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Geometry Fourth Quarter Study Guide
1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement:
All right angles are congruent.
2. Find the measures of angles A, B, and C.
3. The measure of an angle is 64o. What is the measure of its complement? What is the measure of its supplement?
4. Write an equation of the line that passes through point P (2, -3) and is perpendicular to the line x – y = 4.
5. Using the diagram, give the coordinates of M if it is a midpoint.
6. A board 24 inches long is cut into two pieces in the ratio
7. Given:
Prove:
Find the length of each piece.
ABC is isosceles with base AC , BD bisects B
ABD  CBD
8. Name five Theorems or Postulates that you can use to prove that two triangles are congruent.
9. A spotlight is mounted on a wall 7.4 feet above a security desk in an office building. It is used to light an
entrance door 9.3 feet from the desk. To the nearest degree, what is the angle of depression from the spotlight to
the entrance door?
10. Line l passes through the points (–3, 1) and (2, 5). If j l and k
reasoning.
j, what is the slope of k? Explain your
11. Identify the property that makes the statement true. If MP = PQ and PQ = QR, then MP = QR.
12. For each set of numbers, determine whether the numbers represent the lengths of the sides of an acute triangle,
a right triangle, an obtuse triangle, or no triangle.
a)
b) 26, 28, 51
c) 18, 38.5, 42.5
13. Find the values of x and y.
14.
Find the measure of exterior angle
15. Find a, b, and h.
16.
, and
and
bisect each other. Which triangle congruence theorem
or postulate could you use to prove that HML KMJ? Explain.
17. Tong is making a triangular shaped frame out
of three strips of wood. One of the strips is 10
centimeters long and the second one is 15 centimeters
long. What are the possible lengths of the third
side?
18.
Determine whether the figures are similar.
19. Name a ray from Q through P.
20. A photo needs to be enlarged from an original with a length of 9 inches and a width of 7 inches to a size where
the new width is 14 inches. What is the new length? What is the scale factor?
21. Given:
bisects RST . Find QR if
(not drawn to scale)
and
22.
Find the value of x.
23. Explain the difference between inductive and deductive reasoning.
24. In the diagram,
are midsegments of triangle ABC.
Find the values of the variables if
.
25. Line l is the perpendicular bisector of
Find m M.
.
26. The length of one ramp is 16 feet. The vertical rise is 14 inches. Estimate the ramp’s horizontal distance and its
ramp angle.
27. Draw a Venn diagram showing the relationship between squares, rectangles, rhombuses, parallelograms, and
quadrilaterals.
28. Solve for x, given that
29.
. Is
equilateral?
1 and 2 are complementary, and 2 and 3 form a linear pair. If m 1 =
reasoning.
, what is m 3? Explain your
30. Given the following statements, can you conclude that Becky plays basketball on Wednesday night?
(1) If it is Wednesday night, Becky goes to the gym.
(2) If Becky goes to the gym, she plays basketball.
31. Would HL, ASA, SAS, AAS, or SSS be used to justify that
the pair of triangles is congruent?
32. Which lines, if any, can be proved parallel given the following diagram?
For each conclusion, provide the justification.
33. Can the measurements 9.7 meters, 1.1 meters, and 6.9 meters be the lengths of the sides of a triangle?
34.
Given the following, determine whether quadrilateral XYZW must be
a parallelogram. Justify your answer.
.
35. a. Is the statement "If a quadrilateral is a rectangle, then it is a parallelogram" True or False?
b. Write the inverse of the statement in part (a) and tell if it is True or False.
36. Draw and label the angles and the sides of the two special triangles: 45o-45o-90o and 30o-60o-90o.
37. True or False: The median and altitude of a triangle can never be the same line segment.
38. The altitude of an equilateral triangle is 6. What is the length of each side? Find the area of the triangle.
39. According to the Parallel Postulate, if there is a line and a point not on the line, then how many parallels to the
given line can be drawn through the point?
40. Wires are used to stabilize a telephone pole that is 50 feet high. A wire from the top of the pole to the ground is
63 feet long. To the nearest tenth of a foot, how far from the bottom of the pole is the wire anchored in the
ground?
41. Each interior angle of a regular n-gon has a measure of 156o. Find the value of n.
42. If
is an altitude of PQR, what type of triangle is PQR?
43. Find the geometric mean of 8 and 12.
44. Find the number of sides of a convex polygon if the measures of its interior angles have a sum of 2340°.
45. Solve the right triangle:
and
46. Given that PQR ~ PST, explain why
find , b, and c.
.
47. Given that a || b, what is the value of x?
(The figure may not be drawn to scale.)
48.
49. What is the measure of the smallest
interior angle in the quadrilateral?
50.
JKL is an equilateral triangle.
What is the length of KM ?
Find the value of x.
51. A triangle has the given vertices. Classify the triangle by its sides. Then determine if it is a right triangle.
52. List all of the important characteristics of each quadrilateral.
a. square
b. rectangle
c. parallelogram
d. rhombus
e. trapezoid
f. kite
53. Consider an octagonal stop sign.
a. Find the sum of the interior angles of a stop sign.
b. Find the measure of one of the interior angles of a stop sign.
c. Find the measure of an exterior angle of a stop sign.
54. Which lines, if any, can be proved parallel given the following diagram?
55. Point S is between points R and T. P is the midpoint of
relationship between the specified segments. Find ST.
56. Given:
is the perpendicular bisector of
you can conclude.
57. m SQR = (
)° and m PQR = (
Find m SQR and m PQR.
. RT = 20 and PS = 4. Draw a sketch to show the
. Name three things that
)° and m SQP = 70°.
58.
Solve the right triangle.
59. List each type of quadrilateral for which the statement is always true:
The diagonals are congruent.
60. In RSTU , RS is 3 centimeters shorter than ST . The Perimeter of
Find RS and ST .
61. In the diagram,
°,
°,
, and
Is there enough information given to show that quadrilateral ABCD
is an isosceles trapezoid? Explain.
62. If p q, solve for x.
RSTU is 42 centimeters.
.
63. If one angle of a triangle is larger than another angle, then the side opposite the larger
angle is longer than the side opposite the smaller angle. Use this fact to help you list the
sides of triangle STU in order from least to greatest. (The figure may not be drawn to scale.)
64. Two triangles are similar. The height of the smaller triangle is 4 units and the height of the larger triangle is 6
units. If the area of the smaller triangle is 24 square units, what is the area of the larger triangle?
65. Let p be “it is raining”, let q be “it is thundering”, and let r be “we cannot swim”. What is p  q ?
66.
form a linear pair.
67. Find
in the diagram, if
°. Find
° and
.
°.
68. Find the area of the isosceles triangle with side lengths 17 meters, 17 meters, and 30 meters.
69. Calculate the slope of the line. Does it matter which points are used?
Why or why not?
70. A building casts a shadow 260 meters long. At the same time, a pole 3 meters high casts a shadow 15 meters
long. What is the height of the building?
71. Tell whether each pair of triangles is similar. Explain your reasoning.
72. In
and
In
triangles are similar, and if so, write a similarity statement.
and
State whether the
73. Identify the hypothesis and conclusion of the statement: If today is Friday, then yesterday was Thursday.
74. The ratios of the side lengths of triangle ABC are 7:9:12 (AB:AC:BC). Solve for x.
75. True or False: If a quadrilateral is a parallelogram, then opposite angles are complementary.
76. Write a logical conclusion from the following statements:
If it is raining, then we will watch a movie. If we watch a movie, then we will eat popcorn.
77. Find the appropriate symbol to place in the blank. (not drawn to scale)
AB __ AC
78. Find the side lengths of the kite.
79. Find the value of x.
80. State the third congruence statement that is needed to prove that the two triangles are congruent using the given
postulate or theorem.
81. Find the value of each variable.
a)
b)
c)
d)
82. Find RS in C. Explain your reasoning.
83.
84. Find the mG.
is tangent to O at A (not drawn to scale).
Find the length of the radius r, to the nearest tenth.
85. Find the values of the variables.
86. Find the value of x.
a)
b)
c)
87. Write the standard equation of a circle with a center (3, -2) and a point on the circle (23, 19).
88. In the diagram,
is a radius of circle R. Is
89. Graph the equation: ( x  5)2  ( y  3)2  9
91. Find the value of x.
a)
b)
tangent to circle R? Explain.
90. Find the value of x.
c)
92. The rule for this transformation
of
onto
is _________
94. The translation vector is
95. The vertices of
=
93. What are the coordinates of the vertices
when the figure is reflected in line m?
. If the image of A is
are
find the coordinates of point A.
.
Reflect the triangle in the line
.
ABC
Then find the image of
after a dilation with its center at the origin
and a scale factor of 2.
96. A triangle has vertices
.
Find the coordinates of the vertices of the images of
after rotations of 90°, 180° and 270° about the origin.
97. The vertices of
are A(2,5), B(6,5)
and C (3,8) . Find the image of ABC
after the glide reflection:
Translation:
Reflection: in
98. Which transformation(s) are used in the
tessellation below? Which shape, 1, 2, 3, or 4,
belongs in the location indicated
by the arrow?
99. State whether the following figure has line
symmetry, rotational symmetry, both kinds
of symmetry, or neither kind of symmetry.
Geometry Fourth Quarter Study Guide – Answer Key
1. If the angles are right angles, then they are congruent.
If the angles are not right angles, then they are not congruent.
3. 26o, 116o
2. m A = 103°, m B = 77°, m C = 46°
6.
7.
8. SSS, SAS, HL, AAS, ASA
10. – . The slope of line l is
=
4.
y   x 1
5.
1) ABC is isosceles with base AC, BD bisects B (Given)
2) AB  BC (Definition of isosceles triangle)
3) ABD  CBD (Definition of angle bisector)
4) BD  BD (Reflexive Property of Segment Congruence)
5) ABD  CBD(SAS )
9. 39°
. Since j l, the slope of j is also
11. Transitive Property of Equality
13.
If the angles are congruent, then they are right angles.
If the angles are not congruent, then they are not right angles.
12.
x = 11, y =
. Since k
j, the slope of k is the negative reciprocal of
, which is –
.
A. acute triangle, B. obtuse triangle, C. right triangle
14.
89°
15.
a = 14, b =
,h=
16. SAS Congruence Postulate. Since
and
bisect each other,
and
.
because they
are vertical angles. Since you know that 2 pairs of sides and the included angle are congruent, you can use the SAS Congruence Postulate
to prove the triangles are congruent.
17. 5 < x < 25
18. The figures are not similar.
20. new length = 18 inches; scale factor = 2
19.
21. 34
22. 6
23. Inductive reasoning is the process of making a generalization based on a number of similar cases or specific patterns. Deductive
reasoning is the process of following a series of logical steps, often beginning with some given or known information, that leads to a specific
conclusion.
24. X=8, Y= 2, Z=15
25. 64°
26. 7.7,
27. Diagrams vary.
29.
28.
. Since 1 and 2 are complementary, the sum of their measures is
they are supplementary and the sum of their measures is
. So, m 3 =
30. yes
31. AAS
32.
. So, m 2 =
–
=
61o
x = 8; no
. Since
.
, Consecutive Interior Angles Converse
2 and
3 form a linear pair,
33. No
34. Yes. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
35. a. True
b. If a quadrilateral is not a rectangle, then it is not a parallelogram. False
37. False
41.
38.
n  15
42.
,
12 3
47. 71
48. 9
3
39. Exactly one
A right triangle
46. Since PST ~ PQR, PST
Q and PTS
Corresponding Angles Converse Postulate.
36.
43.
4 6
40. 38.3
44. 15
R. Since the pairs of angles are corresponding angles,
49.
40
50. 124°
45.
by the
51. Right scalene triangle
52. The following characteristics for each quadrilateral might be indicated.
a. four congruent sides, four right (congruent) angles, opposite sides parallel, congruent diagonals, diagonals are the perpendicular
bisectors of each other.
b. opposite sides congruent and parallel, four right (congruent) angles, congruent diagonals, diagonals bisect each other
c. opposite sides congruent and parallel, opposite angles congruent, diagonals bisect each other
d. four congruent sides, opposite sides parallel, opposite angles congruent, diagonals are the perpendicular bisectors of each other
e. one pair of opposite sides parallel
f. two pairs of adjacent sides congruent
53. a. 1080°, b. 135°, c. 45°
54. No lines can be proved parallel from the given information.
56. Any three of the following:
=
55.
12
;
,
;
=
A  24.4, C  65.6, AC  12.1
57. m SQR = 20° and m PQR = 50°
58.
59. square, rectangle
60. 12, 9
61. Yes, enough information is given to show ABCD is an isosceles trapezoid. ABCD is a trapezoid because
.
and
are not congruent so
trapezoid ABCD are congruent because
62. 12
63.
67.
. By definition ABCD is a trapezoid. The diagonals of
. So, ABCD is an isosceles trapezoid by Theorem 8.16.
64. 54 square units
66. 107°
69.
is not parallel to
so
65. If it is raining, then it is not thundering.
°
68.
; no; the slope ratio is the same for any two points on a line.
70. 52 meters
71. Yes; The two right angles are congruent, and since parallel lines are given the alternate interior angles are congruent, so the triangles are
similar by the AA Similarity Postulate
72. not similar
73. hypothesis: today is Friday, conclusion: yesterday was Thursday
74. 6
75. False
76. If it is raining, we will eat popcorn.
77.
78.
XY  YZ  5 5;WX  WZ  461
81. a) 9.2
80. F , J ;
79. X= 2.3
c) x = 24, y = 24 2
b) 10
82. RS = 7. In a circle, two chords that are equidistant from the center are congruent
85.
a  20, b  22
88. No; for
86. a) 67o
to be tangent to circle R
find that
89. Graph
and
, therefore
b) 70o
D, G
d) x  7 3 , y  21
2
2
84. 85o
83. 10.5
c) 56o
87.
( x  3)2  ( y  2)2  292
would have to be perpendicular. Using the converse of the Pythagorean Theorem you
is not a right triangle and
90. X= 150o
and
91. a) x=5
92. (x, y) (x – 9, y + 2)
93.
95.
96. 270o:
W'
o
180 :
90o:
98. Part A rotation of 180°, reflection, and translation
Part B shape 1
(1, 3),
X'
(–2, 5),
(-4, 5),
(5, 4),
(4, -5),
Y'
are not perpendicular.
b) x=12
(–6, –1)
(0, 5), (0, 0)
(5, 0), (0, 0)
(0, 5), (0, 0)
99. Line symmetry
c) x= 4
94. (13, –8)
97.