Geometry Practice Test

Geometry Practice Test

You may draw diagrams or use a calculator if necessary. A good score is at least 38 correct.  It’s best to have a friend check your score the first time, so you won’t memorize the answers if you want to take the test again.

1. Two triangles are directly similar if and only if

(a) they are both equilateral

(b) they are both isosceles

(c) they have corresponding sides of identical lengths

(d) they have the same proportions in the same rotational sense

(e) the sum of the measures of their interior angles is equal to 180°

2. A regular polygon

(a) has sides that are all the same length, but interior angles whose measures may differ

(b) has interior angles that all have the same measure, but sides whose lengths may differ

(c) has sides that are all the same length, and interior angles whose measures are all the same

(d) has vertices that all lie along the same line

(e) has interior angles whose measures add up to 360°

3. Which of the following statements is false in Euclidean geometry?

(a) Two different, parallel, straight lines do not intersect

(b) A straight line segment has no perpendicular bisectors

(c) An acute angle measures less than 90°

(d) Three points, not all on the same line, always lie in the same plane

(e) Two intersecting straight lines always lie in the same plane

4. Look at Fig. Test 1-1. Suppose that line segments SQ and PR are perpendicular to each other at their intersection point T . From this, we can be certain

(a) that PQRS is a rhombus

(b) that PQRS is a square

(c) that PQRS is a rectangle

(d) that PQRS does not lie in a single plane

(e) about none of the above 

Fig. Test 1-1 . Illustration for Questions 4, 5, 6, and 7 in the test for Part One.

5. Suppose that in Fig. Test 1-1, line segments PQ and RS are parallel, but line segments PS and QR are not parallel. From this, we can be certain

(a) that PQRS is a rhombus

(b) that PQRS is a square

(c) that PQRS is a rectangle

(d) that PQRS is a parallelogram

(e) about none of the above

6. Suppose that in Fig. Test 1-1, Δ PTQ ≅ Δ RTS and Δ PTS ≅ Δ RTQ . From this, we can be certain

(a) that PQRS is a rhombus

(b) that PQRS is a square

(c) that PQRS is a rectangle

(d) that PQRS does not lie in a single plane

(e) about none of the above

7. Suppose that in Fig. Test 1-1, line segments PS and QR are parallel, and Δ PSQ ≅ Δ RQS . From this, we can be certain

(a) that PQRS is a rhombus

(b) that PQRS is a square

(c) that PQRS is a rectangle

(d) that PQRS is a parallelogram

(e) about none of the above

8. When creating a geometric construction, it is important that

(a) the compass be a precision drafting device, not a dime-store item

(b) the straight edge be calibrated in metric units

(c) markings on the straight edge or ruler be non-existent or ignored

(d) the marking device be a pencil, not a pen

(e) the paper be white and the marking device be black for maximum contrast

9. How far is the point (10,10) from the origin in Cartesian coordinates? Round off the answer to three decimal places.

(a) 7.071 units

(b) 10.000 units

(c) 14.142 units

(d) 20.000 units

(e) 100.000 units

10. A half-open line segment

(a) has zero length

(b) contains both of its end points

(c) contains one, but not both, of its end points

(d) contains neither of its end points

(e) has no end points

11. Suppose you see a graph of a circle in Cartesian coordinates. The circle is centered at the origin (that is, the point where x = 0 and y = 0). The radius of the circle is equal to 10 units. At what point on the circle are the x and y values equal and negative? Round off the values to three decimal places.

(a) (−10.000, −10.000)

(b) (−14.142, −14.142)

(c) (−7.071, −7.071)

(d) There are infinitely many such points

(e) There is no such point

12. What is the slope of the graph of the equation x = 3 in Cartesian coordinates?

(a) 3

(b) −3

(c) 0

(d) It is not defined

(e) More information is needed to determine it

13. Look at Fig. Test 1-2. Suppose that lines TR, QU , and QR are all straight, and that lines TR and QU are parallel. Which of the following statements is false?

(a) TRS and RQU are alternate interior angles

(b) TRS and PQU have the same measure

(c) Line PR is a transversal to lines TR and QU

(d) PRT has the same measure as RQU

(e) All of the above statements are true

Fig. Test 1-2 . Illustration for Questions 13 and 14 in the test for Part One.

14. Look at Fig. Test 1-2. Suppose that lines TR, QU , and QR are all straight and all lie in the same plane. Also suppose that TRQ has the same measure as UQR . Which of the following statements (a), (b), (c), or (d) is not necessarily true?

(a) Lines TR and QU are parallel

(b) TRS has the same measure as UQP

(c) Line SP is a transversal to lines TR and QU

(d) QRS has the same measure as PQR

(e) All of the above statements (a), (b), (c), and (d) are true

15. In a geometric construction, which of the following operations (a), (b), (c), or (d) is not allowed?

(a) Placing the compass down and adjusting it to span the length of an existing line segment

(b) Drawing two lines so they intersect at an angle measuring a certain number of degrees as indicated by a protractor

(c) Drawing a circle of arbitrary radius, centered at an arbitrary point

(d) Drawing a circle of arbitrary radius, centered at a specific point

(e) All of the above operations (a), (b), (c), and (d) are allowed

16. If the radius of a circle is 3 units, but nothing else about it is known, then which of the following (a), (b), (c), or (d) cannot be determined?

(a) The diameter of the circle

(b) The circumference of the circle

(c) The perimeter of the circle

(d) The interior area of the circle

(e) All of the above (a), (b), (c), and (d) can be determined

17. Suppose a field is shaped like a parallelogram. The long sides measure 100 meters each, and the short sides measure 20 meters each. What is the area of the field?

(a) 240 square meters

(b) 200 square meters

(c) 150 square meters

(d) 120 square meters

(e) It is impossible to calculate it without more information

18. Suppose a field is shaped like a parallelogram. The long sides measure 100 meters each, the short sides measure 20 meters each, and the width of the field, as measured at right angles to its long sides, is 15 meters. What is the area of the field?

(a) 2400 square meters

(b) 2000 square meters

(c) 1500 square meters

(d) 1200 square meters

(e) It is impossible to calculate it without more information

19. Suppose a Cartesian-coordinate graph shows two straight, parallel lines. How many solutions exist to the pair of simultaneous equations represented by these lines?

(a) More information is necessary in order to say

(b) None

(c) One

(d) Two

(e) Infinitely many

20. Suppose the diagonals of a plane quadrilateral are equally long and they intersect each other at a right angle. Then we can be certain

(a) that the quadrilateral is a square

(b) that the quadrilateral is a rectangle, but not a square

(c) that the quadrilateral is a rhombus, but not a square

(d) that the quadrilateral is a parallelogram, but not a rectangle or a rhombus

(e) about none of the above

21. Suppose the diagonals of a plane quadrilateral are equally long, they intersect each other at their midpoints, and they intersect each other at a right angle. Then we can be certain

(a) that the quadrilateral is a square

(b) that the quadrilateral is a rectangle, but not a square

(c) that the quadrilateral is a rhombus, but not a square

(d) that the quadrilateral is a parallelogram, but not a rectangle or a rhombus

(e) about none of the above

22. A triangle is obtuse if and only if

(a) all its interior angles are obtuse

(b) all its exterior angles are obtuse

(c) one of its interior angles is obtuse

(d) one of its exterior angles is obtuse

(e) the sum of the measures of its interior angles is obtuse

23. In Euclidean geometry, two lines are parallel if and only if

(a) they are in the same plane, and they do not intersect

(b) they are in the same plane, and they intersect at only one point

(c) they are in different planes, and they do not intersect

(d) they are in different planes, and they intersect at only one point

(e) they intersect at a right angle (90° or π/2 rad)

24. In Fig. Test 1-3, suppose all the sides of the polygon have identical length s (in meters), and all the interior angles have identical measure z (in degrees). This figure is

(a) a regular pentagon

(b) a regular hexagon

(c) a regular septagon

(d) a regular octagon

(e) an irregular polygon 

Fig. Test 1-3 . Illustration for Questions 24, 25, 26, and 27 in the test for Part One.

25. What is the measure of each interior angle z in the polygon of Fig. Test 1-3, assuming the conditions given in Question 24 all hold?

(a) 105°

(b) 120°

(c) 135°

(d) 150°

(e) 165°

26. What is the length of line segment AD in Fig. Test 1-3, assuming the conditions given in Question 24 all hold? (The exponent ½ indicates the square root.)

(a) s + (2 ^1/2 /2)

(b) s + (2 ^1/2 × s )

(c) s + (2 ×2 ^1/2 )

(d) 2 ( s + 2 ^1/2 )

(e) 2 + s ^1/2

27. How can the area of the polygon in Fig. Test 1-3 be found if the value of s is known and all the conditions given in Question 24 hold?

(a) Determine the area x of the square formed by the dashed lines, then determine the area y of Δ ABC , and finally determine x + y

(b) Determine the area x of the square formed by the dashed lines, then determine the area y of Δ ABC , and finally determine x + 4 y

(c) Determine the area x of the square formed by the dashed lines, then determine the area y of Δ ABC , and finally determine x − y

(d) Determine the area x of the square formed by the dashed lines, then determine the area y of Δ ABC , and finally determine x − 4 y

(e) Without more information, this problem cannot be solved

28. Two angles are said to be complementary if and only if

(a) they both have measures of 180°

(b) they are alternate interior angles formed by a transversal to two parallel lines

(c) they are opposite angles formed by the intersection of two lines

(d) the sum of their measures is equal to π/2 rad

(e) they are equal halves of a bisected angle

29. In a geometric construction, which of the following operations (a), (b), (c), or (d) is allowed?

(a) Drawing a line segment 10 centimeters long, as indicated by a ruler

(b) Drawing an angle measuring 29 degrees, as indicated by a protractor

(c) Drawing a circle with a radius of 5 centimeters, as indicated by a ruler

(d) Using a ruler to draw a straight line through two specific points

(e) None of the above operations (a), (b), (c), or (d) is allowed

30. Imagine two triangles Δ ABC and Δ DEF . Suppose the names of the vertices of each triangle go alphabetically in order as you proceed counterclockwise. Further suppose that all three of the following hold true:

• Line segment AB is the same length as line segment DE

• CAB has the same measure as FDE

• ABC has the same measure as DEF

What can we say, with certainty, about Δ ABC and Δ DEF?

(a) They are directly congruent triangles

(b) They are both isosceles triangles

(c) They are both right triangles

(d) They are both acute triangles

(e) Nothing in particular

31. If the diagonals of a parallelogram are both equally long, then that parallelogram is

(a) a square

(b) a rhombus

(c) a rectangle

(d) an irregular quadrilateral

(e) congruent

32. What is the equation of the straight line in Fig. Test 1-4?

(a) y = −4 x − 1

(b) y = 4 x − 1

(c) y = x − 4

(d) y = x + 4

(e) y = 4 x + 4

Fig. Test 1-4 . Illustration for Questions 32, 33, and 34 in the test for Part One.

33. What is the equation of the circle in Fig. Test 1-4?

(a) x ² + y ² = 6

(b) x ² − y ² = 6

(c) x ² + y ² = 36

(d) x ² − y ² = 36

(e) None of the above

34. How many solutions exist for the pair of simultaneous equations represented by the line and the circle in Fig. Test 1-4?

(a) None

(b) One

(c) Two

(d) Three

(e) Infinitely many

35. As the number of sides in a regular polygon increases, the interior area of the polygon

(a) increases without limit

(b) approaches the area of a circle inscribed within the polygon

(c) approaches the area of a square inscribed within the polygon

(d) remains constant

(e) becomes undefined

36. Consider a triangle whose vertex points are D, E , and F . Suppose the measure of DEF is equal to π/2 rad. Further suppose that the length of side DE is 30 meters, and the length of side EF is 40 meters. What is the interior area of Δ DEF?

(a) 1200 square meters

(b) 600 square meters

(c) 500 square meters

(d) 120 square meters

(e) It is impossible to tell without more information

37. Envision again the triangle described in the previous question. What is the perimeter of Δ DEF?

(a) 1200 meters

(b) 600 meters

(c) 500 meters

(d) 120 meters

(e) It is impossible to tell without more information

38. In Fig. Test 1-5, suppose all the points, line segments, lines, and arcs lie in a single plane, arc A has radius u and is centered on point P , and arc B has radius u and is centered on point Q . Which of the following statements logically follows from these facts?

(a) Line segment PT has the same length as line segment RT

(b) Arcs A and B encompass the same angular measure

(c) The length of line segment TQ is equal to u

(d) Line segment PQ has the same length as line segment RS

(e) None of the above

39. In Fig. Test 1-5, suppose line segment PQ is drawn, and then arcs A and B are drawn having equal radii u and centered on points P and Q , respectively. Finally, line RS is drawn through the points at which the arcs intersect each other. This process illustrates a method of

(a) angle duplication

(b) arc duplication

(c) line-segment bisection

(d) angle bisection

(e) none of the above

Fig. Test 1-5 . Illustration for Questions 38, 39, and 40 in the test for Part One.

40. In Fig. Test 1-5, suppose all the points, line segments, lines, and arcs lie in a single plane, arc A has radius u and is centered on point P , and arc B has radius u and is centered on point Q . Based on these facts, which of the following statements is not necessarily true?

(a) Line segment PT has the same length as line segment TQ

(b) Quadrilateral PRQS is a rhombus

(c) Line segment RS has the same length as line segment PQ

(d) Δ PTR is a right triangle

(e) All of the above statements are necessarily true

41. An angle of 3π/4 rad has the same measure as an angle of

(a) 30°

(b) 45°

(c) 90°

(d) 135°

(e) 180°

42. Suppose a straight line is graphed in Cartesian coordinates. If you move 2 units toward the right (that is, you increase the x -value by 2), the graph moves up by 4 units (that is, the y -value increases by 4). What is the slope of the line?

(a) 2

(b) 4

(c) −2

(d) −4

(e) There is not enough information to tell

43. In a geometric construction, which of the following operations (a), (b), (c), or (d) does not require the use of a drafting compass?

(a) Duplicating a line segment

(b) Bisecting a line segment

(c) Drawing a perpendicular bisector to a line segment

(d) Bisecting an angle

(e) All of the above operations (a), (b), (c), and (d) require the use of a drafting compass

44. In a trapezoid

(a) opposite pairs of angles have equal measure

(b) adjacent pairs of angles have equal measure

(c) the triangles formed by the sides and diagonals are all congruent

(d) the sum of the measures of the interior angles is equal to 360°

(e) the sum of the measures of the interior angles is equal to 180°

45. Suppose a square measures exactly 16 centimeters on a side. What is the perimeter of a circle inscribed within this square?

(a) 4π centimeters

(b) 8π centimeters

(c) 16π centimeters

(d) 32π centimeters

(e) It cannot be determined without more information

46. The graph of the equation y = 5 − 6 x ² in Cartesian coordinates is

(a) a straight line with positive slope

(b) a straight line with negative slope

(c) a parabola

(d) a circle

(e) none of the above

47. Suppose there exists a line L in Euclidean geometry. Let P be some point that is not on line L . How many lines can exist that pass through P and are parallel to L?

(a) It is impossible to say without more information

(b) None

(c) One

(d) Two

(e) Infinitely many

Fig. Test 1-6 . Illustration for Questions 48, 49, and 50 in the test for Part One.

48. Suppose, in the triangle shown by Fig. Test 1-6, that the measures of angles x and y are both 60°. Then we can be certain that Δ PQR is

(a) an equilateral triangle

(b) an isosceles triangle, but not an equilateral triangle

(c) a right triangle

(d) a directly congruent triangle

(e) a triangle, but no particular type

49. Examine Fig. Test 1-6. If angles x and y have the same measure, then we can be certain that Δ PQR is

(a) an equilateral triangle

(b) an isosceles triangle

(c) a right triangle

(d) a directly congruent triangle

(e) a triangle, but no particular type

50. Suppose, in the triangle shown by Fig. Test 1-6, that the sum of the measures of angles x and y is equal to π/2 rad. Then we can be certain that Δ PQR is

(a) an equilateral triangle

(b) an isosceles triangle

(c) a right triangle

(d) a directly congruent triangle

(e) a triangle, but no particular type

Answers

1. d

2. c

3. b

4. a

5. e

6. e

7. d

8. c

9. c

10. c

11. c

12. d

13. a

14. e

15. b

16. e

17. e

18. c

19. b

20. e

21. a

22. c

23. a

24. d

25. c

26. b

27. d

28. d

29. d

30. a

31. c

32. c

33. e

34. c

35. b

36. b

37. d

38. e

39. c

40. c

41. d

42. a

43. e

44. d

45. c

46. c

47. c

48. a

49. b

50. c