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Name: ________________________ Class: ___________________ Date: __________ ID: A GA Milestone Review Unit 1 ____ 1. The map shows a linear section of Highway 35. Today, the Ybarras plan to drive the 360 miles from Springfield to Junction City. They will stop for lunch in Roseburg, which is at the midpoint of the trip. If they have already traveled 55 miles this morning, how much farther must they travel before they stop for lunch? a. b. 125 mi 145 mi c. d. 180 mi 305 mi ____ 2. K is the midpoint of JL . JK = 6x and KL = 3x + 3 . Find J K, KL, and J L. a. J K = 1, KL = 1, J L = 2 c. J K = 12, KL = 12, J L = 6 b. J K = 6, KL = 6, J L = 12 d. J K = 18, KL = 18, J L = 36 ____ 3. BD bisects ∠ABC , m∠ABD = (7x − 1)°, and m∠DBC = (4x + 8)°. Find m∠ABD. a. m∠ABD = 22° c. m∠ABD = 40° b. m∠ABD = 3° d. m∠ABD = 20° ____ 4. Two angles with measures (2x 2 + 3x − 5)° and (x 2 + 11x − 7)° are supplementary. Find the value of x and the measure of each angle. a. x = 5; 60°; 30° c. x = 5; 60°; 120° b. x = 6; 85°; 95° d. x = 4; 40°; 90° ____ 5. Two lines intersect to form two pairs of vertical angles. ∠1 with measure (20x + 7)° and ∠3 with measure (5x + 7y + 49)° are vertical angles. ∠2 with measure (3x − 2y + 30)° and ∠4 are vertical angles. Find the values x and y and the measures of all four angles. a. x = 6; y = 10; 127°; 127°; 28°; 28° c. x = 5; y = 5; 107°; 107°; 73°; 73° b. x = 8; y = 11, 167°; 167°; 13°; 13° d. x = 7; y = 9; 147°; 147°; 33°; 33° → 1 Name: ________________________ ____ 6. Write a justification for each step, given that EG = FH . EG = FH EG = EF + FG FH = FG + GH EF + FG = FG + GH EF = GH a. b. c. d. ____ ID: A Given information [1] Segment Addition Postulate [2] Subtraction Property of Equality [1] Angle Addition Postulate [2] Subtraction Property of Equality [1] Substitution Property of Equality [2] Transitive Property of Equality [1] Segment Addition Postulate [2] Definition of congruent segments [1] Segment Addition Postulate [2] Substitution Property of Equality 7. Find m∠ABC . a. b. m∠ABC = 40° m∠ABC = 45° c. d. 2 m∠ABC = 35° m∠ABC = 50° Name: ________________________ ____ 8. Write and solve an inequality for x. a. b. ____ ID: A c. d. x>2 x<2 x>1 x < −2 9. Write a justification for each step. m∠JKL = 100° m∠JKL = m∠JKM + m∠MKL 100° = (6x + 8)° + (2x − 4)° 100 = 8x + 4 96 = 8x 12 = x x = 12 a. b. c. d. [1] Substitution Property of Equality Simplify. Subtraction Property of Equality [2] Symmetric Property of Equality [1] Transitive Property of Equality [2] Division Property of Equality [1] Angle Addition Postulate [2] Division Property of Equality [1] Angle Addition Postulate [2] Simplify. [1] Segment Addition Postulate [2] Multiplication Property of Equality 3 Name: ________________________ ID: A ____ 10. A gardener has 26 feet of fencing for a garden. To find the width of the rectangular garden, the gardener uses the formula P = 2l + 2w , where P is the perimeter, l is the length, and w is the width of the rectangle. The gardener wants to fence a garden that is 8 feet long. How wide is the garden? Solve the equation for w, and justify each step. P = 2l + 2w 26 = 2(8) + 2w 26 = 16 + 2w −16 = −16 10 = 2w 10 2w = 2 2 5=w w=5 a. b. Given equation [1] Simplify. Subtraction Property of Equality Simplify. [2] Simplify. Symmetric Property of Equality [1] Substitution Property of Equality [2] Division Property of Equality The garden is 5 ft wide. [1] Simplify [2] Division Property of Equality The garden is 5 ft wide. c. d. [1] Substitution Property of Equality [2] Subtraction Property of Equality The garden is 5 ft wide. [1] Subtraction Property of Equality [2] Simplify The garden is 5 ft wide. ____ 11. Find m∠RST . a. b. m∠RST = 108° m∠RST = 24° c. d. m∠RST = 156° m∠RST = 72° ____ 12. Three vertices of parallelogram WXYZ are X(–2,–3), Y(0, 5), and Z(7, 7). Find the coordinates of vertex W. a. (4, 0) c. (5, 0) b. (9, 15) d. (5, –1) 4 Name: ________________________ ID: A ____ 13. Fill in the blanks to complete the two-column proof. Given: ∠1 and ∠2 are supplementary. m∠1 = 135° Prove: m∠2 = 45° Proof: Statements 1. ∠1 and ∠2 are supplementary. 2. [1] 3. m∠1 + m∠2 = 180° 4. 135° + m∠2 = 180° 5. m∠2 = 45° a. b. c. d. Reasons 1. Given 2. Given 3. [2] 4. Substitution Property 5. [3] [1] m∠2 = 135° [2] Definition of supplementary angles [3] Subtraction Property of Equality [1] m∠1 = 135° [2] Definition of supplementary angles [3] Substitution Property [1] m∠1 = 135° [2] Definition of supplementary angles [3] Subtraction Property of Equality [1] m∠1 = 135° [2] Definition of complementary angles [3] Subtraction Property of Equality ____ 14. A video game designer is modeling a tower that is 320 ft high and 260 ft wide. She creates a model so that 1 the similarity ratio of the model to the tower is 500 . What is the height and the width of the model in inches? a. b. c. d. height = 0.64 in.; width = 0.52 in. height = 3840 in.; width = 3120 in. height = 7.68 in.; width = 6.24 in. height = 160,000 in.; width = 130,000 in. 5 Name: ________________________ ID: A ____ 15. Use the given flowchart proof to write a two-column proof of the statement AF ≅ FD . Flowchart proof: AB = CD ; BF = FC Given AB + BF = AF FC + CD = FD Segment Addition Postulate AB + BF = FC + CD Addition Property of Equality AF = FD AF ≅ FD Substitution Definition of congruent segments Complete the proof. Two-column proof: Statements 1. AB = CD ; BF = FC 2. [1] 3. [2] 4. AF = FD 5. AF ≅ FD a. b. c. d. [1] [2] [1] [2] [1] [2] [1] [2] Reasons 1. Given 2. Addition Property of Equality 3. Segment Addition Postulate 4. Substitution 5. Definition of congruent segments AB + BF = AF ; FC + CD = FD AF = FD AF = FD AB + BF = FC + CD AB = CD ; BF = FC AB + BF = FC + CD AB + BF = FC + CD AB + BF = AF ;FC + CD = FD 6 Name: ________________________ ID: A ____ 16. Use the given two-column proof to write a flowchart proof. Given: ∠1 ≅ ∠4 Prove: m∠2 = m ∠3 Two-column proof: Statements 1. ∠1 ≅ ∠4 2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary. 3. ∠2 ≅ ∠3 4. m∠2 = m ∠3 Reasons 1. Given 2. Definition of linear pair 3. Congruent Supplements Theorem 4. Definition of congruent segments Complete the proof. Flowchart proof: ∠1 ≅ ∠4 Given [1] ∠2 ≅ ∠3 Definition of linear pair a. b. c. d. [2] [1] ∠1 and ∠2 are supplements; ∠3 and ∠4 are supplementary [2] Congruent Complements Theorem [1] ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary [2] Congruent Supplements Theorem [1] ∠2 ≅ ∠3 [2] Definition of congruent segments [1] Definition of congruent segments [2] Congruent Supplements Theorem 7 m∠2 = m ∠3 Definition of congruent segments Name: ________________________ ID: A ____ 17. Use the given paragraph proof to write a two-column proof. Given: ∠BAC is a right angle. ∠1 ≅ ∠3 Prove: ∠2 and ∠3 are complementary. Paragraph proof: Since ∠BAC is a right angle, m∠BAC = 90° by the definition of a right angle. By the Angle Addition Postulate, m∠BAC = m∠1 + m∠2 . By substitution, m∠1 + m∠2 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the definition of congruent angles. Using substitution, m∠3 + m∠2 = 90°. Thus, by the definition of complementary angles, ∠2 and ∠3 are complementary. Complete the proof. Two-column proof: Statements 1. ∠BAC is a right angle. ∠1 ≅ ∠3 2. m∠BAC = 90° 3. m∠BAC = m∠1 + m∠2 4. m∠1 + m∠2 = 90° 5. m∠1 = m∠3 6. m∠3 + m∠2 = 90° 7. ∠2 and ∠3 are complementary. Reasons 1. Given 2. Definition of a right angle 3. [1] 4. Substitution 5. [2] 6. Substitution 7. Definition of complementary angles a. c. b. [1] Substitution [2] Definition of congruent angles [1] Angle Addition Postulate [2] Definition of congruent angles d. 8 [1] Angle Addition Postulate [2] Definition of equality [1] Substitution [2] Definition of equality Name: ________________________ ID: A ____ 18. Find the value of x. a. b. x=6 x=4 ____ 19. Determine whether triangles a. b. c. d. c. d. EFG and The triangles are congruent because (x, y) → (−x, y). The triangles are congruent because (x, y) → (−y, −x). The triangles are congruent because (x, y) → (x, −y). The triangles are congruent because (x, y) → (−y, x). x=2 x=8 PQR are congruent. EFG can be mapped to PQR by a reflection: EFG can be mapped to PQR by a rotation: EFG can be mapped to PQR by a reflection: EFG can be mapped to PQR by a rotation: 9 Name: ________________________ ID: A ____ 20. Given that ∆ABC ≅ ∆DEC and m∠E = 23°, find m∠ACB. a. b. m∠ACB = 77° m∠ACB = 67° c. d. m∠ACB = 23° m∠ACB = 113° c. d. m∠K = 79° m∠K = 39° ____ 21. Find m∠K . a. b. m∠K = 63° m∠K = 55° 10 Name: ________________________ ID: A ____ 22. Apply the transformation M to the triangle with the given vertices. Identify and describe the transformation. M: (x, y) → (x – 6, y + 2) E(3, 0), F(1, –2), G(5, –4) a. c. This is a translation 6 units left and 2 units up. b. This is a translation 6 units left. d. This is a translation 2 units left and 6 units up. This is a translation 6 units left and 2 units down. 11 Name: ________________________ ID: A ____ 23. Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation. M: (x, y) → (–x, –y) A(–3, 6), B(–3, 1), C(1, 1), D(1, 6) a. c. This is a reflection over the x-axis. This is a rotation of 180° about the origin. b. d. This is a rotation of 90° clockwise about the origin. This is a rotation of 180° about the origin. 12 Name: ________________________ ID: A ____ 24. Find m∠E and m∠N , given m∠F = m∠P, m∠E = (x 2 )°, and m∠N = (4x 2 − 75)°. a. b. m∠E = 25°, m∠N = 25° m∠E = 25°, m∠N = 65° c. d. m∠E = 65°, m∠N = 25° m∠E = 65°, m∠N = 65° ____ 25. Given: P is the midpoint of TQ and RS . Prove: ∆TPR ≅ ∆QPS Complete the proof. Proof: Statements 1. P is the midpoint of TQ and RS . Reasons 1. Given 2. TP ≅ QP, RP ≅ SP 3. [2] 4. ∆TPR ≅ ∆QPS 2. [1] a. c. b. [1]. Definition of midpoint [2] ∠TPR ≅ ∠QPS [3] SAS [1] Definition of midpoint [2] RT ≅ SQ [3] SSS 3. Vertical Angles Theorem 4. [3] d. 13 [1] Definition of midpoint [2] ∠PRT ≅ ∠PSQ [3] SAS [1] Definition of midpoint [2] ∠TPR ≅ ∠QPS [3] SSS Name: ________________________ ID: A ____ 26. What additional information do you need to prove ∆ABC ≅ ∆ADC by the SAS Postulate? a. b. AB ≅ AD ∠ACB ≅ ∠ACD c. d. ∠ABC ≅ ∠ADC BC ≅ DC ____ 27. Determine if you can use ASA to prove ∆CBA ≅ ∆CED. Explain. a. b. c. d. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. No other congruence relationships can be determined, so ASA cannot be applied. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Adjacent Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by SAS. 14 Name: ________________________ ID: A ____ 28. For these triangles, select the triangle congruence statement and the postulate or theorem that supports it. a. b. ∆ABC ≅ ∆JLK , HL ∆ABC ≅ ∆JKL, HL c. d. ∆ABC ≅ ∆JLK , SAS ∆ABC ≅ ∆JKL, SAS ____ 29. Two Seyfert galaxies, BW Tauri and M77, represented by points A and B, are equidistant from Earth, represented by point C. What is m∠A? a. b. m∠A = 65° m∠A = 115° c. d. 15 m∠A = 50° m∠A = 60° Name: ________________________ ID: A ____ 30. Given ∆ABC with AB = 3 , BC = 5 , and CA = 6, find the length of midsegment XY . a. b. XY = 3 XY = 1.5 c. d. XY = 2.5 XY = 2 ____ 31. Point O is the centroid of ∆ABC , BY = 3.3 and CO = 3 . Find BO . a. b. BO = 2.2 BO = 1.1 c. d. BO = 3.3 BO = 3 → ____ 32. Given that YW bisects ∠XYZ and WZ = 4.23, find WX . a. b. WX = 4.23 WX = 8.46 c. d. 16 WX = 45° WX = 90° Name: ________________________ ID: A ____ 33. The diagram shows a new kind of triangular bread. Where should the baker place her hand while spinning the dough so that the triangle is balanced? a. ÁÊ 1, 1 ˜ˆ Ë ¯ c. b. ÊÁ 1, 0 ˆ˜ Ë ¯ d. ÁÁÁÊ ÁË ÁÁÁÊ ÁË 1 2 3 2 ˆ , 1 ˜˜˜˜ ¯ ˆ˜˜ , 1 ˜˜ ¯ ____ 34. The diagram shows the parallelogram-shaped component that attaches a car’s rearview mirror to the car. In parallelogram RSTU, UR = 25, RX = 16, and m∠STU = 42.4o. Find ST, XT, and m∠RST. a. b. ST = 16, m∠RST = 42.4°, XT = 25 ST = 25, m∠RST = 47.8°, XT = 16 c. d. ST = 25, m∠RST = 137.6°, XT = 16 ST = 5, m∠RST = 137.6°, XT = 4 c. d. MP = 20 MP = 6 ____ 35. MNOP is a parallelogram. Find MP. a. b. MP = 25 MP = 30 17 Name: ________________________ ID: A ____ 36. An artist designs a rectangular quilt piece with different types of ribbon that go from the corner to the center of the quilt. The dimensions of the rectangle are AB = 10 inches and AC = 14 inches. Find BX . a. b. BX = 7 inches BX = 10 inches c. d. BX = 5 inches BX = 14 inches c. d. SU = 5 SU = 3 ____ 37. TRSU is a rhombus. Find SU . a. b. SU = 7 SU = 1 ____ 38. Apply the dilation D to the polygon with the given vertices. Name the coordinates of the image points. D: (x, y) → (3x, 3y) J(1, 4), K(6, 4), L(6, 1), M(1, 1) a. b. J´(12, 3), K´(12, 18), L´(3, 18), M´(3, 3) J´(–3, –12), K´(–18, –12), L´(–18, –3), M´(–3, –3) c. d. 18 J´(3, 12), K´(18, 12), L´(18, 3), M´(3, 3) J´(3, 12), K´(18, 12), L´(6, 1), M´(1, 1) Name: ________________________ ID: A ____ 39. To find out how wide a river is, Jon and Sally mark an X at the spot directly across from a big rock on the other side of the river. Then they walk in a straight line along the river, perpendicular to the straight line between the X and the rock. After walking for 20 feet Jon stops while Sally continues along the straight line for another 10 feet. Then she makes a 90 degree turn and walks for 30 feet. When she stops and looks at the rock she sees that the straight line from her to the rock passes through Jon. What is the distance from X to the rock? a. b. 30 feet 50 feet c. d. 60 feet 63 feet c. d. NP = 1.6 NP = 2 ____ 40. Find NP. a. b. NP = 1 NP = 1.25 19 Name: ________________________ ID: A ____ 41. An artist used perspective to draw guidelines in her picture of a row of parallel buildings. How many centimeters is it from Point B to Point C? a. b. 1 cm 3.75 cm c. d. 4 cm 2.4 cm c. d. BD = 10 BD = 12 ____ 42. Find BD . a. b. BD = 5 BD = 22 20 Name: ________________________ ID: A ____ 43. Given that ∆KON ∼ ∆LOM, find the coordinates of L and the scale factor. 4 3 a. L (6, 0) and scale factor is 2 c. L (9, 0) and scale factor is b. L (9, 0) and scale factor is 3 d. L (6, 0) and scale factor is 3 ____ 44. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.) a. b. m∠1 = 95° m∠1 = 80° c. d. 45. Find the value of x in the rhombus. 21 m∠1 = 85° m∠1 = 75° Name: ________________________ ID: A 46. Find the value of x so that m Ä n. 47. Find the value of n in the triangle. 22 ID: A GA Milestone Review Unit 1 Answer Section 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. A B D B D D C A B A D D C C D B B A C B A A B A A B C B A B A A A C B A A C C B B D 1 ID: A 43. 44. 45. 46. 47. B C 0.5 17 11 2