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Virginia Standards of Learning End-of-Course Geometry

Congruence, similarity, circles, coordinates, right triangles, area, and volume.

These questions are original. This is not an official Virginia Standards of Learning test, and FreeExam is not affiliated with the Virginia Department of Education.

  1. 1.On a tour of Capitol Square in Richmond, a guide states that if a polygon is a square, then it has four right angles. Which statement is the converse?

  2. 2.A ranger on Skyline Drive posts that if a trail is icy, then it is closed for the day. Which statement is the contrapositive?

  3. 3.A sign in Luray Caverns states that if a formation is a stalactite, then it hangs from the ceiling. Which statement is the inverse?

  4. 4.At the Yorktown battlefield, a docent uses these true statements: if a cannon stands on the redoubt, then it faces the river. This cannon stands on the redoubt. What follows by the Law of Detachment?

  5. 5.Two notices at Monticello are true. If a room opens onto the garden, then it gets afternoon light. If a room gets afternoon light, then its shutters are left open. What follows by the Law of Syllogism?

  6. 6.A student at Historic Jamestowne claims that every quadrilateral with four equal sides is a square. Which figure is a counterexample?

  7. 7.At a Maymont workshop, 18 students sketch triangles, 14 sketch circles, and 6 sketch both. How many students sketch at least one of those two shapes?

  8. 8.On the Belle Isle footbridge, goose counts on four successive pilings are 4, 7, 10, and 13. Which statement is a reasonable inductive conclusion from this pattern alone?

  9. 9.A Shenandoah sign says a marker is a summit marker if and only if it names the highest point on that ridge. Which description matches that biconditional?

  10. 10.At Great Falls Park, these premises are true: if the overlook gate is latched, then the spur trail is closed. The spur trail is not closed. Which conclusion is valid?

  11. 11.On a King Street walk in Alexandria, a guide gives two facts: every brick townhouse on this block has black shutters, and the corner house is a brick townhouse on this block. Which conclusion follows?

  12. 12.A logic chart for the Mill Mountain star in Roanoke uses P for “the gate is open” and Q for “the path is lit.” When is the conditional “if P, then Q” false?

  13. 13.Two triangular trail signs at Natural Bridge match on two sides and on the angle formed by those sides. Which congruence theorem applies?

  14. 14.Boardwalk triangles at First Landing State Park match on one angle, the side between that angle and a second angle, and the second angle. Which theorem proves congruence?

  15. 15.Railing triangles on the High Bridge Trail have side lengths 9 feet, 14 feet, and 15 feet in both triangles. Which statement is true?

  16. 16.Two garden triangles at Mount Vernon have two pairs of congruent angles, and a side opposite one of those angles in the first triangle matches the corresponding side in the second. Which theorem gives congruence?

  17. 17.Two right triangles on a Fort Monroe seawall have hypotenuses of 25 feet and a pair of corresponding legs of 7 feet. Which theorem proves the triangles congruent?

  18. 18.Congruent trail triangles at Pocahontas State Park match angle K with angle P and side KL with side PQ. Angle K measures 52 degrees and KL is 18 meters. What must be true?

  19. 19.Beside Douthat Lake, triangle ABD has angles of 28 degrees and 81 degrees. Triangle QRS has angles of 28 degrees and 71 degrees. What is the third angle of triangle ABD, and how does it compare with the third angle of triangle QRS?

  20. 20.A pony-pen triangle on Chincoteague has remote interior angles of 47 degrees and 68 degrees. What is the exterior angle at the third vertex?

  21. 21.A footbridge triangle on the New River Trail has two sides of 7 meters and 11 meters. Which inequality must the third side, x meters, satisfy?

  22. 22.A shelter triangle in Grayson Highlands has angles of 35 degrees, 55 degrees, and 90 degrees. Which side is the longest?

  23. 23.Two bracing boards cross on a footbridge in Lexington. One angle of a vertical pair measures 124 degrees. What is the measure of the angle vertically opposite it?

  24. 24.Along Colonial Parkway, a ramp is parallel to a service road, and a footpath crosses both. One corresponding angle measures 58 degrees. What is the matching corresponding angle?

  25. 25.Two roof triangles along the James River in Richmond each contain a 32 degree angle and a 71 degree angle. Why are the triangles similar?

  26. 26.A classroom model of a Cape Henry lighthouse is 15 inches tall, and the display tower it copies is 45 feet tall. What is the scale factor from the model to the tower when both heights are in inches?

  27. 27.Two similar triangular sails on Smith Mountain Lake have corresponding sides in the ratio 2:5, smaller to larger. The smaller base is 8 feet. What is the corresponding larger base?

  28. 28.At a Blue Ridge wayside, a segment joins the midpoints of two sides of a triangular overlook and is parallel to the third side. That third side is 22 meters. How long is the midsegment?

  29. 29.In a Fredericksburg street diagram, a line parallel to side GH of triangle FGH meets FG at P and FH at Q. FP = 6 m, PG = 9 m, and FQ = 8 m. What is QH?

  30. 30.A sketch for a mural near the Elizabeth River in Norfolk is dilated by a scale factor of 3.5. A border that is 12 cm on the sketch becomes how long on the image?

  31. 31.Two similar garden plots at Lewis Ginter Botanical Garden have similarity ratio 2:3, smaller to larger. What is the ratio of their areas?

  32. 32.Two similar storage pyramids at a Virginia Beach marina have similarity ratio 1:3, smaller to larger. What is the ratio of their volumes?

  33. 33.A flagpole in Byrd Park casts a 15-foot shadow while a 6-foot post casts a 5-foot shadow. How tall is the flagpole?

  34. 34.On a Williamsburg fence plan, triangle WLM is similar to larger triangle WRS, with vertices matched in that order. WL = 8, WR = 14, and LM = 10. Side LM corresponds to RS. What is RS?

  35. 35.An altitude to the hypotenuse of a right triangle beside Crabtree Falls splits that hypotenuse into segments of 5 m and 20 m. How long is the altitude?

  36. 36.On a sail plan for a boat on the Rappahannock, a segment parallel to the 30-foot base cuts one side into 8 feet above the segment and 12 feet below it. How long is that parallel segment?

  37. 37.A right-triangle path on Belle Isle has legs of 8 meters and 15 meters. What is the hypotenuse?

  38. 38.A guy wire at Kiptopeke State Park is 15 meters long and meets the ground 9 meters from the base of a vertical pole. How tall is the pole?

  39. 39.A square plaza in Charlottesville is crossed by a diagonal walk, forming a 45-45-90 triangle with each leg 14 feet. How long is the diagonal?

  40. 40.A ramp triangle at Natural Tunnel has angles of 30, 60, and 90 degrees. The side opposite 30 degrees is 9 feet. How long is the side opposite 60 degrees?

  41. 41.A 30-60-90 window brace in Staunton has a hypotenuse of 20 inches. How long is the side opposite the 30 degree angle?

  42. 42.A support cable at Massanutten makes a 28 degree angle with level ground and is 40 feet long. Which expression gives the height of the upper end above the ground?

  43. 43.From a dock on Lake Anna, a 24-foot ramp rises at 32 degrees to a pier. Which expression gives the horizontal distance from the lower end to the point directly below the upper end?

  44. 44.A surveyor in Winchester stands 45 meters from a monument. From eye level with the base, the angle of elevation to the top is 37 degrees. Which expression gives the monument's height?

  45. 45.A ladder at Leesylvania State Park reaches 9 feet up a wall, and its base rests 12 feet from the wall. Which expression gives the angle between the ladder and the ground?

  46. 46.A right triangle on the Appalachian Trail near McAfee Knob has legs of 20 feet and 21 feet. What is the hypotenuse?

  47. 47.A rescue ladder in Virginia Beach reaches 20 feet up a pier wall, with its base 15 feet out from the wall. How long is the ladder?

  48. 48.A path at Breaks Interstate Park rises 8 feet for every 15 feet of horizontal run. What is the cosine of the angle of elevation?

  49. 49.In a circular garden at Norfolk Botanical Garden, a central angle measures 72 degrees. What is the measure of the arc it intercepts?

  50. 50.An inscribed angle on a plaza circle in Richmond intercepts an arc of 136 degrees. What is the measure of the inscribed angle?

  51. 51.In a fountain circle at Maymont, a chord lies 9 feet from the center and the radius is 15 feet. How long is the whole chord?

  52. 52.From a point outside a circular pool in Reston, a tangent segment of 15 meters meets the circle. The radius to the point of contact is 8 meters and is perpendicular to the tangent. How far is the external point from the center?

  53. 53.From a bench outside a circular flower bed in Abingdon, two tangent segments touch the bed. One of them is 11 feet long. How long is the other tangent from that same bench?

  54. 54.A curved walk on the Lawn in Charlottesville follows a circle of radius 12 meters through a central angle of 60 degrees. What is the arc length?

  55. 55.A sprinkler in a Harrisonburg park waters a sector of radius 12 feet with a central angle of 40 degrees. What is the area of the sector?

  56. 56.A circular harbor model in Norfolk has center (−4, 1) and radius 6. Which equation matches the circle?

  57. 57.A window at Montpelier is a semicircle, and an inscribed angle has sides that meet the endpoints of the diameter. What is the measure of that inscribed angle?

  58. 58.Two chords cross inside a circle at a pavilion in Lynchburg and intercept opposite arcs of 84 degrees and 46 degrees. What is the measure of an angle formed by the chords?

  59. 59.From a point outside a circular amphitheater in Vienna, two secants intercept a far arc of 150 degrees and a near arc of 30 degrees. What is the angle between the secants?

  60. 60.Two chords meet inside a circular window in Alexandria. One chord is split into 5 cm and 8 cm. The other is split into 4 cm and a segment of length x. What is x?

  61. 61.A triangular flower bed at Lewis Ginter Botanical Garden has a base of 18 feet and a height of 7 feet. What is its area?

  62. 62.A parallelogram platform beside Barter Theatre in Abingdon has a base of 15 feet and a height of 8 feet measured perpendicular to that base. What is its area?

  63. 63.A trapezoidal plot beside the Virginia Creeper Trail has parallel sides of 9 meters and 15 meters and a height of 6 meters. What is its area?

  64. 64.A circular sand area at First Landing State Park has a radius of 8 meters. What is its area?

  65. 65.A rhombus kite at a Chincoteague festival has diagonals of 12 inches and 16 inches. What is its area?

  66. 66.At Hungry Mother State Park, an 18-foot by 10-foot rectangle has a semicircle of radius 5 feet attached outside it along the 10-foot side. What is the total area?

  67. 67.An equilateral triangular sign in Front Royal has sides of 10 inches. Which expression equals its area?

  68. 68.A square courtyard in Manassas has sides of 12 meters, and a circle inscribed in the square touches all four sides. What is the area inside the square but outside the circle?

  69. 69.A fountain border in Roanoke is the region between two circles with the same center. The outer radius is 9 feet and the inner radius is 4 feet. What is the area of the border?

  70. 70.A triangular lot in Lexington has two sides of 14 meters and 10 meters with an included angle of 30 degrees. What is its area?

  71. 71.A rectangular lawn in Fairfax measures 24 yards by 15 yards. A triangular herb bed with base 8 yards and height 5 yards is cut out of it. How much lawn remains?

  72. 72.A kite-shaped banner for a parade in Ghent, a neighborhood in Norfolk, has diagonals of 13 feet and 8 feet. What is its area?

  73. 73.A rectangular crab crate at a Suffolk dock is 6 feet long, 4 feet wide, and 2.5 feet high. What is its volume?

  74. 74.A cylindrical corn crib near Winchester has radius 4 meters and height 9 meters. What is its volume?

  75. 75.A conical pile of sand at the Virginia Beach oceanfront has radius 6 feet and height 7 feet. What is its volume?

  76. 76.A spherical buoy near Cape Charles has a radius of 3 feet. What is its volume?

  77. 77.A square pyramid roof at a wayside in Shenandoah River State Park has base edges of 9 meters and a height of 4 meters. What is its volume?

  78. 78.A hemispherical dome model at the Science Museum of Virginia has radius 6 meters. What is its volume?

  79. 79.A marker at Yorktown is a cylinder of radius 3 feet and height 8 feet, topped by a cone with the same radius and height 4 feet. What is the total volume?

  80. 80.A model cistern in Lexington is similar to the full cistern, with similarity ratio 1:4 from model to full cistern. The model holds 3 liters. What is the capacity of the full cistern?

  81. 81.A water trough at a Bedford farm is a triangular prism. Each end is a right triangle with legs 9 inches and 12 inches, and the trough is 15 inches long. What is its volume?

  82. 82.A cylindrical rain barrel in Crozet holds 45π cubic feet and has radius 3 feet. What is its height?

  83. 83.A packing crate in Newport News measures 8 feet by 5 feet by 3 feet. What is its surface area?

  84. 84.A conical tent at Grayson Highlands has radius 5 feet and height 12 feet. What is its lateral surface area?

  85. 85.A path runs from (0, 0) to (6, 8) on a coordinate map of the Richmond canal walk. What is the distance?

  86. 86.A trail segment in Shenandoah National Park has endpoints (3, 8) and (11, 2). What is the midpoint?

  87. 87.A drainage line on a Farmville map passes through (−2, 5) and (4, 17). What is the slope?

  88. 88.A curb on Duke of Gloucester Street in Williamsburg has slope −2/3. A parallel curb passes through (0, 4). Which equation matches the parallel curb?

  89. 89.A property line in Loudoun County has slope 4/7. What is the slope of a line perpendicular to it?

  90. 90.Two docks on a Claytor Lake chart are plotted at (1, −2) and (9, 4). What is the distance between them?

  91. 91.On a Blacksburg map, segment AB runs from A(2, 1) to B(14, 10). A lamp divides AB in the ratio 1:2, with the shorter part starting at A. Where is the lamp?

  92. 92.A ferry lane on a Tangier Island chart passes through (0, 3) and (5, 13). Which equation matches the lane?

  93. 93.A rectangular herb bed in Staunton has vertices (2, 3), (11, 3), (11, 8), and (2, 8). What is its area?

  94. 94.A park circle in Petersburg has center (2, −1) and radius 10. Which point lies on the circle?

  95. 95.On a map of the New River at Fries, a horizontal bridge passes through the point (−5, 6). Which equation represents the bridge?

  96. 96.A triangular picnic area in Floyd has vertices (0, 0), (9, 0), and (0, 12). What is its perimeter?

  97. 97.A rectangular target at a Richmond fair is 12 inches by 8 inches. A circular bull's-eye of radius 3 inches lies completely inside it. A dart hits a random point on the rectangle. What is the probability it hits the bull's-eye?

  98. 98.A 24-meter gangway at Yorktown includes an 8-meter stretch of new decking. A visitor steps on a random point of the gangway. What is the probability that the point is on the new decking?

  99. 99.A circular spinner at a festival in Norfolk has one prize sector of 45 degrees. What is the probability that the spinner stops in the prize sector?

  100. 100.A square garden in Leesburg has sides of 10 meters. A rectangular pond inside it measures 5 meters by 4 meters. A leaf lands at a random point in the garden. What is the probability it lands in the pond?

  101. 101.A circular patio in Virginia Beach has radius 10 meters, and a grill circle of radius 6 meters lies entirely on it. A napkin lands at a random point on the patio. What is the probability it lands outside the grill circle?

  102. 102.A 15-inch by 8-inch poster in a Charlottesville gallery contains a triangular logo with base 6 inches and height 4 inches. A pin hits a random point on the poster. What is the probability it hits the logo?

  103. 103.A 40-foot mooring line at a Gloucester dock is frayed from the 10-foot mark to the 18-foot mark. A knot is tied at a random point on the line. What is the probability the knot is on the frayed part?

  104. 104.A circular game board in Harrisonburg has a shaded sector of 72 degrees. A token stops at a random point, with probability proportional to area. What is the probability it stops in the shaded sector?

  105. 105.A rectangular field in Orange County measures 50 meters by 20 meters, and a circular stage of radius 5 meters sits inside it. What is the probability that a random point of the field lies on the stage?

  106. 106.On a 20-mile route through the New River Valley, a rider boards at a random point from mile 0 to mile 20. The reduced-fare zone runs from mile 5 to mile 13. What is the probability the rider boards in that zone?

  107. 107.A rectangular courtyard in Winchester measures 12 meters by 9 meters. A right triangle painted in one corner has legs of 6 meters and 6 meters. A coin lands at a random point in the courtyard. What is the probability it lands on the triangle?

  108. 108.A circular clock on the courthouse lawn in Staunton is used as a random spinner. The arc from 12 to 3 is a 90 degree sector. What is the probability the spinner stops outside that sector?