Logic and

1
Logic and Geometric Proof
Volume and Area Formulas
Trigonometry
Angle Relations Constructions Lines
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2
Which of the following best describes deductive
reasoning?
A. using logic to draw conclusions based on
accepted statements B. accepting the meaning of
a term without definition C. defining
mathematical terms to correspond with physical
objects D. inferring a general truth by examining
a number of specific examples
3
In the diagram below,
Which of the following conclusions does not have
to be true?
4
Theorem A triangle has at most one obtuse
angle.
Eduardo is proving the theorem above by
contradiction. He began by assuming that in
triangle ABC, that angle A and B
are both obtuse. Which theorem
will Eduardo use to reach a contradiction?

• If two angles of a triangle are equal , the sides
  opposite the angles are equal.
• If two supplementary angles are equal, the angles
  each measure 90 degrees.
• The largest angle in a triangle is opposite the
  longest side.
• The sum of the measures of the angles of a
  triangle are 180 degrees.

5
In the triangle below, n is a whole number. What
is the smallest possible value for n?
n
n
15
A. 1 B. 7 C. 8 D. 14
6
In the figure at right, AB gt BC
If we assume that the measure of angle A equals
the measure of angle C, it follows that AB BC.
This contradicts the given statement that AB gt
BC. What conclusion can be drawn from this
contradiction?
7
Two lines that lie in a plane always intersect
in exactly one point.
Which of the following best describes a
counterexample to the assertion above?

• Coplanar lines
• Parallel lines
• Perpendicular lines
• Intersecting lines

8
Which figure can serve as a counterexample to the
conjecture below?
If one pair of opposite sides of a quadrilateral
is parallel, then the quadrilateral is a
parallelogram.

• Rectangle
• Rhombus
• Square
• Trapezoid

9
Given TRAP is an isosceles trapezoid with
diagonals RP and TA. Which of the following
must be true?
10
A conditional statement is written below
If a quadrilateral has perpendicular diagonals,
then it is a rhombus.
Which diagram is a counterexample to the
statement above?
11
Which triangles must be similar?

• Two obtuse triangles
• Two scalene triangles with congruent bases
• Two right triangles
• Two isosceles triangles with congruent vertex
  angles.

12
Which of the following facts would be sufficient
to prove that triangles ABC and DBE are similar?
13
In Parallelogram FGHI, diagonals IG and FH are
drawn and intersect at point M. Which of the
following statements must be true?

• Triangle FGI must be an obtuse triangle.
• Triangle HIG must be an acute triangle.
• Triangle FMG must be congruent to triangle HMG.
• Triangle GMH must be congruent to triangle IMF.

14
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15
In the accompanying diagram, parallel lines l and
m are cut by transversal t.
Which statement about angles 1 and 2 must be
true?
16
What values of a and b make quadrilateral MNOP a
parallelogram?
17
Quadrilateral ABCD is a parallelogram. If
adjacent angles are congruent, which statement
must be true?

• Quadrilateral ABCD is a square.
• Quadrilateral ABCD is a rhombus.
• Quadrilateral ABCD is a rectangle.
• Quadrilateral ABCD is an isosceles trapezoid.

18
For the quadrilateral shown below, what is

• 53o
• 137o
• 180o
• 233o

19
If ABCD is a parallelogram, what is the length of
segment BD?

• 10
• 11
• 12
• 14

20
A right circular cone has radius 5 in and height
8 in. What is the lateral area of the cone?
(Lateral area of where l slant height)
21
Figure ABCD is kite. What is the area of ABCD in
square centimeters?

• 120
• 154
• 168
• 336

22
If a cylindrical barrel measures 22 inches in
diameter, how many inches will it roll in 8
revolutions along a smooth surface?

• 121 pi inches
• 168 pi inches
• 176 pi inches
• 228 pi inches

23
The four sides of this figure will be folded up
and taped to make an open box. What will be the
volume of the box?

• 50 cm3
• 75 cm3
• 100 cm3
• 125 cm3

24
A classroom globe has a diameter of 18 inches.
Which of the following is the approximate surface
area, in square inches, of the globe?

• 113.0
• 226.1
• 254.3
• 1017.4

25
The rectangle shown below has length 20 meters
and width 10 meters. If four triangles are
removed from the rectangle as shown, what will be
the area of the remaining figure?

• 136 m2
• 144 m2
• 168 m2
• 184 m2

26
If RSTW is a rhombus, what is the area of
triangle WXT ?
27
What is the area, in square units, of the
trapezoid shown below?

• 37.5
• 42.5
• 50
• 100

28
The figure below is a square with four congruent
parallelograms inside. What is the area, in
square units, of the shaded portion?

• 60
• 84
• 114
• 129

29
The perimeters of two squares are in a ratio of
4 to 9. What is the ratio between the areas of
the two squares?

• 2 to 3
• 4 to 9
• 16 to 27
• 16 to 81

30
Lea made two candles in the shape of right
rectangular prisms. The first candle is 15 cm
high, 8 cm long, and 8 cm wide. The second candle
is 5 cm higher but has the same length and width.
How much additional wax was needed to make the
taller candle?

• 320 cm3
• 640 cm3
• 960 cm3
• 1280 cm3

31
Two angles of a triangle have measures of 55
and 65. Which of the following could not be a
measure of an exterior angle of the triangle?

• 115
• 120
• 125
• 130

32
The sum of the interior angles of a polygon is
the same as the sum of its exterior angles. What
type of polygon is it?

• quadrilateral
• hexagon
• octagon
• decagon

33
What is the measure of angle x?

• 35
• 60
• 85
• 95

34
If the measure of an exterior angle of a regular
polygon is 120, how many sides does the polygon
have?

• 3
• 4
• 5
• 6

35
What is the value of x? A. 40 B. 50 C. 80 D. 90
36
What is the measure of angle 1?

• 34
• 56
• 64
• 92

37
What is the measure of angle WZX?

• 80
• 90
• 100
• 110

38
A diagram from a proof of the Pythagorean theorem
is pictured below. Which statement would not be
used in the proof of the Pythagorean theorem?

• The area of a triangle equals 1/2 ab.
• The four right triangles are congruent.
• The area of the inner square is equal to half of
  the area of the larger square.
• The area of the larger square is equal to the sum
  of the areas of the smaller square and the four
  congruent triangles.

39
A right triangles hypotenuse has length 5. If
one leg has length 2, what is the length of the
other leg?
40
A new pipeline is being constructed to re-route
its oil flow around the exterior of a national
wildlife preserve. The plan showing the old
pipeline and the new route is shown below. About
how many extra miles will the oil flow once the
new route is established?

• 24
• 68
• 92
• 160

41
Marsha is using a straightedge and compass to do
the construction shown at right. Which best
describes the construction Marsha is doing?

• A line through P parallel to line l
• A line through P intersecting line l
• A line through P congruent to line l
• A line through P perpendicular to line l

42
Given angle A What is the first step in
constructing the angle bisector of angle A?
43
Scott is constructing a line perpendicular to
line l from point P. Which of the following
should be his first step?
44
The diagram shows triangle ABC. Which statement
would prove that triangle ABC is a right triangle?

• (slope AB)(slope BC) 1
• (slope AB)(slope BC) -1
• Distance from A to B distance from B to C
• Distance from A to B -(distance from B to C)

45
Figure ABCD is a parallelogram. What are the
coordinates of the point of intersection of the
diagonals?
46
What type of triangle is formed by the points
A(4 2), B(6,-1), and C(-1 3)?

• right
• equilateral
• isosceles
• scalene

47
In the figure below, if sin x 5/13, what are
cos x and tan x?
48
In the figure below, sin A 0.7. What is the
length of side AC?

• 14.7
• 21.7
• 30
• 32

49
Approximately how many feet tall is the
streetlight?

• 12.8
• 15.4
• 16.8
• 23.8

50
In the diagram, measure of angle A 32o and AC
10. Which equation could be used to find x in
triangle ABC?
51
The diagram shows an 8-foot ladder leaning
against a wall. The ladder makes a 53o angle with
the wall. Which is closest to the distance up the
wall the ladder reaches?

• 3.2
• 4.8
• 6.4
• 9.6

52
Triangle JKL is shown below Which equation
should be used to find the length of JK ? .
53
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54
What is the value of x in the triangle below?
55
A square is circumscribed about a circle. What
is the ratio of the area of the circle to the
area of the square?
56
If AE 5, BE 12, and CE 6, what is the length
of side DE?

• 7
• 9
• 10
• 13

57
What is the measure of angle CBR?

• 50
• 65
• 90
• 130

58
What is the measure of angle ABC?

• 20
• 40
• 55
• 70

59
The vertices of triangle ABC are A(2, 1), B(3, 4)
and C(1, 3). If triangle ABC is translated 1
unit down and 3 units to the left to create
triangle DEF, what are the coordinates of the
vertices of triangle DEF?

• D(0, 1), E(1, 2), F(1, 3)
• D(0, -1), E(0, 3), F(-2, -2)
• D(-2, 2), E(0, 3), F(-1, 0)
• D(-1, 0), E(0, 3), F(-2, 2)

60
If triangle ABC is rotated 180o about the origin,
what are the coordinates of A ?

• (-5, -4)
• (-5, 4)
• (-4, 5)
• (-4, -5)

61

• (1, -3)
• (2, 1)
• (6, 1)
• (8, -3)

62
Answer A!
63
Answer B!
64
Answer C!
65
Answer D!