Download CP GEOMETRY Final Exam Study Packet

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CP GEOMETRY
Final Exam Study Packet
*Figures not drawn to scale unless a scale is indicated*
Formulas you will be given on the exam:
Chapter 1: Foundations of Geometry
Collinear
Coplanar
Angle bisector:
Segment bisector (midpoint):
Complementary:
Supplementary:
Adjacent angles:
Linear pair:
Vertical angles:
Segment Addition Postulate:
Angle Addition Postulate:
Points on the same line
Points on the same plane
A ray that cuts an angle into two congruent halves.
A point that cuts a segment into two congruent halves.
Add up to 90°
Add up to 180°
Angles that share a side and do not overlap
A pair of angles that are adjacent and supplementary
Non-adjacent angles formed by two intersecting lines, they are congruent
The length of a whole segment is equal to the sum of its parts’ lengths
The measure of a whole angle is equal to the sum of its parts’ measures
Distance Formula: d
( x2
x1 ) 2
Midpoint of a Segment:
x1
x2 y1
,
1.
2
( y2
y2
2
y1 ) 2 or plot the points and use Pythagorean Theorem (a2 + b2 = c2)
or find average of the x’s and of the y’s
is the angle bisector of CAD.
m CAB = (6x + 8)°, and m CAD = (15x + 2)°.
What is the value of x?
A.
B.
3.
intersects
at E. If
is the perpendicular
bisector of
, which of the following MUST be
true?
2
3
A.
80
21
C.
B.
D.
14
C.
3
170
D.
21
2. What is the midpoint of the segment whose
endpoints are (6, 8) and (4, 0)?
A. (5, 4)
4. m A = (4x + 6)°, and its supplement has
measure (6x – 16)°. What is the value of x?
A. 10
B. 11
C. 17
D. 19
B. (1, 3)
C. (3, 6)
D. (3, 2)
5. What is the intersection of
A. H
B. F
C.
D.
and
?
Chapter 2: Geometric Reasoning
conditional
converse
inverse
contrapositive
original
pq
reverse order
qp
put “not” in front of both
~p  ~q
reverse order AND put “not” in front of both
~q  ~p
“If I have gotten a good grade,
then I am smiling.”
“If I am smiling,
then I have gotten a good grade.”
“If I have not gotten a good grade,
then I am not smiling.”
“If I am not smiling,
then I have not gotten a good grade.”
Biconditional Statement:
Conjecture:
Counterexample:
“If and only if” (combines a statement and its converse)
A statement based on observation
An example that proves a conjecture or statement wrong.
Law of Detachment:
If p  q is a valid conditional statement, and p is true, then q is true.
(“If I am happy, then I am smiling.” I am happy. Therefore, I am smiling.)
If p  q and q  r are valid conditional statements, then p  r is a valid statement.
(“If I have gotten a good grade, then I am happy. If I am happy, then I am smiling.”
So “If I have gotten a good grade, then I am smiling” is a valid statement.)
Law of Syllogism:
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
We can add the same thing to both sides of an equation.
We can subtract the same thing from both sides of an equation.
We can multiply both sides of an equation by the same thing.
We can divide both sides of an equation by the same thing (except 0).
Reflexive Property of Equality
Symmetric Property of Equality
Transitive Property of Equality
x=x
If x = 5 , then 5 = x.
If x = y and y = 5, then x = 5.
Substitution Property
Distributive Property of Equality
Commutative Property
Associative Property
If x = 5, then we can “plug in” 5 for x anywhere we want to.
We can distribute things outside the parentheses to inside: 5(x + 2) = 5x + 10
We can add or multiply in any order: 3 + 5 = 5 + 3
We can move parentheses around: 3 + (5 + 1) = (3 + 5) + 1
Reflexive Property of Congruence
Symmetric Property of Congruence
Transitive Property of Congruence
Definition of Congruence
A
A
If A
B, then
If A
B and
When you go from
B
A.
B
C, then A
to = or = to .
C.
6. “All Blingots are Whozits.”
Based on this statement, which of the
following must be a valid statement?
A. If Carl is a Whozit, then he is a Blingot.
B. If Carl is not a Blingot, then he is not a
Whozit.
C. If Carl is not a Whozit, then he is not a
Blingot.
10. Fill in the proof below.
Given: A and B are complementary.
m A = 45°
Prove: m A = m B
A and B are
complementary.
m A + m B = 90°
45° + m B = 90°
D. None of the above.
m B = 45°
7. Which of the following is the converse of the
statement, “If an angle is acute, then its
measure is between 0 and 90 degrees”?
m A=m B
Given
Definition of
Complementary
Substitution
Property of Equality
_____________________
Transitive
Property of Equality
A. “If an angle’s measure is between 0 and
90 degrees, then it is acute.”
A. Definition of Complementary
B. “If an angle’s measure is not between 0
and 90 degrees, then it is not acute.”
C. Subtraction Property of Equality
C. “If an angle is not acute, then its measure
is not between 0 and 90 degrees”
D. None of the above.
B. Definition of Congruence
D. Transitive Property of Equality
11. AB = CD. Which property says that CD = AB?
A. Reflexive Property of Equality
8. “If something is a Bork, then it is either
Glubous or Blatious.”
Based on this statement, which of the
following must be a valid statement?
A. If something is not a Bork, then it is
neither Glubous nor Blatious.
B. If something is Glubous, then it is a Bork.
C. If something is not Glubous, then it is not
a Bork.
D. None of the above.
9. “If a Planzer has a Boon, then the Planzer is
either Yoppy or Flinky.”
B. Subtraction Property of Equality
C. Symmetric Property of Equality
D. Transitive Property of Equality
12. What can you deduce from the statements: “If you
pass the final exam, then you will pass the class. If
you pass the class, then you will make your parents
happy.”
A. If you make your parents happy, then you will
pass the class.
B. If you pass the class, then you will make your
parents happy.
Which of the following could serve as a
counterexample to the statement above?
C. If you do not pass the final exam, then you will
not pass the class.
A. A Planzer that is not Yoppy.
D. If you pass the final exam, then you will make
your parents happy.
B. A Planzer that does not have a Boon.
C. Something Yoppy that is not a Planzer.
D. A Planzer with a Boon that is neither
Yoppy nor Flinky.
Chapter 3: Parallel and Perpendicular Lines
When parallel lines are cut by a transversal, then …
Vertical s
are congruent
Corresponding s
are congruent
//
Parallel Lines:
Perpendicular Lines:
Skew Lines:
Slope of a line:
Alternate Interior
are congruent
s
Alternate Exterior
are congruent
15. In the diagram below, l || m || n. If m 2 = 84°, what
is m 12?
1
3
(19b)°
(16b – 10)°
9
11
n
C.
B.
10
3
D.
38
7
A. 84°
190
3
C. 104°
5 6
7 8
m
n
16. What is the slope of a line perpendicular to a line
going through the points (0, -8) and (2, 4)?
2
4
10
12
D. 106°
j
1
l
B. 96°
14. In the diagram below, which of the following
pairs of angles are alternate interior angles?
3
2
4
5 6
7 8
m
20
7
Consecutive Interior s
are supplementary
Coplanar lines that never intersect, slopes are the same
Coplanar lines that intersect to form right angles, slopes are opposite reciprocals
Noncoplanar lines that never intersect
y 2 - y1
rise
or
x 2 - x1
run
13. In the diagram below, m || n.
What is the value of b?
A.
s
1
6
m
A.
n
C. -6
A.
3 and
5
B.
3 and
4
C.
3 and
6
D.
1 and
8
B. 6
D.
1
6
Chapters 4 and 5: Triangles
Scalene triangle:
Isosceles triangle:
Equilateral triangle:
no congruent sides
at least two congruent sides
three congruent sides
Acute triangle:
Right triangle:
Obtuse triangle:
Equiangular triangle:
all angles are acute
one angle is right
one angle is obtuse
all angles are congruent (60°)
Triangle Sum Theorem: The three angles of a triangle add up to 180°.
Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two remote, interior angles.
Third Angles Theorem: If 2 angles in a triangle are congruent to 2 angles in another triangle, 3 rd angles are congruent.
Base angles of an isosceles triangle: the congruent angles (opposite the congruent sides)
SSS
CPCTC:
SAS
ASA
AAS
HL
Corresponding Parts of Congruent Triangles are Congruent
Median
a segment whose
endpoints are a vertex and
midpoint of a triangle.
Medians meet at the
Centroid (balancing
point, 1:2 ratio)
Altitude
a perpendicular segment
from a vertex to the
opposite side of a triangle.
Altitudes meet at the
Orthocenter
Perpendicular Bisector
a segment that is
perpendicular to another
segment at its midpoint.
Perpendicular bisectors
meet at the Circumcenter
(equidistant to vertices)
Angle Bisector
a segment that divides an
angle in half.
Angle bisectors meet at the
Incenter (equidistant to
sides)
If a point is on the perpendicular bisector of a segment, then it is equidistant to the endpoints of the segment.
If a point is on the angle bisector of a segment, then it is equidistant to the sides of the angle.
Triangle Inequality Theorem:
short + short > long
(makes a triangle)
The largest angle of a triangle is opposite its longest side. Likewise, the smallest angle is opposite the shortest side.
Midsegment of a Triangle: Segment connecting the midpoints of the sides. Half the length of the side it is parallel to.
17. Which of the following would be enough to
prove ΔABC ΔDEF by SAS?
A. BC
EF , AC
DF ,
A
D
B. AB
DE , BC
EF ,
A
D
C. AB
DE , AC
DF ,
B
E
D. AB
DE , AC
DF ,
A
D
18. Based on the drawing below, what is true?
19
C
A
12
9
20. Given that ΔABC ΔDCB, which of the following
is NOT necessarily true?
A
B
C
D
A. AC || BD
B. AC
BD
C. m CBD = m CBA
D. m BCA = m CBD
B
A. m A < m B < m C
B. m A < m C < m B
21. Which of the following could be the side lengths of
a triangle?
C. m B < m C < m A
A. 2, 5, 3
D. m B < m A < m C
B. 5, 13, 7
C. 8, 4, 4
D. 4, 11, 8
19. Based on the diagram below, which of the
following must be true?
22. Find the value of x so that
HL .
MNO
D
C
B
A
A. m CAD = 45°
A. 5
B. 11
B. AB
BD
C. 4
D. 7
C. DB
DC
D.
CDB is a right angle.
PRO by
Chapter 6: Polygons and Quadrilaterals
triangle
3 sides
quadrilateral
4 sides
pentagon
5 sides
hexagon
6 sides
heptagon
7 sides
octagon
8 sides
nonagon
9 sides
decagon
10 sides
The sum of the measures of the interior angles of a polygon with n sides is (n – 2) • 180°.
The sum of the measures of the exterior angles of a polygon with n sides is 360°.
Quadrilateral:
Parallelogram:
4 sides
2 pairs of parallel sides ( opposite sides are congruent, opposite angles are congruent,
consecutive angles are supplementary, diagonals bisect each other)
Rectangle:
Parallelogram with four congruent (right) angles ( diagonals are congruent)
Rhombus:
Parallelogram with four congruent sides ( diagonals are perpendicular, diagonals bisect angles)
Square:
A rectangle and a rhombus
Trapezoid:
Exactly 1 pair of parallel sides (midsegment is average of the bases and parallel to bases)
Isosceles Trapezoid: Trapezoid with congruent legs ( diagonals are congruent, base angles are congruent)
Kite:
2 pairs of consecutive congruent sides ( diagonals are perpendicular, one pair opposite angles
congruent)
23. What is the measure of each interior angle of
a regular octagon?
A. 45°
26. Refer to rhombus FISH. If m FIX = 18°, what is
m FIS?
F
I
X
B. 135°
C. 180°
H
D. 1,080°
S
A. 18°
24. What is m
A. 30°
NMR in the square MNOP?
N
M
B. 90°
C. 36°
D. 6°
C. 45°
D. 60°
B. 9°
R
P
O
27. OL is the midsegment. If CA = 16 and OL = 12,
find RM.
25. Which of the following are properties of a
rhombus?
A
C
I. Diagonals are congruent
L
O
II. Diagonals are perpendicular
III. Diagonals bisect each other.
A. 28
A. I and II
B. I and III
B. 4
C. II and III
D. I, II, and III
C. 2
D. 8
R
M
Chapter 7: Similarity
scale factor =
new
old
The angles are congruent; the sides are proportional.
AA Similarity
SSS Similarity
SAS Similarity
Ratio of perimeters of similar figures is = scale factor
Ratio of areas of similar figures is = (scale factor)2
A line is parallel to a side of a triangle if and only if it divides the sides it intersects proportionally.
An angle bisector divides the opposite side proportionally to the other two sides.
28. What is the value of x in the figure below?
6
A.
25
64
B.
5
8
4
x
8
A. 3
B. 4
C.
30. If the scale factor of two similar figures is 5 : 8,
what is the ratio of their perimeters?
C.
10
4
D.
10
2
16
3
D. 12
29. The two polygons below are similar as they
appear. What is the value of x?
9
6
6
31. The two trapezoids below are similar. To the
nearest tenth, what is the value of x?
10
x
x
6
A.
2
3
B. 4
C. 6
D. 9
A. 2.1
B. 2.9
C. 4.2
D. 6
8
Chapter 8: Right Triangles & Trigonometry
Pythagorean Theorem: a2 + b 2 = c 2
Must be a right triangle, a and b are legs, c is hypotenuse
Pythagorean Triple:
3 numbers that satisfy the Pythagorean Theorem example: 3, 4, 5 and 5, 12, 13
Similar Right Triangles:
Geometric Mean = product
Special Right Triangles:
45
30
x 2
2x
x 3
x
45
60
x
Trigonometry:
SohCahToa
sin x
opposite
hypotenuse
32. What is the value of x in the figure below?
1
x
3
x
cos x
adjacent
hypotenuse
tan x
opposite
adjacent
34. The angle of elevation of a rope tied from a stake in
the ground to the top of a 3-foot-tall fence is 72°.
To the nearest tenth of a foot, how far is the stake in
the ground from the base of the fence?
sin 72° = 0.951
cos 72° = 0.309 tan 72° = 3.078
A. 1 foot
B. 3.2 feet
A. 2 2
C. 6.4 feet
B. 8
D. 9.7 feet
C. 6 2
35. Which of the following represents the value of x?
D. 9
33. Find the value of x to the nearest tenth.
A. 14 sin 38°
B. 14 cos 38°
A. 57.4
B. 28.3
C. 0.5
D. 32.6
7
13
C.
x
D.
x
14
14
sin 38 
14
cos 38 
38
Chapter 10: Area, Perimeter, and
Circumference
Area of parallelogram, rectangle, and square = bh
b b
dd
Area of a trapezoid = 1 2 h
Area of a rhombus and kite = 1 2
2
2
bh
Area of a triangle =
Area of a circle = πr2
Circumference of a circle =2πr
2
36. For what value of x will the area of the
trapezoid below be 48 square units?
39. What is the area of the shaded region below?
4
x
8
A. 4
B. 8
A. 8 m 2
C. 6
D. 12
B. 64 m 2
C. 16 m 2
37. What is the height of a triangle whose area is
50 cm2 and base is 20 cm?
A. 30 cm
B. 5 cm
C. 10 cm
D. 12.5 cm
D. 32 m 2
40. What is the area of the trapezoid below?
8
38. What is the area of a kite with d1 = (x + 2)
and d2 = (2x + 4)?
6
60
A. 2x2 + 8x + 8
B. x2 + 4x + 4
C. 3x + 6
D. 1.5x + 3
A. 24 square units
B. 48 square units
C. 33 3 square units
D. 66 3 square units
Chapter 11: Surface Area and Volume
Length is measured in units (cm)
Area is measured in square units (cm2)
Volume is measured in cubic units (cm3)
Ratio of SA of similar solids= (scale factor)2
Ratio of Volumes of similar solids = (scale factor)3
41. What is the volume of the prism below?
44. What is the surface area of the right cylinder below?
A. 18π cm2
B. 36π cm2
6 cm
D. 15π cm2
9 cm
4 cm
A. 36 5 cm3
B. 108 cm
6 cm
C. 54π cm2
3 cm
45. What is the radius of a sphere with surface area
12π cm2?
3
3 cm
C. 72 5 cm3
A.
D. 216 cm3
B. 3 cm
C. 9 cm
42. The figure below shows a right square
pyramid placed on top of a cube. What is the
surface area of the resulting solid?
A. 450 in2
7 in
D. 4 3 cm
46. The solids below are similar. Which of the
following is the ratio of their surface areas?
B. 531 in2
C. 567 in2
D. 918 in2
9 in
43. What is the volume of a sphere with diameter
12 feet?
A. 288π ft3
15 cm
A.
B.
5
4
C.
25
16
D.
125
64
B. 864π ft3
C. 1152π ft3
5
2
D. 2304π ft3
12 cm
Transformations
translation
rotation
47. The segment below is reflected in the x-axis.
What are the coordinates of A’?
reflection
dilation
49. What is the image of the point (4, -5) if it is
translated by (x – 3, y + 2)?
A. (7, -3)
B. (1, -7)
C. (1, 3)
D. (1, -3)
A. (1, -2)
50. What is the image of the point (0, -3) if it is
reflected over the y-axis?
B. (-1, -2)
C. (1, 2)
A. (-3, 0)
D. (-2, 1)
B. (3, 0)
C. (-3, 0)
48. What type of transformation turns the dashed
figure into the solid figure?
A. dilation
B. reflection
C. rotation
D. translation
D. (0, -3)