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Final Exam Review Packet
Geometry
Regents Review
Final Review by topic worksheets
Final Review Mixed worksheets
Final Review #1 (by topic)
Geometry
Name ______________________________
Date _______________ Block ________
Exterior angles of a polygon
1) Two angles of a triangle have measures of 80 and 40. Which is not the measure of
an exterior angle of the triangle?
(a) 1200
(b) 1000
(c) 1100
(d) 1400
2) If the measure of an exterior angle of a regular polygon is 720, then the polygon is:
(a) a decagon (b) an octagon (c) a pentagon (d) a square
3) The number of degrees in the measure of one exterior angle of a square is:
(a) 600
(b) 1800
(c) 2700
(d) 900
______________________________________________________________________
Similarity and proportions
4) The sides of a triangle have lengths 3, 5, and 7. In a similar triangle, the shortest side
has length x-3, and the longest side has length x+5. Find the value of x.
C
5) In the diagram, CDE ~ CAB . If CD = 8,
CE = 6, and EB = 5, find AD.
E
D
C
6) In the diagram, ADE ~ ABC . Given that AD = 4,
DB = 3, and EC = 4.5, find AE.
B
A
E
A
D
B
________________________________________________________________________
Complementary and supplementary angles
7) Two complementary angles are in the ratio of 7:2. Find the number of degrees in
the smaller angle.
8) If two angles are supplementary and the measure of <A is 12 less than twice the
measure of <B, find the larger of the two angles.
9) Angles A and B are complementary. If the measure of <B is 2 greater than three
times the measure of <A, find the smaller of the two angles.
1
Ratio of areas
10) The ratio of the radii in two circles is 3:7. What is the ratio of the area of the smaller
circle to the larger circle?
11) The ratio of the corresponding sides of two similar polygons is 1:4. Find the ratio of
their areas.
12) If the ratio of the corresponding sides of two similar polygons is 2:3, and the area of
the larger triangle is 243, find the area of the smaller triangle.
_______________________________________________________________________
Geometric probability
13) Find the probability that a penny tossed at
random onto the figure will land in the shaded
region. The length of a side of the square is 4 cm.
Round to the nearest hundredth.
14) Find the probability that a dart tossed at
random onto the figure will land in the shaded
region. The radius of the circle is 2.
Round to the nearest hundredth.
15) Find the probability that a dart tossed at
random onto the figure will land in the
shaded region.
________________________________________________________________________
Segment addition
16) Given that A, B, and C are collinear with A-C-B, if AB = 25, AC = 3x+1, and CB = x,
find the value of x.
17) Given P, Q, and R are collinear with P-Q-R, with PQ = 2x, QR = 3x, and
PR = 25. Find the value of x.
18) If E, D, and F are collinear with E-D-F, ED = 5x, DF = 3x, and EF = 25, find x.
2
Final Review #2 (by topic)
Geometry
Name _____________________
Date ________ Block _____
Quadratic functions
Write the coordinates of the vertex and state the equation of the axis of symmetry for
each parabola.
1) y  x 2  4 x  3
2) y   x2  2 x  8
3) y   x2  2 x  3
______________________________________________________________________
Parallelograms
4) In parallelogram ABCD, AB = 5x  4 and CD = 2 x  14 . Find the value
of x.
5) In parallelogram ABCD, m<A = 2x, and m<B = 2x+20. Find the value
of x.
6) If the degree measures of two consecutive angles of a parallelogram are
represented by x+40 and 2x-10, find the value of x.
______________________________________________________________________
Pythagorean theorem
7) If the hypotenuse of a right triangle is 10 and one leg is 6, find the length of the
other leg.
(a) 64
(b) 16
(c) 8
(d) 4
8) In an isosceles right triangle, one leg is 3. Find the length of the hypotenuse.
(a) 3 2
(b) 6
(c) 3
(d) 3
9) In a right triangle, one leg has a length of 3 and the hypotenuse has a length of 10.
What is the length of the other leg?
(a) 91
(b) 7
(c) 109
(d) 91
3
Indirect proof
10) To prove indirectly that AB  CD , what assumption must be made?
11) To prove indirectly that BD is not the perpendicular bisector of AC , what
assumption must be made?
12) To prove indirectly that
ABC  EFG , what assumption must be made?
_______________________________________________________________________
Surface area
13) If the surface area of a cube is 96 cubic centimeters, what is the length of a side
of the cube?
(a) 3 cm
(b) 4 cm
(c) 5 cm
(d) 6 cm
14) A cereal box is a rectangular prism 30 cm high. The sides of the base measure 8
cm and 25 cm. Find the surface area of the box.
15) What is the surface area of a cube with a side length of 4?
______________________________________________________________________
Midpoint
16) Find the coordinates of the midpoint of the segment whose endpoints are (-2, 3)
and (4, -3).
17) Line segment AB has midpoint M. If the coordinates of A are (2, 3) and the
coordinates of M are (-1, 0), what are the coordinates of B?
18) Find the midpoint of the line segment formed by the points (5, 4) and (-3, -4).
4
Final Review #3 (by topic)
Geometry
Name ___________________
Date _________ Block _______
Triangle inequality
1) In ABC , AB = 14, and BC = 9. AC may not be equal to:
(a) 5
(b) 13
(c) 23
(d) 25
2) If the lengths of 2 sides of a triangle are 4 and 8, the length of the third side may
NOT be:
(a) 5
(b) 6
(c) 7
(d) 4
3) Which of the following sets may represent the lengths of the sides of a triangle?
(a) {2, 4, 6}
(b) {4, 7, 12}
(c) {7, 12, 5}
(d) {8, 10, 14}
______________________________________________________________________
Transformations
4) Find the coordinates of the image of point T(-7, 3) under a reflection in the origin.
5) What are the coordinates of R’, the image of R(-4, 3) after a reflection in the x-axis?
6) What are the coordinates of N’, the image of N(5, -3) under a translation such that
 x, y    x  3, y  4 ?
_______________________________________________________________________
5
Equations of circles
7) What are the coordinates of the center of a circle represented by the equation
2
2
 x  2   y  3  49 ?
8) State the equation of a circle which has a radius of 5 and has a center with the
coordinates (-2, 6).
9) A circle whose center is a point (1, 2) passes through a point (4, -2). What is the
length of the radius?
________________________________________________________________________
Midsegments
10) A triangle has sides of 3, 5, and 10. What is the perimeter of a triangle formed by
connecting the midpoints of the sides of the triangle?
11) If a triangle has sides of 15, 20, and 25, which of the following could be the length
of a midsegment of the triangle?
(a) 15
(b) 10
(c) 9
(d) 12
12) In a triangle, which of the following is always parallel to a side of the triangle?
(a) median
(b) altitude
(c) midsegment (d) hypotenuse
________________________________________________________________________
Rational Equations
Solve the following equations.
13)
r 2 r 3 1


4
3
2
14)
5
3

 13
x 2x
15)
3
x 1
1
2x
x
6
Final Review #4 (by topic)
Name ______________________
Geometry
Date ______________ Block _____
Triangle congruence
1) In the accompanying diagram, RL  LP ,
LR  RT , and M is the midpoint of TP . Which
method could be used to prove TMR  PML ?
(a) SAS
(b) AAS
(c) HL
(d) SSS
2) In the accompanying diagram, ACE, BCD ,
A  E and C is the midpoint of AE . Which
theorem justifies ABC  EDC ?
(a) SSS
(b) SAS
(c) ASA
(d) SSA
3) In the diagram of isosceles triangle ABC,
<ACB is the vertex angle, CM  AB , and M is
the midpoint of AB . Which statement can not
be used to justify ACM  BCM ?
(a) HL
(b) AAS
(c) SSS
(d) AAA
_____________________________________________________________________
Special triangles
4) If the shortest side of a 30-60-90 triangle has length x, then the hypotenuse has
length:
(a) x (b) x 2
(c) x 3
(d) 2x
5) If the leg of a 45-45-90 triangle has length z, then the hypotenuse has length:
(a) z (b) z 2
(c) z 3
(d) 2z
6) If a rhombus has a side length of 2 and a 60 degree angle, what are the lengths of
the diagonals of the rhombus?
(a) 2 and 2 3
(b) 1 and 3
(c) 2 and 2 2
(d) 1 and 2
7
Final Review #5 (mixed)
Geometry
Name ______________________
Date _________ Block _________
1) An equilateral triangle has a side length of 20. What is the length of the altitude?
(a) 40
(b) 10 3
(c) 20 2
(d) 20
2) In Triangle ABC, if AB < BC < AC, then which of the following statements is false?
(a) m A  m C (b) m A  m B (c) m B  m C (d) m B  m A
3) Which of the following statements is always true?
(a) The diagonals of a parallelogram are congruent.
(b) The diagonals of a parallelogram bisect the angles of the parallelogram.
(c) The diagonals of a parallelogram bisect each other.
(d) The diagonals of a parallelogram are perpendicular to each other.
4) The degree measures of two supplementary angles are 3x-17 and 5x+21. Find the
measures of both angles.
5) Find the probability that a dart tossed at random onto the figure will land in the
shaded region. State the answer as a fraction in simplest form.
6) The diagonals of two rectangles which are similar measure 5 and 15 respectively. If
the area of the smaller rectangle is 27, find the area of the larger rectangle.
7) Which whole number, when substituted for x, makes the following statement true?
3x  6 and x  4
8) In parallelogram LMNO, an exterior angle at vertex O measures 72. Find the
measure of angle L of the parallelogram.
9) To prove indirectly that YT is a median, what assumption must be made?
8
10) Given the equation y  2 x 2  8 , find the vertex and the axis of symmetry.
11) Given line segment AB with C between A and B, if AC = x+1, CB = 2x, and AB = 19,
find the value of x.
12) The length of the hypotenuse of a right triangle is 17 meters and the length of one
leg is 15 meters. What is the length of the other leg?
13) What is the length of a side of a cube with a surface area of 150?
14) If the midpoint of a line segment is (1.5, -1) and one of the endpoints is (-2, 7), find
the other endpoint of the line segment.
15) A translation maps A(2, 5) onto A’(-3, 7). What are the coordinates of the point (3, 0) under the same translation?
16) What is the radius of a circle whose center is at the origin and that passes through
the point (4, 0)?
17) A triangle has sides of 7, 10, and 20. What is the perimeter of a triangle formed by
connecting the midpoints of the sides of the triangle?
18) Given
XYZ  STU , name all congruent sides and all congruent angles.
19) In a right triangle ABC, CD is drawn perpendicular to hypotenuse AB . If AB = 16,
and DB = 4, find BC.
9
Final Review #6 (mixed)
Geometry
Name _____________________
Date __________ Block _____
Find the value of x.
x
1)
2)
3)
70o
160 o
60
4)
5)
6)
88 o
x
x
5
x
o
4
40 o
x
4
16
6
120 o
______________________________________________________________________
7) Given a circle with a radius of 5 and a center of (2, -3), write the equation of the
circle.
8) Given circle O with A and B on the circle and m<AOB = 80, find the length of arc
AB (round to the nearest tenth if necessary).
9) Given: TU is tangent to Circle P at point T; mQR  90 , mRT  150 , and mQS  50 .
Find m STU , m 1 , and m 2 .
10
10) The vertices of quadrilateral ABCD are A(1, 1), B(3, 4), C(9, 1), and
D(7, -2).
a) Prove that ABCD is a parallelogram.
b) Prove that ABCD is not a rectangle.
11) Quadrilateral DEFG has vertices D(-4, 0), E(0, 1), F(4, -1), and G(-4, -3).
a) Prove that DEFG is a trapezoid.
b) Prove that DEFG is or is not an isosceles trapezoid.
11
Final Review #7 (mixed)
Geometry
Name _____________________
Date __________ Block _____
1. What is the equation of the circle with a radius of 7 and a center of
(3, -5)?
2. The larger of two supplementary angles is 40 more than three times the smaller
angle. Find the measure of the larger angle.
3. If a triangle has sides of 30, 15, and 41, what is the perimeter of the triangle formed
by connecting the midpoints of the sides of the triangle?
4. What is the image of (5, -1) after a reflection in the y-axis?
5. Midpoint M of segment AB has coordinates (4, -3). If the coordinates of A are (2, 0),
what are the coordinates of B?
6. In parallelogram MATH, m<T = x+20 and m<H = 2x+10. Find the value for m<A.
7. In the diagram, ABC ~ AEG . If AE = 10,
EB = 4, and GC = 5, find the value of AG.
8. In triangle TAP, m<T = 50, m<A = 100. Which side of the triangle is the smallest?
9. A square has a side of 16. What is the length of a diagonal of the square?
10. Find the probability that a dart tossed at random will land in the shaded area of
the figure.
12
Mixed Review 1:
1. Graph: ( x  1)2  ( y  3)2  9
2. Write a statement that is
logically equivalent to ”If an angles
is a straight angle, then its measure
is 180 degrees”?
_________________________________
_________________________________
_________________________________
__________________________
__________________________
3. Given parallelogram ABCD. If m A  x2  14 and m B  10x  50 , find the positive
value of x.
4. Intersecting lines a and b are in plane R. Line m is perpendicular to both
lines a and b. Line m also satisfies which of the following conditions?
(a)
Line m is parallel to line a and b.
(b)
Line m is skew to line a and b.
(c)
Line m is perpendicular to plane R.
(d)
Line m is parallel to plane R
Draw a picture or explain your answer in words.
13
A
P
6
D 1 2 E
3
4
C
B
5.
Given: AD  AE; PX  QX ; PD  EQ
Prove: BD  CE
Q
5
X
14
Mixed Review 2:
1. MN MN is the median of
Trapezoid ABCD.
A
M
D
B
N
C
2. Write the statement that is
the inverse of ”If a quadrilateral is
a rhombus, then its’ diagonals are
perpendicular”?
_________________________________
If AB=10, DC=14, then MN=_______
_________________________________
_________________________________
__________________________
__________________________
3. Two parallel lines below are cut by a transversal, find the value of x.
4. The measures of the angles of a quadrilateral are in the ratio 2:4:5:7.
Find the measure of the angles.
15
5.
Given: DAC  BCA
Prove: ABCD is a parallelogram
A
D
B
C
16
Mixed Review 3:
1.
2.
3. Find the slope of a line that passes through the points (-6, 8) and (2, -4).
4.
5.
17
Mixed Review 4:
1.
2.
3.
4. If the endpoints of the diameter of a circle are A (5, 2) and B (-3, 4), find the
coordinates of the center of the circle.
5.
6.
18
Mixed Review 5:
1. If B, C and D are collinear, and m  ACD = 50, what can you say about the
measure of angle ACB?
(1) mACB  50 (2) mACB  50
(3) mACB  50
(4) mACB  40
2. What is the total number of points of intersection of the graphs of the
equations y  x 2  5 and y  x . Draw a sketch to prove your answer.
3. Write the equation of the perpendicular bisector of line segment with
endpoints A(2, 6) and B(-2, 0).
4. Write the contrapositive of the statement:
“If you do your homework, then you will do well on the test”
19
5. Given:
ABC is an isosceles triangle with base AC
Segment BE is not a median
B
Prove:
Segment BE is not an angle bisector
A
E
C
20
Rectangle
- opp. sides || and 
- diag. bisect each
other & are 
- all right s
Parallelogram
- opp. sides || and 
- diag. bisect each other
- opp s 
Rhombus
- opp. sides || and 
- diag. bisect each other
- diag. bisect the s of rhom.
- all sides 
Parallel lines
|| lines  corresponding s 
|| lines  alternate interior s 
|| lines  alternate exterior s 
|| lines  same-side interior s supplementary
Trapezoid
- 1 pr. opp sides ||
Coordinate Geometry
Distance =
 x2  x1 
2
  y2  y1 
2
 x  x2 y1  y2 
Midpoint:  1
,
2 
 2
y  y1
slope = 2
x2  x1
slope-intercept form of line: y = mx + b
point-slope form of a line: y  y1  m  x  x1 
circle equation:  x  h    y  k   r 2
center: (h, k) and radius = r
parabola / quadratic equation:
2
y  ax 2  bx  c;
axis of symmetry : x  
Thms. to prove
s are 
ASA
SAS
AAS
SSS
HL (rt )
Thms. to prove
s similar
AA similarity
SAS similarity
SSS similarity
Altitude to Hypotenuse of Rt.
2
b
2a
y  a  x  h   k ; vertex   h, k 
2
Right Triangle
Trigonometry
opp
sin  
hyp
adj
cos  
hyp
opp
tan  
adj
Square
- opp sides ||
- all sides 
- diag.  & bisect each other
- all right s
Ratios for special
Right s
30-60-90
: 1: 3 : 2
45-45-90
: 1:1: 2
Triangle Inequalities
- any 2 sides of have a sum greater than the
3rd side
- the larger angle of a is opposite the larger
side of the
- exterior
of a is greater than either remote
interior
of the
seg1 on hyp.
alt

alt
seg 2 on hyp.
seg1 on hyp.
adj leg

adj leg
whole hyp.
Logic
conditional: p  q
converse: q  p
inverse: ~ p  ~ q
contrapositive: ~ q  ~ p
Indirect Proof
Assume the opposite of “prove”
statement and continue a direct
proof method until there is a
contradiction (usually of the
given information)
21
Circle Rules -- Angles:
1) central angle = meas. arc
1
2) inscribed angle = arc
2
1
(sum arcs)
2
3)
formed by 2 chords =
4)
formed with vertex outside circle =
1
(difference arcs)
2
Circle Rules – Segments:
1) radius bisect chord  chord  to radius
2) 2 intersecting chords: products of the segments on
each chord are =
3) 2 secants:
(ext. seg)(whole secant) = (ext seg)(whole secant)
4) tangent/secant:
(tangent seg)2 = (ext. secant)(whole secant)
Polygon Names:
Triangle – 3 sides
Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon – 6 sides
Octagon – 8 sides
Decagon – 10 sides
Regular Polygon: all sides
congruent and all interior
angles congruent
Triangles – by sides:
Scalene – all sides different lengths
Isosceles – 2 congruent sides
Equilateral – all 3 sides congruent
Polygon Angles – (n-sided)
Sum of interior s = 180  n  2 
Sum of exterior
Triangles – by angles:
Acute – all angles are acute
Right – one right angle
Obtuse – one obtuse angle
SurfaceArea of Prism=(Perimeter)(height)+2(BaseArea)
SurfaceAreaofPyramid=(Perimeter)(slantheight)+BaseArea
s = 360
Regular Polygons…
180 (n  2)
1 interior
=
n
360
1 exterior
=
n
Volume of Prism = (Base Area)(height)
Volume of Pyramid =
1
(Base Area)(height)
3
cylinder
cone
sphere
l
SA  2 r 2  2 rh
SA   r 2   rl
V  ( BaseArea)(height )
V  13 ( BaseArea)(height )
V   r 2h
V  13  r 2 h
SA  4 r 2
V  43  r 3
22