
5 PROOF Write a paragraph proof. 6. Given: bisects BED; BCE and ECD are right angles. 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, 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: bisects It is given that W Y, and bisects WZY. By the definition of angle bisector, WZX YZX. The Angle-Side-Angle Congruence Postulate tells us that It is given that W Y, and bisects WZY. By the definition of angle bisector, WZX YZX. The Angle-Side-Angle Congruence Postulate tells us that esolutions Manual - Powered by Cognero Page 5
6 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 1. JKL MLK, JLK MKL (Given) 2. (Reflexive Prop.) 3. (ASA) 4. (CPCTC) 1. JKL MLK, JLK MKL (Given) 2. (Refl. Prop.) 3. (ASA) 4. (CPCTC) esolutions Manual - Powered by Cognero Page 6
7 PROOF Write a two-column proof. 9. Given: V is the midpoint of 1. V is the midpoint of (Given) 2. (Midpoint Theorem) 3. VWX VYU (Alternate Interior Angles Theorem) 4. VUY VXW (Alternate Interior Angles Theorem) 5. (AAS) 1. V is the midpoint of (Given) 2. (Midpoint Theorem) 3. VWX VYU (Alt. Int. Thm.) 4. VUY VXW (Alt. Int. Thm.) 5. (AAS) esolutions Manual - Powered by Cognero Page 7
8 10. Given: 1. (Given) 2. SPM QPR (Vertical angles are congruent.) 3. SMP QRP (Alternate Interior Angle Theorem) 4. (AAS) 1. (Given) 2. SPM QPR (Vert. s are.) 3. SMP QRP (Alt. Int. Thm.) 4. (AAS) esolutions Manual - Powered by Cognero Page 8
9 11. CCSS ARGUMENTS Write a flow proof. Given: A and C are right angles. ABE CBD, esolutions Manual - Powered by Cognero Page 9
10 12. PROOF Write a flow proof. Given: bisects JML; J L. esolutions Manual - Powered by Cognero Page 10
11 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. 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. Since and HJ = 1350, FG = If the regatta is to be 1500 m, the lake is not long enough, since 1350 < 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 < ALGEBRA Find the value of the variable that yields congruent triangles. 14. Since, the corresponding sides are congruent. Therefore, By the definition of congruence, BC = WX. x = 3 esolutions Manual - Powered by Cognero Page 11
12 15. Since, the corresponding sides are congruent. Therefore, By the definition of congruence, HJ = QJ. y = 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 bisects CBD and CAD, prove that b. If and FCA EDA, prove that c. If BHG BEA, HGJ EAD, and JGB DAB, prove that a. Given: bisects CBD and CAD. 1. bisects CBD and CAD. (Given) 2. ABC ABD, CAB DAB (Definition of bisect) 3. (Reflexive Property) 4. (ASA) b. Given: FCA EDA esolutions Manual - Powered by Cognero Page 12
13 1. FCA EDA (Given) 2. (CPCTC) 3. CAF DAE (Vertically opposite angles are congruent.) 4. (ASA) c. Given: BHG BEA, HGJ EAD, JGB DAB 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) a. Given: bisects CBD and CAD. esolutions Manual - Powered by Cognero Page 13
14 1. bisects CBD and CAD. (Given) 2. ABC ABD, CAB DAB (Def. of bisect) 3. (Refl. Prop.) 4. (ASA) b. Given: FCA EDA 1. FCA EDA (Given) 2. (CPCTC) 3. CAF DAE (Vert. s are.) 4. (ASA) c. Given: BHG BEA, HGJ EAD, JGB DAB esolutions Manual - Powered by Cognero Page 14
15 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 We are given that is perpendicular to is perpendicular to and C is the midpoint of. Since is perpendicular to m CED = 90. Since is perpendicular to m BAC = 90. CED BAC because all right angles are congruent. from the Midpoint Theorem. ECD ACB because they are vertical angles. Angle-Side-Angle postulate gives us that because corresponding parts of congruent triangles are congruent. We are given that is perpendicular to is perpendicular to and C is the midpoint of Since is perpendicular to m CED = 90. Since is perpendicular to m BAC = 90. CED BAC because all right angles are congruent. from the Midpt. Thm. ECD ACB because they are vertical angles. Angle-Side-Angle gives us that because corresponding parts of congruent triangles are congruent. esolutions Manual - Powered by Cognero Page 15
16 18. Given: F J, 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. 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. esolutions Manual - Powered by Cognero Page 16
17 PROOF Write a two-column proof. 19. Given: K M, KPL MRL 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) 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) esolutions Manual - Powered by Cognero Page 17
18 20. Given: 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) 1. (Given) 2. QRV SRW (Vert. s are.) 3. (SAS) 4. VQR SWR (CPCTC) 5. QRT URW (Vert. s are.) 6. (ASA) 7. (CPCTC) esolutions Manual - Powered by Cognero Page 18
19 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. 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) 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) esolutions Manual - Powered by Cognero Page 19
20 22. OPEN ENDED Draw and label two triangles that could be proved congruent by ASA. Sample answer: Sample answer: 23. CCSS CRITIQUE Tyrone says it is not possible to show that Lorenzo disagrees, explaining that since ADE ACB, and A A by the Reflexive Property, Is either of them correct? Explain. Tyrone is correct. Lorenzo showed that all three corresponding angles were congruent, but AAA is not a proof of triangle congruence. Tyrone; Lorenzo showed that all three corresponding angles were congruent, but AAA is not a proof of triangle congruence. esolutions Manual - Powered by Cognero Page 20
21 24. REASONING Find a counterexample to show why SSA (Side-Side-Angle) cannot be used to prove the congruence of two triangles. 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. C Z. However,. is obtuse while is acute so Sample answer: C Z. 25. CHALLENGE Using the information given in the diagram, write a flow proof to show that esolutions Manual - Powered by Cognero Page 21
22 esolutions Manual - Powered by Cognero Page 22
23 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. esolutions Manual - Powered by Cognero Page 23
24 esolutions Manual - Powered by Cognero Page 24
25 27. Given: is perpendicular to 1 2. Which theorem or postulate could be used to prove A AAS B ASA C SAS D SSS Given: By the definition of perpendicular lines, And by the Reflexive property. Therefore, by ASA postulate, The correct choice is B. That is, B 28. SHORT RESPONSE Write an expression that can be used to find the values of s(n) in the table. 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 As the input value n increases from 0 to, the output value s(n) increases from 3.00 to 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 y- intercept is at (0, 3.00), so the value of b in mx + b is 3 Therefore, the expression for s(n) could be esolutions Manual - Powered by Cognero Page 25
26 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 When a number greater than 1 is multiplied with 7, we get a negative number less than 7. For example: 4 7 = < = < = < 7 The correct choice is J. J 30. SAT/ACT A 15 B 21 C 25 D 125 E 225 The correct choice is A. A esolutions Manual - Powered by Cognero Page 26
27 Determine whether Explain. 31. A(6, 4), B(1, 6), C( 9, 5), X(0, 7), Y(5, 3), Z(15, 8) 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. AB = BC = AC = XY = YZ = XZ = The corresponding sides have the same measure and are congruent. by SSS. esolutions Manual - Powered by Cognero Page 27
28 32. A(0, 5), B(0, 0), C( 2, 0), X(4, 8), Y(4, 3), Z(6, 3) 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. AB = 5, BC = 2, AC = XY = 5, YZ = 2, XZ = the corresponding sides have the same measure and are congruent. by SSS. esolutions Manual - Powered by Cognero Page 28
29 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. Corresponding sides of triangles RST and JKL are congruent. Since. Solve the equation for y. Since RT = JL. Solve for x. x = 19; y = 3 esolutions Manual - Powered by Cognero Page 29
30 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. 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: y = 4x + 5 Copy and complete each truth table. 35. esolutions Manual - Powered by Cognero Page 30
31 36. PROOF Write a two-column proof for each of the following. 37. Given: , 1 3 (Given) (Transitive Property) 3. (If alternative interior angles are congruent, then the lines are parallel.) , 1 3 (Given) (Trans. Prop.) 3. (If alt. int. s are lines are.) esolutions Manual - Powered by Cognero Page 31
32 38. Given: MJK KLM LMJ and KLM are supplementary. 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.) 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
4-5 Proving Triangles Congruent - ASA, AAS. PROOF Write the specified type of proof. 1. two-column proof Given: bisects ABD and ACD.
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