geometry final review name: per geometry final review name: _____ per: ___ chapter 1 vocab word...
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Geometry Final Review Name: ____________________________ Per: ___
Chapter 1
Vocab Word Definition
Acute angle
Adjacent angles
Angle bisector
Collinear
Line
Linear pair
Midpoint
Obtuse angle
Plane
Pythagorean theorem
Ray
Right angle
Supplementary angles
Complementary angles
Vertical angles
Perpendicular lines
Straight angle
Segment bisector
Segment addition postulate
Distance formula
Midpoint formula
Angle addition postulate
Congruent
Example Problems
Find the distance between F( -1 , 4 ) and G( 6 , -2 )
Find the midpoint of the segment between H( -3 , 2 ) and K( 8 , -1 )
Find x, leave in simplified radical form
x 14
9
2
Given that ⃗⃗⃗⃗ ⃗ ⃗⃗⃗⃗ ⃗ ⃗⃗ ⃗⃗ ⃗
Multiple Choice Practice
If two angles form a linear pair, they are _______.
A. Congruent B. Supplementary C. Complementary D. vertical
The measure of the supplement of an angle is 30 less than four times the measure of the complement of the angle. Find the measure of the angle.
A. 130 B. 60 C. 40 D. 50
Find the coordinates of the midpoint on for L( 10 , 8 ) and M( 2 , 6 )
A. ( 6 , 7 ) B. ( 8 , 2 ) C. ( 12 , 14 ) D. ( 5 , 2 )
Find m C if C ≌ D, m C = 3x – 5, and D = 2x + 5
A. 65 B. 10 C. 30 D. 25
Chapter 2
Vocab Word Definition
Conditional statement
Hypothesis
Conclusion
Conjecture
Counterexample
Deductive reasoning
Inductive reasoning
If-then statement
Inverse
Converse
Contrapositive
Law of detachment
Law of syllogism
T
S R Q
U
3
Important Theorems
Theorem What does it say
2-2: Supplement Theorem
2-4
2-5
2-6
2-7
Example Problems
Determine if each conjecture is true or false, explain your answer
Write each conditional in if-then form Write the converse, inverse and contrapositive
Given: X, Y, and Z are collinear and XY = YZ.
Conjecture: Y is the midpoint of
Every cloud has a silver lining If a rectangle has four congruent sides, then it is a square. Converse: ____________________ _____________________________ _____________________________ Inverse: _____________________ ____________________________ ____________________________ Contrapositive: _______________ ____________________________ ____________________________
Given: 1 and 2 are supplementary
Conjecture: 1 ≌ 2
A rectangle has four right angles
Determine if statement (3) follows from statement (2) and (1) by the law of detachment or the law of syllogism. If it does state which law was used. If it does not, write invalid.
(1) All pilots must pass a physical examination. (2) Kris Thomas must pass a physical examination (3) Kris Thomas is a pilot
(1) If a student is enrolled in Nipomo High, then the student has an ID number. (2) Jenny Jones is enrolled at Nipomo High. (3) Jenny Jones has an ID number.
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Multiple Choice Practice
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
"Two lines in a plane always intersect in
exactly one point." Which of the following best describes a counterexample to the assertion above?
A. coplanar lines B. parallel lines C. perpendicular lines D. intersecting lines
Find the value of x. A. 7 B. 14 C. 13.3 D. 26
Identify the hypothesis of the following if-then statement. If Ralph does his math homework, then he will get a good grade on the quiz.
A. He will not get a good grade on the quiz. B. Ralph may do his math homework. C. He will get a good grade on the quiz. D. Ralph does his math homework.
Which statement follows from statements (1) and (2) by the Law of Syllogism? (1) If an object is a square, then it is a rhombus. (2) If an object is a rhombus, then it is an equilateral.
A. An object is a rhombus. B. If an object is an equilateral, then it is a square. C. If an object is a square, then it is an equilateral. D. An object is a square.
Which statement is the inverse of the statement angles with the same measure are congruent?
A. If two angles do not have the same measure, then they are not congruent.
B. If two angles are not congruent, then they do not have the same measure.
C. If two angles have the same measure, then they are congruent.
D. If two angles are congruent, then they have the same measure.
Chapter 3
Vocab Word Definition
Parallel Lines
Alternate interior angles
Alternate exterior angles
Consecutive angles
Corresponding angles
4x – 5 3x + 2
5
Skew lines
transversal
Slope of a line
Important Theorems
Theorem What does it say
Corresponding angles postulate
Alternate interior angle theorem
Consecutive interior angle theorem
Alternate exterior angles theorem
Perpendicular transversal theorem
Converse of Corresponding angles postulate
Converse of Alternate interior angle theorem
Converse of Consecutive interior angle theorem
Converse of Alternate exterior angles theorem
Converse of Perpendicular transversal theorem
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Postulate 3-2
Postulate 3-3
Example Problems
In the figure,
4 = _____ 7 = _____
3 = _____ 6 = _____
5 = _____ 8 = _____
Find the value of x, y, and z.
Find the slope of the line parallel to the line through ( -3 , 0 ) and ( -4 , 5 ).
Find the slope of the line perpendicular to the line through ( 2 , -9 ) and ( -1 , 4 ).
Find the value of x so that
Determine whether each statement is true or false. Explain your reasoning
1. 6 and 11 are alternate interior angles
2. 4 and 9 are alternate exterior angles
3. 7 and 11 are corresponding angles
Multiple Choice Practice
Find x
What is the m 2?
1
2
3 4
5
6
7 8
X Y Z
S T
42◦ 3x◦
( y + 7 )◦ z◦
( 5x - 8 )◦
( 3x + 20 )◦ l
m
n k
l
m
1 3 4
5 6
7
10 11 12 13 14
2
8 9
A. -7
B. 7
C. -91
D. 132
A. 15◦
B. 75◦
C. 90◦
D. 105◦
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Complete the following proof. 1 b
Given: 2 1 3 Prove: b || w w
2 Statements Reasons
1. 2 1 1. Problem #5
2. 1 3 2. Problem #6
3. 2 3 3. Problem #7 4. b || w 4. Problem #8
5. A) Given B) Prove C) Alternate Exterior Angles Theorem D) Alternate Exterior Angles Converse 6. A) Given B) Corresponding Angles Theorem C) Alternate Exterior Angles Converse D) Vertical Angles are congruent
Chapter 4
Vocab Word Definition
Acute triangle
Equilateral triangle
Obtuse triangle
Equiangular triangle
Isosceles triangle
Scalene triangle
Right triangle
Exterior angle
CPCTC
7.
A) Transitive Property
B) Reflexive Property C) Corresponding Angles Theorem D) Corresponding Angles Converse
8.
A) Prove B) Corresponding Angles Converse C) Corresponding Angles Theorem D) Lines are always parallel
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Important Theorems
Theorem What does it say
Angle sum theorem
Third angle theorem
Exterior angle theorem
SAS
SSS
ASA
AAS
Isosceles Triangle theorem
Thm: 4-7 Converse of the isosceles
triangle theorem
Example Problems
Use the distance formula to classify the triangle by the measures of its sides.
ABC with vertices A( 6 , 4 ), B( -2 , 4 ), and C( 2 , 7 )
Find the value of x
Find x
20◦ 15◦
x◦ 45◦
(3x + 16)◦ 112◦
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If find the measure of each angle
1 = _____ 2 = _____
3 = _____ 4 = _____
5 = _____ 6 = _____
Complete the congruence statement
ABX ≌ __________
Determine which theorem or postulate can be used to prove the triangles are congruent, if it is not possible state that.
Multiple Choice Practice
Use the proof to answer the question below.
Given AB BC; D is the midpoint of AC
Prove: ∆ABD ∆CBD
Statement Reason
1. AB BC 1. Given 2. D is the midpoint of AC 2. Given
3. AD CD 3. Def of midpoint
4. BD BD 4. Reflexive Prop
5. ∆ABD ∆CBD 5. ? What reason can be used to prove that the triangles are congruent?
A. AAS B. ASA C. SAS D. SSS
In the figure below, AC DF and A D.
Which additional information would be enough to
prove that ∆ABC ∆DEF?
A. AB DE
B. AB BC
C. BC EF
D. BC DE
Find the values of x and y:
A. x = 32, y = 74 B. x = 74, y = 54 C. x = 74, y = 106 D. x = 32, y = 106
Given H L and HJ JL. Which of the following is true?
A. ∆HIJ ∆JKL by SAS
B. ∆HIJ ∆KLJ by ASA
C. ∆HIJ ∆KLJ by SAS
D. ∆HIJ ∆LKJ by ASA
43◦
78◦
1
2 3
4 5
6
56◦
A
B
D
C
C
A B
F
D E
106
x
y y
I
J
K
L H
P Q
R S
B
X
A
D C
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, which term does not describe the triangle?
A. Isosceles B. Obtuse C. Acute D. Equilateral
Which of the following is not the way to prove two triangles congruent? 1. SSS
2. SAS
3. CPCTC
4. SSA
5. SAA
6. AAA
Chapter 5
Vocab Word Definition
Altitude of a triangle
Angle bisector of a triangle
Median of a triangle
Perpendicular bisector of a triangle
Important Theorems
Theorem What does it say
HL
HA
LL
LA
Triangle inequality Theorem
A B
C
x + 4 3x - 2
7
A. 1, 2, and 5 B. 3, 4, and 6 C. 3 only D. 4 only
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Example Problems
State the additional information needed to prove the triangles are congruent by the given theorem
LL HA HL
Find the value of x so that
MNP ≌ ONP by LL
Find the values of x and y so that
ABC ≌ DEF by HA In ABC identify the following: Median: ____ Altitude: ____ Angle bisector: ____ Perpendicular bisector: ____
Multiple Choice Practice
Which of the following sets of numbers could represent the lengths of the sides of a triangle? A. 2, 2, 5 B. 3, 3, 5 C. 4, 4, 8 D. 5, 5, 15
Two sides of a triangle have lengths of 7 and 13. The third side has a length that is _____? A. 6 < 3rd side < 13 B. 6 < 3rd side < 20 C. 13 < 3rd side < 20
D. 6.5 < 3rd side < 19.5
Which may not contain a vertex of a triangle?
A. Perpendicular bisector B. Altitude C. Median D. Angle bisector
Multiply a number by 5, subtract 6, multiply the result by 3, then add 8. If the final result is 80, what number did you start with?
A. 6 B. 8 C. 30 D. 72
V W
Q
X Y
Z A B
C D
S
V
T
R U
P
M N
O 2x - 4
A
B
E
D
F
C
(6y
– 2
)◦
C
A B
D
E
F G
H 8
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Chapter 6
Vocab Word Definition
trapezoid
parallelogram
quadrilateral
Rhombus
Square
Rectangle
Median of a trapezoid
Isosceles trapezoid
diagonal
Important Theorems
Theorem What does it say
6-1
6-2
6-3
6-4
6-5
6-6
6-7
6-8
13
6-9
6-10
6-11
6-12
6-13
6-14
6-15
6-16
Name: Name: Name: Name: Name:
Properties:
Properties:
Properties:
Properties:
Properties:
Parallelograms 1. 2. 3. 4. 5.
Directions: 1. Consider each of the six polygons above. Write the name and properties of each in the spaces provided. 2. Label the figures to show all congruent sides, diagonals, angles, and segments. 3. Label congruent angles, and show any special relationships such as supplementary angles or parallel sides.
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Example Problems
In the parallelogram find the values of x, y and z
Determine if the quadrilateral is a parallelogram
Find the values of x and y that makes the quadrilateral a parallelogram
Use the rhombus below to find x
m 5 = 2( x + 1 )
m 3 = 4( x + 1 )
PQRS is an isosceles trapezoid with
bases
use the following to solve If TV = x + 7 and PS + QR = 5x + 2, find x
True or false. Explain your answer. If the diagonals of a quadrilateral bisect each other, then it is a rectangle.
Multiple Choice Practice
What values of a and b make MNOP a parallelogram?
If ABCD is a parallelogram, then what is the length of BD?
120◦
35◦ x◦
y◦
z◦ 12
12
8
8
5x + 4y x◦
y◦ 7x + 22
3
4
6
1
5
P T
Q
S V
R
A 10
B 11
C 12
D 14
15
Find the perimeter of the rectangle: (2x + 5) in A. 95 in 30 in B. 60 in C. 25 in (3x – 15) in D. 150 in
There is always a pair of non-congruent sides in a
A. Parallelogram B. Trapezoid C. Rhombus D. quadrilateral
Chapter 7
Vocab Word Definition
Geometric mean
Similar polygons
Important Theorems
Theorem What does it say
AA similarity
SSS Similarity
SAS Similarity
Thm: 7-4 Triangle proportionality
Thm: 7-5
Thm: 7-6
Thm: 7-7
Thm: 7-8
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Thm: 7-9
Thm: 7-10
Thm: 7-11
Example Problems
Determine if the triangles are similar, if they are, tell how you know
Solve for x
70◦
30◦
85◦
10
8
6
12
15 9
8
10
10
8
8 24
8
4
6
x 5
10
5 x - 3
x
17
. If BP = 8, AP = 6, DF = 2x + 1 and EY = 2x – 4, find DY.
If ∆STV~∆PQM, the perimeter of ∆PQM is 28, Find the perimeter of ∆STV.
Multiple Choice practice problems
Find x
A. -2 B. 12 C. -12 D. 6
In . If EI = 8, IF = 4, and EH = 5, find HG.
A. 1 B. 2 C. 2.5 D. 10
?
A.
B.
C.
D.
B
E
A P C
D Y F
S
T
V x – 1.5
x - 4 10
P
Q
M
8
x + 6
8
16
x
E
I H
G F
R
N
P S Q
L O M
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Chapter 8
Vocab Word Definition
Sine
Cosine
Tangent
Law of sines
Law of cosines
Pythagorean triple
Geometric mean
Important Theorems
Theorem What does it say
Converse of the Pythagorean therorem
Thm: 8-6
Thm: 8-7
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Trigonometry Flow Chart:
START:
How many side lengths
do you already have?
Draw and Label
the Triangle
EXAMPLE:
5 in.
x
____ : ____ : ____
Visit the
Chief!
____ : ____ : ____
25o
20
START:
How many angles do
you already have?
Draw and Label the Triangle
Example:
Trigonometry Flow Chart:
Example:
Chosen
Angle
A
B C
C
13
12
21
Example problems What is the area in square inches of the triangle below?
In the figure below, sin A = 0.7 What
is the length of ?
A right triangle’s hypotenuse has a length of 5. If one leg has length 2, what is the length of the other leg? (Hint: Draw the triangle.)
What is the value for x in the triangle below?
A new road is being constructed to relieve traffic congestion in a residential neighborhood. The plan for the old road and the new road is shown below. How many fewer miles will the commuters travel on the new road?
An 8-foot ladder is leaning against a wall. Approximately how far up the wall does the ladder reach?
Multiple choice practice problems
Which equation should be used to find the length of ? A 13 ft ladder is leaning against a wall. The top of the ladder touches the wall 12 ft above the ground. The bottom of the ladder is 5 ft from the bottom of the wall. What is the sine of the angle formed by the ground and the base of the ladder?
22
If RSTW is a rhombus, what is the area of ΔWXT?
What is the approximate height, in feet, of the tree in the figure below?
What is the approximate value of x in the triangle below?
A 36
B 363
C 48
D 183
A 3.4 units
B 4.2 units
C 4.9 units
D 7.3 units
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Chapter 9
Vocab Word Definition
Minor arc
Major arc
Central angle
chord
Tangent of a circle
Inscribed angle
Intercepted arc
Secant
Semicircle
Radius
diameter
Important Theorems
Theorem What does it say
Thm:9-1
Thm: 9-2
Thm: 9-3
Thm: 9-4
Thm: 9-5
Thm: 9-6
Thm: 9-7
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Thm: 9-10
Thm: 9-11
Thm: 9-12
Thm: 9-13
Thm: 9-14
Thm: 9-15
Thm: 9-16
Vocabulary Using Correct Symbols, give an example of:
1. Minor Arc: ________ 2. Major Arc: ________ 3. Semi-Circle: ________ 4. Diameter: ________ 5. Radius: ________ 6. Chord: ________ 7. Inscribed Angle: ________ 8. Central Angle: ________
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Example problems
Arcs and Angles Segments Theorems
ON
ON
IN
OUT
Criss-Cross
Outside x Whole
Outside2
Equation of a Circle
Equation: _________________________
Center: ____________
Radius: ____________
Two Tangents From the
Same Point
A Diameter Perpendicular to a Chord
A Tangent and a Radius
Quadrilateral Inscribed in a Circle
• x
y
•
•
x
3x
8y
84o
•
8
10 16
x 8
10 x° 82°
46°
42°
x°
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Multiple Choice Practice Problems Refer to circles B and D in the figure below. If BC = 5 and CD = 5, find AE.
A. 20 B. 15 C. 10 D. 25
What is the measure of the angle formed by the hands of a clock at 4 o'clock?
A. 60 B. 120 C. 30 D. 90
In the figure below, is a diameter of the circle. If US = 9, find SV.
A. 4.5 B. 27 C. 18 D. 9
Find the value of y.
A. 19 B. 11
C.
D.
Find the value of y to the nearest tenth
A. 7.2 B. 7.6 C. 7.5 D. 8.0
Find an equation of the circle that has a diameter with endpoints at (6, 10) and (-2, 4). Hint: find the center of the circle by finding the center f the diameter.
A. (x - 2)2 + (y - 7)2 = 25 B. (x + 2)2 + (y - 4)2 = 100 C. (x - 4)2 + (y - 3)2 = 53 D. (x - 6)2 + (y - 10)2 = 100
30
20
x
10
x
25
12
x°
86°
A
B
C
D
E
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Chapter 10
Vocab Word Definition
apothem
Concave polygon
Convex polygon
Regular polygon
Important Theorems
Theorem What does it say
Interior Angle Sum Theorem
Exterior Angle Sum Theorem
Example Problems Find the measure of one interior angle of a regular 24-gon
Find the measure of one exterior angle of a regular 16-gon
Find the interior angle sum of a regular dodecagon
Is there a regular polygon with an interior angle sum of 9000°? If so, what is it?
Find the area of the following octagon apothem = 14.1 side = 11.7
Find the area of the shaded region
6
8
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Multiple Choice Practice Problems A trapezoid has an area of 80.75 sq. in., and its two bases are 7 and 12 inches long. Find the height of the trapezoid
A. 7 in. B. 8.75 in. C. 8.5 in. D. 9.2 in.
Find the area of a regular octagon with an apothem of 8.5.
E. 239 F. 296 G. 182 H. 340
Find the area of this figure.
A. 104 sq. units B. 98 sq. units C. 200 sq. units D. 168 sq. units
Find the area of the shaded region between the circle of diameter 4 feet and the equilateral triangle. Round to the nearest tenth.
A. 9.4 ft2 B. 10.0 ft2 C. 8.7 ft2 D. 12.6 ft2
Chapter 11
Vocab Word Definition
cylinder
sphere
cone
Prism
Pyramid
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Important Formulas
Solid Lateral Area Surface Area Volume
Prism
Cylinder
Pyramid
Cone
Sphere
Example Problems – Find surface area and volume of each solid
A sphere with a diameter of 6.2 in Sketch the net of the solid
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Multiple Choice Practice Problems Find the surface area of a cylinder with a radius of 2 meters and a height of 6 meters.
A. 125.7 m2 B. 100.5 m2 C. 113.1 m2 D. 141.4 m2
Find the volume of the right prism.
A. 144 cm3
B. 192 cm3
C. 264 cm3
D. 216 cm3
Find the volume to the nearest tenth of a sphere with a radius of 4.2 cm.
A. 485.6 cm3 B. 225.4 cm3 C. 235.6 cm3 D. 310.3 cm3
Find the surface area of the solid. Round to the nearest tenth.
A. 56.4 in2 B. 112.9 in2. C. 494.8 in2 D. 876.7 in2