Geometry—First Semester
Review for Final Exam Name: ______
Problems 1-7: Complete the statement of each postulate:
1. The intersection of two lines is a ______.
2. The intersection of two planes is a ______.
3. Through any two points there is exactly ______.
4. Through any three noncollinear points there is exactly ______.
5. If two points are in a plane, then the line containing them is ______.
6. If two segments have the same length, then the segments are ______.
7. If point R is between points P and Q, then ______.
Problems 5-8: Refer to these pictures:
Picture A Picture B Picture C Picture D Picture E
5. Which of these pictures demonstrates that the intersection of three planes can be a single point?
6. Which of these pictures demonstrates that three lines might not be coplanar?
7. Which of these pictures demonstrates that the intersection of three planes could be a line?
8. Which of these pictures demonstrates that a line and a point not on it determines a single plane?
9. What geometric figure is the intersection of a line with a plane?
10. How many different planes can contain 3 noncollinear points?
11. Give four names for the angle in the picture on the right.
12. Which three points in the figure on the right
are collinear?
Problems 13-15:
The figure on the right is a 3-dimensional drawing of a shed with no doors or windows. Its roof, sides and floor are parts of seven different planes. Planes FBC and EAD (containing the front and back walls) are parallel, as are planes ABF and DCG (containing the left and right walls). No other planes determined by the roof, walls or floor of this shed are parallel.
13. Name a point other than D, C or G that is coplanar with points D, C and G.
14. Name the intersection of plane CDE with BF.
15. What geometric object is the intersection of plane ABC with plane GHP ?
16. Towns A, B and C are located along a straight highway. Town B is between A and C, and the distance from A to C is 83 miles. BC is 7 miles less than twice AB. How far is it from town B to town C?
17. If AB = CD, AC = x, BC = 5, BD = 2x – 7,
then AD = ?
18. In which of the following diagrams is it NOT true that AB + BC > AC?
A. B.
C. D.
E.
Problems 19-20: Refer to these diagrams of adjacent angles:
19. If and , then what is the measure of ?
20. Which of the following are true (more than one)?
(a) (b)
(c) (d)
Problems 21-24: Refer to this figure:
21. Name a straight angle. 22. Name a right angle.
23. Name an obtuse angle. 24. Name all angles that are adjacent to
25. In this figure, and
What is ?
Problems 26-27: Refer to the diagram on the right of
two intersecting lines.
26. If and
then x = ?
27. If and ,
then y = ?
28. How many triangles the same size and shape of
the triangle on the left will fit in the shaded square?
29. What is the area of the shaded square in problem 36?
Problems 30-34: Find the area of each figure:
30. Rectangle: Perimeter = 48 inches
Base = 17 inches
31. Triangle: Base = 7 inches
Height = 4 inches
32. Parallelogram: Base = 14 ft
Height = 8 ft
33. Trapezoid: Bases = 5 feet and 11 feet
Height = 6 feet
34. Circle: Diameter = 12 cm
35. If each square in the grid is 1 cm2,
what is the area of the shaded region?
36. What is the area of the following trapezoid, if its perimeter is 58 inches?
Problems 37-44: Circle the letter of the correct answer.
37. The process of drawing logically certain conclusions by using an argument is known as
A. inductive reasoning B. deductive reasoning
C. contrapositive D. converse
38. The process of making a conjecture based on a series of observed patterns is known as
A. inductive reasoning B. deductive reasoning
C. contrapositive D. converse
39. When a conjecture has been proved, it becomes a
A. postulate B. definition C. peacock D. theorem
40. By interchanging the hypothesis and conclusion of a conditional statement, we form another conditional statement that is called the
A. conclusion B. counterexample C. biconditional D. converse
41. A type of statement formed by combining a conditional statement and its converse into a single statement is called a
A. conclusion B. counterexample C. biconditional D. converse
42. If a definition is written as a conditional statement, then
A. its converse must also be true B. its converse is false
C. it is also a theorem D. it is also a postulate
43. The statement, “A stranger is lethargic if and only if he or she feels tired” means
A. If a stranger is lethargic then he or she feels tired.
B. If a stranger feels tired then he or she is lethargic.
C. If a stranger is lethargic then he or she feels tired, and conversely.
D. If someone feels tired and lethargic, then he or she is a stranger.
44. Which of the following is the converse of the statement, “if a quadrilateral is a square, then its four sides are congruent”?
A. If a square is a quadrilateral, then its four sides are congruent.
B. If the four sides are congruent, then it is a quadrilateral square.
C. If four sides are congruent, then the square is a quadrilateral.
D. If the four sides of a quadrilateral are congruent, then it is a square.
45. What is the name of the property of congruence that says a figure is congruent to itself?
(For example, )
______
46. Which of the following is correct?
A. A rhombus is a square B. A square is a rhombus
47. What is the definition of a rhombus?
48. What is the definition of a parallelogram?
49. What figure is a counterexample that shows the following statement is false?
“If a quadrilateral is a rectangle, then it cannot be a rhombus.”
Problems 50-53: Circle all choices that are true.
50. In a rhombus,
A. the diagonals are congruent. B. the diagonals bisect each other.
C. the diagonals are perpendicular. D. each pair of opposite sides are parallel.
E. each pair of consecutive angles are supplementary.
51. In a parallelogram,
A. the diagonals are congruent. B. the diagonals bisect each other.
C. the diagonals are perpendicular. D. each pair of opposite sides are congruent.
E. each pair of consecutive angles are supplementary.
52. In an isosceles trapezoid,
A. the diagonals are congruent. B. the diagonals bisect each other.
C. the diagonals are perpendicular. D. each pair of opposite sides are parallel.
E. each pair of consecutive angles are supplementary.
53. In a rectangle,
A. the diagonals are congruent. B. the diagonals bisect each other.
C. the diagonals are perpendicular. D. each pair of opposite sides are parallel.
E. each pair of consecutive angles are supplementary.
54. Suppose quadrilateral ABCD is a parallelogram and . Then what is ?
55. Suppose quadrilateral PQRS is a rhombus, its diagonals intersect at point X, and . Then what is ?
56. In problem 55, what is the measure of ?
57. Find the measure of :
58. In the figure on the right, AB = AC,
AE = AD, and .
What is the measure of ?
Problems 59-64: State whether the triangles are congruent by SAS, ASA, SSS, SAA, HL, or possibly not congruent:
59. 60.
61. 62.
63. 64.
Problems 65-67: A has coordinates (3, –7) and B has coordinates (–2, –10).
65. What are the coordinates of the midpoint of AB?
66. What is the slope of AB?
67. What is the slope of any line perpendicular to AB?
Problems 68-70: Graph the quadrilaterals with the following vertices, find the slopes of the sides (write the slopes above each side), then refer to the slopes to answer the questions.
ABCD: A = (–6, 2) B = (0, –5) C = (7, 1) D = (2, 8)
EFGH: E = (–8, 0) F = (–3, 8) G = (5, 4) H = (0, –4)
KLMN: K = (9, 7) L = (0, 9) M = (–6, –2) N = (3, –4)
PQRS: P = (–6, 5) Q = (4, 10) R = (8, 2) S = (–2, –3)
68. Which of the quadrilaterals are parallelograms?
69. Which of the quadrilaterals are rectangles?
70. Which of the quadrilaterals are squares?
71. Draw with AB = 14 cm, BC = 10 cm, and AC = 6 cm. Then give the measure of each angle.
Problems 72-75: In , and Find the indicated angle measures.
72. 73.
74. 75.
76. Find the value of x:
77. Mark the diagram and supply the missing statements or reasons for the proof:
Given: , P bisects
Prove:
Proof:
Statement: / Reason:/ Given
______/ Alternate Interior Angle Theorem
P bisects
Definition of “bisects”
Vertical Angle Theorem
78. Mark the diagram and supply the missing statements or reasons for the proof:
Given: and
Prove: P is the midpoint of
Proof:
Statement: / Reason:Definition of “midpoint”
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