1

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.

4

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

6

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◦

7

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◦

9

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

11

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

12

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.

14

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

16

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

18

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

19

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

23

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

24

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: ________

25

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°

26

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

27

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

28

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

29

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

30

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