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New Jersey Student Learning Assessments Geometry

Congruence, similarity, circles, coordinates, trigonometry, area, and volume.

These questions are original. This is not an official New Jersey Student Learning Assessment, and FreeExam is not affiliated with the New Jersey Department of Education.

  1. 1.On a Hoboken ferry map, a dock corner at (-2, 1) is translated 4 units right and 3 units up. Where does that corner land?

  2. 2.A chart of Sandy Hook reflects a buoy across the y-axis. The buoy starts at (5, -3). What are the image coordinates?

  3. 3.A kiosk corner on the Atlantic City boardwalk sits at (2, 5). The drawing is rotated 90 degrees counterclockwise about the origin. Where does that corner go?

  4. 4.Planners reflecting a Rutgers garden across the x-axis move a bench that starts at (-4, 6). What is the image of the bench?

  5. 5.A trail bolt on a Palisades map is at (3, -7). The map is rotated 180 degrees about the origin. Where is the bolt after the turn?

  6. 6.A mosaic tile in Morristown is translated 5 units right and then reflected across a vertical line. Which statement about the image tile is true?

  7. 7.A speaker on a Red Bank stage turntable starts at (-1, 4) and rotates 90 degrees clockwise about the origin. Where is the image of that point?

  8. 8.Every point on a Goethals Bridge approach drawing is moved 6 units left and 2 units down. A lamp starts at (1, 5). Where does the lamp finish?

  9. 9.A Cape May ferry logo is reflected across the line y = x. One corner of the logo is at (2, 9). What is the image of that corner?

  10. 10.A Newark proof about triangle NJC starts this way. Given: NJ = NC. Step 1: NJ = NC, because it is given. Step 2: Triangle NJC is isosceles, by the definition of an isosceles triangle. Step 3: Angle J is congruent to angle C. Which statement justifies step 3?

  11. 11.Two roads in Trenton cross and form vertical angles 1 and 2. Step 1: The roads intersect, because it is given. Step 2: Angle 1 and angle 2 are vertical angles, by definition. Step 3: Angle 1 is congruent to angle 2. Which statement justifies step 3?

  12. 12.Roof trusses in Montclair give triangles ABC and DEF with AB = DE, angle B = angle E, and BC = EF. Angle B sits between sides AB and BC, and angle E sits between sides DE and EF. Which theorem shows that the triangles are congruent?

  13. 13.Triangles MAP and LEG are sketched for a Camden bridge. Angle M equals angle L, angle A equals angle E, and side MP equals side LG. Side MP is opposite angle A, and side LG is opposite angle E. Which congruence theorem fits these given facts?

  14. 14.A Paterson park proof reaches these last lines. Step 4: Triangle ABD is congruent to triangle CBD. Step 5: AD = CD. Which statement justifies step 5?

  15. 15.On a Cherry Hill worksheet, triangles PQR and SQR share side QR. The proof includes the statement QR = QR. Which reason supports that statement?

  16. 16.Along a straight curb in Jersey City, angle 1 and angle 2 form a linear pair. Step 1 records that the linear pair is given. Step 2 says angle 1 + angle 2 = 180 degrees. Which statement justifies step 2?

  17. 17.Two right triangles mark a Newark library patio. Each has a right angle, the hypotenuses are both 13 feet, and a pair of corresponding legs are both 5 feet. Which theorem shows the triangles are congruent?

  18. 18.On a path in Paterson, M is the midpoint of segment AB. The next line of a proof says AM = MB. Which statement justifies that line?

  19. 19.At the Secaucus rail yard, line m is parallel to line n, and transversal t crosses both. The angle above line m and to the right of t measures 68 degrees. What is the corresponding angle above line n and to the right of t?

  20. 20.In Montclair, Oak Street is parallel to Maple Street, and a cut-through crosses both. The interior angle west of the cut-through and south of Oak Street measures 115 degrees. What is the alternate interior angle, east of the cut-through and north of Maple Street?

  21. 21.Parallel rails in Hoboken are crossed by a walkway. The interior angle east of the walkway and south of the north rail measures 72 degrees. What is the consecutive interior angle east of the walkway and north of the south rail?

  22. 22.In Ocean City, two parallel boardwalk edges are cut by a rope. The exterior angle north of the northern edge and west of the rope measures 41 degrees. What is the alternate exterior angle, south of the southern edge and east of the rope?

  23. 23.On a Trenton map, two parallel lines are cut by a transversal. A pair of corresponding angles measures (3x + 12) degrees and 78 degrees. What is the value of x?

  24. 24.On a Camden bridge drawing, same-side interior angles between parallel lines measure (2x + 10) degrees and (4x - 10) degrees. What is x?

  25. 25.On a Newark street diagram, a transversal cuts two lines, and one pair of alternate interior angles measures 54 degrees and 54 degrees. Which conclusion follows from that pair?

  26. 26.A Garden State Parkway sketch shows parallel lines cut by a transversal. One same-side interior angle measures 125 degrees. What is the other same-side interior angle?

  27. 27.Corresponding angles on a parallel-line diagram of a Paterson bus route measure (5x - 8) degrees and (3x + 16) degrees. What is the measure of each of those angles?

  28. 28.A triangular garden in Princeton has two angles of 47 degrees and 63 degrees. What is the third angle?

  29. 29.At a Hoboken lot, an exterior angle of a triangle measures 118 degrees. One remote interior angle is 49 degrees, and the other remote interior angle is x degrees. What is x?

  30. 30.An isosceles sail on a Barnegat Bay boat has a vertex angle of 40 degrees. What is the measure of each base angle?

  31. 31.A classroom model of a yield sign in Ewing is an equilateral triangle. What is the measure of each interior angle?

  32. 32.In a Morristown triangle, the remote interior angles for one exterior angle measure 35 degrees and 72 degrees. What is that exterior angle?

  33. 33.A right-triangle ramp in Jersey City has a right angle and one acute angle of 28 degrees. What is the other acute angle?

  34. 34.The angles of a triangular plaza in New Brunswick measure 2x degrees, 3x degrees, and 4x degrees. What is the largest angle?

  35. 35.In Cape May, one side of a triangle is extended, forming an exterior angle next to an interior angle of 56 degrees. The two angles make a straight line. What is the exterior angle?

  36. 36.An isosceles triangle painted on a wall in Asbury Park has base angles of 54 degrees each. What is the vertex angle?

  37. 37.A model of a tower in Newark uses the scale 1 centimeter represents 80 centimeters. The model tower is 4 centimeters tall. How tall is the real tower?

  38. 38.Similar triangles on a Delaware Water Gap map have corresponding sides of 6 meters and 15 meters. What is the scale factor from the smaller triangle to the larger triangle?

  39. 39.Similar triangular roofs in Montclair have a scale factor of 3 from the smaller roof to the larger roof. A side of the smaller roof is 7 feet. What is the corresponding side of the larger roof?

  40. 40.In a Princeton sketch, one triangle has angles of 35 degrees and 80 degrees, and a second triangle also has angles of 35 degrees and 80 degrees. Which reason shows that the triangles are similar?

  41. 41.Similar banners in Atlantic City have corresponding sides in the ratio 4 to 7, smaller to larger. A side of the shorter banner is 12 inches. What is the corresponding side of the longer banner?

  42. 42.A scale drawing of a rectangular lawn in Hoboken uses the rule that 1 centimeter on the drawing represents 2 centimeters on the lawn. The lawn covers 15 square centimeters on the drawing. What is the area of the actual lawn?

  43. 43.At the same moment in Trenton, a 6-foot post casts a 4-foot shadow and a flagpole casts an 18-foot shadow. The triangles formed with the ground are similar. How tall is the flagpole?

  44. 44.Two similar triangular paths around a lake in Sussex County have a scale from the larger path to the smaller path of 5 to 2. The larger perimeter is 40 meters. What is the smaller perimeter?

  45. 45.In a Cherry Hill classroom triangle, a midsegment joins the midpoints of two sides. The third side measures 18 centimeters. How long is the midsegment?

  46. 46.A Jersey City ramp is the hypotenuse of a right triangle. For angle R, the opposite side is 3 meters, the adjacent side is 4 meters, and the hypotenuse is 5 meters. What is sin R?

  47. 47.A kite string in Cape May forms a right triangle with sides 3 feet, 4 feet, and hypotenuse 5 feet. Angle K is next to the side of length 3 feet. What is cos K?

  48. 48.A ladder against a Paterson warehouse forms a right triangle. For angle L, the opposite side is 8 feet, the adjacent side is 6 feet, and the hypotenuse is 10 feet. What is tan L?

  49. 49.For angle A in a Sandy Hook dune diagram, sin A = 5/13, and the hypotenuse is 26 meters. What is the length of the side opposite angle A?

  50. 50.For roof angle B in Princeton, cos B = 8/17, and the side adjacent to angle B is 24 feet. What is the hypotenuse?

  51. 51.A trail angle C at the Palisades has tan C = 3/4. The side adjacent to angle C is 20 meters. What is the side opposite angle C?

  52. 52.For angle D at a Hoboken pier, the opposite side is 12 feet, the adjacent side is 5 feet, and the hypotenuse is 13 feet. What is tan D?

  53. 53.In a right triangle for a Camden bridge, angle E and angle F are the two acute angles, and sin E = 7/25. What is cos F?

  54. 54.A wire from the top of a pole in Edison to the ground is the hypotenuse of a right triangle. The sine of the angle between the wire and the ground is 4/5, and the wire is 15 feet long. How high is the pole?

  55. 55.A path on a coordinate plane in Liberty State Park runs from (1, 2) to (4, 6). What is the distance?

  56. 56.An Atlantic City grid shows a segment from (-2, 5) to (6, 1). What is the midpoint of the segment?

  57. 57.Two buoys in Raritan Bay are charted at (3, -4) and (3, 8). What is the distance between the buoys?

  58. 58.A path between two Newark libraries runs from (0, -6) to (10, 4) on a city grid. What is the midpoint of the path?

  59. 59.Two trail markers near the Delaware Water Gap sit at (1, 1) and (6, 13) on a ranger grid. What is the distance between the markers?

  60. 60.A Hudson River ferry segment on a chart runs from (-8, -3) to (4, 7). What is the midpoint?

  61. 61.A seawall in Long Branch runs horizontally from (-5, 2) to (9, 2) on a planner grid. What is the distance from one end to the other?

  62. 62.The midpoint of a segment on a New Brunswick campus map is (3, 5). One endpoint is (1, 2). What is the other endpoint?

  63. 63.A drone over the Meadowlands flies in a straight line from (0, 0) to (8, 15) on a coordinate grid. What is the distance of that flight?

  64. 64.A circular fountain in Branch Brook Park has center (2, -3) and radius 5. Which equation matches the fountain?

  65. 65.A circular pond in Cape May has equation (x + 4)^2 + (y - 1)^2 = 36. What is the center of the pond?

  66. 66.A carousel model in Asbury Park follows (x - 1)^2 + (y + 2)^2 = 49. What is the radius?

  67. 67.A radar circle centered at the origin on a Sandy Hook chart has equation x^2 + y^2 = 25. Which point lies on the circle?

  68. 68.A circular stage in Red Bank has a diameter with endpoints (-2, 1) and (6, 7). Which equation represents the circle?

  69. 69.A circular garden bed in Edison has equation x^2 + y^2 - 6x + 4y - 12 = 0. What are the center and radius?

  70. 70.A sprinkler in Edison covers the circle (x - 2)^2 + (y + 1)^2 = 25. Which point is outside the watered circle?

  71. 71.A circular pond on a Morristown map is centered at the origin and passes through (9, 12). What is the radius?

  72. 72.A central angle of a circular window in Princeton measures 80 degrees. What is the measure of the arc intercepted by that angle?

  73. 73.An inscribed angle on a mural in Newark intercepts an arc of 110 degrees. What is the measure of the inscribed angle?

  74. 74.An inscribed angle on a Cape May compass rose measures 37 degrees. What is the measure of its intercepted arc?

  75. 75.In a circular plaza in Jersey City, an inscribed angle intercepts a semicircle because its sides pass through the endpoints of a diameter. What is the measure of the inscribed angle?

  76. 76.A clock garden in Morristown is a full circle split into three arcs. Two of the arcs measure 95 degrees and 140 degrees. What is the measure of the remaining arc?

  77. 77.A central angle of a circular medallion in Princeton measures (2x + 10) degrees and intercepts an arc of 86 degrees. What is x?

  78. 78.Two inscribed angles on a Trenton seal intercept the same arc of 64 degrees. What is the measure of each inscribed angle?

  79. 79.On a circular window in Hoboken, a central angle and an inscribed angle intercept the same arc. The central angle measures 150 degrees. What is the inscribed angle?

  80. 80.Between two points on a circular patio in Cape May, the minor arc measures 80 degrees. What is the major arc between those same points?

  81. 81.A quarter-circle sprinkler in a Hoboken community garden has radius 10 feet. Use 3.14 for pi. What is the area of the watered sector?

  82. 82.A painted pizza sign on the Asbury Park boardwalk is a 60 degree sector of a circle with radius 6 inches. Use 3.14 for pi. What is the area of the sign?

  83. 83.Volunteers in Cape May planted a semicircular flower bed with radius 6 feet. Use 3.14 for pi. What is the area of the bed?

  84. 84.A paper fan used as a Princeton theater prop is a 90 degree sector with radius 4 inches. Use 3.14 for pi. What is the area of the fan?

  85. 85.A circular badge for a Newark parade has a 72 degree colored wedge and radius 5 centimeters. Use 3.14 for pi. What is the area of the wedge?

  86. 86.Groundskeepers marked a 45 degree wedge of a circular field in the Meadowlands. The radius is 20 meters. Use 3.14 for pi. What is the area of the wedge?

  87. 87.A stage light in Red Bank covers 270 degrees of a circular floor with radius 8 feet. Use 3.14 for pi. What is the area of the lit sector?

  88. 88.The town seal sketched in a Trenton workshop is a circle of radius 15 centimeters with a 120 degree colored sector. Use 3.14 for pi. What is the area of the colored sector?

  89. 89.A shipping crate at Port Newark is a rectangular prism 4 feet long, 3 feet wide, and 5 feet high. What is the volume of the crate?

  90. 90.A storage bin in a Trenton classroom is a rectangular prism 12 feet long, 2 feet wide, and 3 feet high. What is the volume?

  91. 91.A tent in a Jersey City camp is a triangular prism. Each triangular end has base 6 feet and height 4 feet, and the tent is 10 feet long. What is the volume?

  92. 92.A cheese display in Hoboken is a triangular prism. The triangular face has base 8 centimeters and height 5 centimeters, and the prism is 9 centimeters long. What is the volume?

  93. 93.A cylindrical water tank in Jersey City has radius 3 meters and height 10 meters. Use 3.14 for pi. What is the volume?

  94. 94.A cylindrical silo in Sussex County has radius 5 feet and height 4 feet. Use 3.14 for pi. What is the volume?

  95. 95.A monument in Trenton is a square pyramid with a base 6 meters on each side and a height of 5 meters. What is the volume?

  96. 96.A glass pyramid in a Princeton museum has a rectangular base 10 feet by 6 feet and a height of 9 feet. What is the volume?

  97. 97.A cone-shaped sand pile in Atlantic City has radius 4 feet and height 6 feet. Use 3.14 for pi. What is the volume?

  98. 98.A funnel in a Newark lab is a cone with radius 2 centimeters and height 9 centimeters. Use 3.14 for pi. What is the volume?

  99. 99.A spherical buoy off Sandy Hook has radius 6 feet. Use 3.14 for pi. What is the volume of the buoy?

  100. 100.A globe in a Montclair classroom is a sphere of radius 9 inches. Use 3.14 for pi. What is the volume of the globe?

  101. 101.In a Jersey City geometry class of 40 students, 18 ride the bus and passed the quiz, 6 ride the bus and did not pass, 10 walk and passed, and 6 walk and did not pass. A student is picked at random from those who ride the bus. What is the probability that this student passed?

  102. 102.At a Newark museum, 12 adults bought a map, 18 adults did not, 8 children bought a map, and 12 children did not. A visitor is chosen at random from the people who bought a map. What is the probability that the visitor is a child?

  103. 103.A Cape May survey of 80 residents counted 15 who own a bike and live near the beach, 5 who own a bike and live inland, 25 who have no bike and live near the beach, and 35 who have no bike and live inland. A resident is chosen from those who live near the beach. What is the probability that this resident owns a bike?

  104. 104.In a Paterson cafeteria count of 30 students, 7 juniors chose salad, 8 juniors chose pizza, 5 seniors chose salad, and 10 seniors chose pizza. A student is chosen from those who picked salad. What is the probability that the student is a junior?

  105. 105.A Princeton library count of 60 books shows 9 fiction books overdue, 21 fiction books on time, 6 nonfiction books overdue, and 24 nonfiction books on time. A book is chosen from the overdue books. What is the probability that it is fiction?

  106. 106.At Secaucus station, 8 northbound riders have a bike, 42 northbound riders do not, 12 southbound riders have a bike, and 38 southbound riders do not. A rider is chosen from those with a bike. What is the probability that the rider is southbound?

  107. 107.Of 24 garden-club members in Morristown, 6 grew tomatoes and entered the fair, 6 grew tomatoes and did not enter, 3 did not grow tomatoes and entered, and 9 did not grow tomatoes and did not enter. A member is chosen from those who entered the fair. What is the probability that this member grew tomatoes?

  108. 108.Of 36 students in an Edison school club, 10 are in band and in honors, 8 are in band but not honors, 6 are in honors but not band, and 12 are in neither. A student is chosen from the honors students. What is the probability that the student is in band?