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Name: _________________________ Geometry, Final Review Packet

I. Vocabulary – match each word on the left to its definition on the right.

Word Letter Definition

Acute angle C A. Meeting at a point

Angle bisector M B. An angle with a measure greater than 90° and less than 180°

Congruent H C. Angle with a measure of greater than 0° and less than 90°

Endpoint E D. An angle with a measure of 90°

Intersecting A E. The point at the beginning or end of a segment

Line J F. Two points on a line and all the points in between them

Line segment F G. A point on a segment that is equidistant from both endpoints

Midpoint of a segment G H. Having the same size and shape

Obtuse angle B I. Two lines that intersect and form a right angle at the point of intersection

Parallel P J. A one dimensional figure extending in two directions forever

Perpendicular I K. Two angles that share a vertex and no sides and that are formed by intersecting lines

Point L L. A location in space (zero dimensions)

Right angle D M. A line, segment or ray that cuts an angle into two congruent angles.

Side Q N. Angles with a sum of 180°

Supplementary angles N O. The point where two lines, rays, or segments meet.

Vertex O P. Two lines in the same plane that never intersect.

Vertical angles K Q. One of the line segments that makes up a polygon.

Match each word on the left to its diagram on the right.

1. Trapezoid

2. Parallelogram

3. Rectangle

4. Prism

5. Cylinder

6. Cone

7. Pyramid

8. Sphere

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II – Drawing – for each description or symbolic statement, draw a figure.

1. Acute angle RED with angle bisector ⃗⃗⃗⃗ ⃗

2. ̅̅ ̅̅ with midpoint M

3. ̅̅ ̅̅ ⃡⃗⃗⃗ ⃗

4. ⃡⃗⃗⃗ ⃗⃗ ⃡⃗⃗⃗ ⃗

5. Vertical angles

6. Supplementary angles

III – Midpoint and Distance – find the length and the midpoint of both segments below.

1. Endpoint (-5, 6) and endpoint (-5, 10) Length = 4 Midpoint = ( –5, 8)

2. Endpoint (-1, -3) and endpoint (-1, 14) Length = 17

Midpoint = (–1, 5.5 , 1)2

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E

R

D

F T OM

C D

F

G

M N

P Q

DA

EB

C

K

J

G H

3

IV -- For each of the following symbolic or written statement, draw a labeled diagram to match.

1. Parallelogram FROG

2. Isosceles with base ̅̅ ̅̅ ̅

3. Trapezoid HGFT with ̅̅ ̅̅ ̅̅̅̅

4. Right with hypotenuse ̅̅ ̅̅

V – Classifying Polygons -- Classify and name each of the polygons below. Use the most specific

classification possible. (Remember that polygons are named by stating the letters of their vertices in

order.)

1.

Isosceles Triangle

2.

Rectangle KNML

3.

Obtuse isosceles triangle OPQ

4.

Right Triangle SRT

5.

Equilateral Triangle UWV

6.

Parallelogram XAZY

7.

Square ADCB

8.

Hexagon

9.

Parallelogram IJKL

O

FG

R

M

K N

L

114°

P

Q

O

R

S T60° 60°

60°

V

U W

XA

Y Z

CB

A D

K

I L

J

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VI -- Determine whether the three side lengths given could form a triangle. Write yes or no.

1. 4 miles, 5 miles, 6 miles 3. 2 in, 2 in, 2 in

YES YES

2. 8 km, 5 km, 1 km 4. 6 cm, 5 cm, 2 cm

NO YES

VII – Algebra – Solve for x

1.

x=8.5

2.

x=8

VIII – Parallel line vocabulary

1. Jenisteen and Abria and Amari have houses that are corresponding angles.

2. Prasuna and _Olivia_____ have houses that are alternate exterior angles.

3. Danny and _Jenisteen_________ have houses that are vertical angles.

4. Adam and __Abria__________________ have houses that are consecutive interior

angles (also called “Same Side Interior” angles).

5. Ashley and _Alvia_____________ have houses that are alternate interior angles.

6. Amari and __Meaghan_________ have houses that are alternate interior angles.

4x

(3x - 1.5) cm(2x + 7) cm

N

MO

(8x-22)°

(4x + 10)°Q

P

R

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IX -- Use the angle measures to determine whether or not each pair of LINES is parallel, not

parallel, or cannot be determined.

1.

NOT PARALLEL-

corresponding angles not

2.

NOT PARALLEL

AEA not

3.

PARALLEL

Both given angles are supplementary to the same

angle

4.

CANNOT BE

DETERMINED

5.

PARALLEL

AIA are

6.

PARALLEL

Corresponding angles are

X – Pythagorean Theorem

1.

6.874?

2.

?=6

3.

x = 130

4.

x=21

74°

66°

91°

89°

91°

89°

91° 89°

89°

89°

89°

89°

?

7 cm

5 cm

?

8 cm

10 cm

X

120

50X

3528

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XI – Proportions – Solve these problems.

1.

2.

975

4515

451575

x

x

2.4

2084

45712

x

x

x

3. If 5 boxes cost $7.50, how much will 2 boxes cost?

00.3$

25

5.7

x

x

4. A flagpole casts a shadow that is 27 feet long. A person standing nearby casts a shadow 8 feet long. If

the person is 6 feet tall, how tall is the flagpole?

Object height Shadow length 6 8 x 27

Set up the Proportion (other ways to set this up are OK):

25.20

1628

27

86

x

x

x

5. The Smiths paid $80 for 480 square feet of wallpaper. They need an additional 120 square feet. How

much will the additional wallpaper cost?

Sq feet of wall paper $$ 480 80 120 x

Set up the Proportion:

20$

9600480

80

120

480

x

x

x

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XII – Transformations

1.

2.

3.

4.

Identify the transformation performed below:

5.

4,4, yxyx or Right 4, Down4

6.

Reflection over the x-axis

8

7.

180° Rotation (either direction)

8.

Translate: down 1, right 3 or

1,3, yxyx

9. Draw quadrilateral JKLM with vertices J(-5,3), K(-4,5), L(-3,3) and M(-4,1). Then find the

coordinates of the vertices of the image after the translation (x, y) → (x + 6, y – 2).

1,2

1,3

3,2

1,1

M

L

K

J

10. Draw parallelogram ABCD with vertices A(-3,3), B(2,3), C(4,1) and D(-1,1). Then find the

coordinates of the vertices of the image after a reflection across the x-axis, and draw the image.

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XIII – Similar Figures – Find the missing measures of these similar figures.

1.

AL = 6 RA = 10 RG = 4 KN= 6

Work Space

2.

NY = 21 YC = 42 CM = 27 MB = 30

Work Space

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Are these triangles similar?

3. YES 4. NO

XIV – Area and Volume

Find the area and perimeter of the rectangle.

A = 228 m2

P =62 m

Area = 96 yd2, find the base and the Perimeter 0f the rectangle.

b = 8 yd P =40 yd2

Find the area and perimeter of the parallelogram.

A =96 in2

P =42 in.

Area = 2508 cm2. Find the base and the perimeter of the parallelogram.

2508=44b b = 57 cm P =57*2+48*2=210 cm

Find the area of the triangle.

A = (1/2)*8*5=20 cm2

Area = 39 cm2, find the height of the triangle.

39=(1/2)*13*h h =6 cm

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Find the area of the trapezoid.

A = 6061462

1 cm2

Area = 180 m2. Find the length of the missing base.

9242

1180 b

b = 16 m

If the circumference of a circle is 12 in, find the radius and the area.

r 212 r =6 A =pi*r2 = 36*pi = 113

Find the area of the circle, the area of the triangle, and the area of the shaded region. Use 3.14 for pi and round to the nearest tenth of a centimeter.

A (circle) = 51.2 cm2

A (triangle) I think this is too hard. Each angle measures 60°. If h is the height of the triangle,

142

360sin

h ; therefore, h = 37 and

Area= 37142

1 =84.9 cm2

Area of shaded region = 84.9 – 51.2 = 33.7 cm2

Find the lateral surface area, surface area, and volume of the right circular cylinder. All the measures on the diagram are in centimeters. Show your work. LA =80*pi cm = 251 SA =112*pi cm2= 351.7

V =160*pi cm3= 502.7

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Find lateral surface area, surface area, and volume of the rectangular prism. Show your work.

LA = 23525.2325.24 ft

SA = LA + 2(3)(4) = 59 ft2

V = (2.5)(3)(4)=30 ft3

Find the surface area and volume of the sphere. Leave your answers in terms of pi.

SA = 36*pi = 113.1 V = 36*pi = 113.1

Find the total surface area and volume of the combined shape below. SA = You will first need to find the slant height l, of the cone using the Pythagorean Theorem: l2 =52 + 122 ,or l = 13. SA of cone: pi*(5)(13)=65 *pi SA of cylinder: 2 pi*(5)(6) + pi*(5)2 = 85 pi Total surface area = 150*pi ft2= 471.2

V =

Volume of cone: 1001253

1 2

Volume of cylinder: 75652

Total volume = 175*pi ft3= 549.8

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Find the slant height ( ), lateral surface area, surface area, and volume of the right circular cone. All measures are in centimeters. Leave your answer in terms of pi. (Hint: use the Pythagorean theorem to find the slant height.)

= 22.98576 22

LA =2 *pi*(6)(7)=84*pi≈263.9

SA = 7.43784856

V = 9.26384763

1 2

Find the slant height ( ), lateral surface area, surface area, and volume of the right square pyramid. All measures are in centimeters. (Hint: use the Pythagorean theorem to find the slant height.)

= 85.994 22

LA =2(8)(9.85)=157.6 SA =157.6+64=221.6 V = (1/3)(64)9=192

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XV – Lines

1. Find the slopes of the two lines. Then, determine whether each pair of lines is parallel, perpendicular,

or neither. Justify your answer.

2x – 3y = 12 and 4 = –3x – 2y

Converting the lines to slope-intercept form gives:

43

2 xy and 2

2

3 xy Therefore, the lines are perpendicular because their slopes are

negative reciprocals of each other.

2. Graph the following lines on the grid. Then tell if they are parallel or perpendicular:

y = 4 and x = –2

The lines are perpendicular.

3. Graph the line on the coordinate plane: 13

1 xy

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4. a. Write the equation for the following line:

b. Write an equation for a line perpendicular to the line in part a.

Many possible answers: 22

1 xy (any line with slope = ½)

5. a. Write the equation for the following line:

b. Write the equation for any line perpendicular to the line from part a.

y = –3x +2 (or any line with slope = –3)

6. a. Write the equation for the following line:

b. Write an equation for a line parallel to the line above that goes through the point (2,2)

y = 2

y = –2x +2

33

1 xy

y = 4

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7. Use lines p and q shown to the right to answer the questions.

a. What is the slope of each line? (Use the points shown

to find your answers.)

slope of line p = ______3_______

slope of line q = ______-2/6 = -1/3_______

b. Are the lines perpendicular? How do you know?

Yes, because their slopes are negative reciprocals

8. Determine the slope of the line through the given points:

a. A(3, 4) and B (-9, 12)

m= –2/3

b. C(-4, -5) and D (10, 12)

m = 17/14

XVI – Triangle Congruence

1. Are the following triangles congruent? Why? If so, write the congruence statement.

YES: by HL, ΔABD ΔCBD

2. Are these triangles congruent? Why?

a. b. c.

YES: HL YES: AAS NO: AAA

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d. e. f.

YES: SSS YES: ASA NO: SA

g. h. i.

YES: SSS YES: SAS YES: ASA

3. ∆CGI ∆MPR.

a. Draw a picture of this situation.

b. If mC = 27 and mG = 63, what is mR? 90°

c. If PR = 6 cm, what do you know about ∆CGI ? GI = 6

XVII – Proof

Directions: Fill in each of the missing steps (statements or reasons) in the proofs below.

1. Given: ⃗⃗⃗⃗⃗⃗ ;

Prove:

Statement Reason

1. ⃗⃗⃗⃗⃗⃗ 2. GIVEN

3. ROQPOQ 2. Definition of bisects

4. 3.GIVEN

5. OQOQ 4. Same Side (Reflexive property)

6. 5. AAS

I

C

GP

M

R

QO

R

P

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2. Given: ; Q is the midpoint of ̅̅̅̅

Prove: ̅̅̅̅ ̅̅ ̅̅ ;

Statement Reason

1. 1. Given

2. ̅̅̅̅ 2. Given

3. QPSQ

3. Definition of midpoint

4. TQPRQS 4. Vertical angles are congruent

5.

5. ASA

6. ̅̅̅̅ ̅̅ ̅̅

6. CPCTC

3.

TP

Q

R S

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Directions: Find all of the mistakes in the proofs below.

4.

CORRECTED PROOF: CHECK CAREFULLY!

Statement Reason

1. ̅̅ ̅̅ ̅̅ ̅̅ ; 1.Given

2. ̅̅ ̅̅ bisects 2. Given

3. ̅̅ ̅̅ ̅̅ ̅̅ 3. Same Side (Reflexive Property)

4. Definition of angle bisector

5. SAS

5.

CORRECTED PROOF: CHECK CAREFULLY!

Statement Reason

1. ̅̅ ̅̅ ̅̅ ̅̅ 1. GIVEN

2. ̅̅ ̅̅ 2. Given

3. ̅̅ ̅̅ ̅̅ ̅̅ 3. DEFINITION OF MIDPOINT

4. ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ 4. Given

5. 5. Definition perpendicular lines

6. 6. All right angles are congruent

7. 7. SAS

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6.

Statement Reason

1. ̅̅ ̅̅ ̅̅ ̅̅ 1. Given

2. ̅̅ ̅̅ ̅̅ ̅̅ 2. Given

3. ̅̅ ̅̅ ̅̅ ̅̅ 3. Definition of segment bisector

4. ̅̅ ̅̅ ̅̅ ̅̅ 4. Same Side (Reflexive Property)

5. 5. SSS

XVIII – Triangle Inequalities

1. List the angles in order from smallest to largest.

a. b.

ANSWER: KLM ,, ANSWER: CBA ,,

2. List the sides in order from smallest to largest.

ANSWER: FH, GF, GH

3. Find the longest side of triangle ABC, with , , and

Find all the angles by solving: 18070203102 xx

That gives x = 20. The largest angle is angle C (80 degrees). Therefore the longest side is

opposite. ANSWER: side AB.

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XIX – Algebra disguised as geometry- Set up an equation, then find the missing variable.

1. AC = 44 2. HJ x= -6 7

ANSWER: x = 8 ANSWER: x = 5

3.

 

ÐDEF =111 4.

 

ÐJKM =117 Find the measure of ÐLKM .

ANSWER: x = 8 ANSWER: x = 11

5. 6.

ANSWER: x = 5 ANSWER: x = 60

3x+4

2x

C

B

A

12

11

J

I

H

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7. B is the midpoint of

 

AC. Find x. 8. Suppose

 

RQ bisects

 

ÐPQS and

 

ÐPQS = 98 .

ANSWER: x = 3 ANSWER: x = 8; y = 15

9. 10.

FOR THIS PROBLEM YOU NEED TO SAME HERE: x = 8 (angles

ASSUME THE LINES ARE PARALLEL supplementary if lines are parallel)

OR THERE IS NO WAY TO DO IT. IF YOU

MAKE THAT ASSUMPTION, AIA are congruent

and:

x = 5

163x+7

CB

A

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XX -- Polygon Sum

1. Regular Octagon (8-gon)

a. What is the sum of the interior angles?

108018028 °

b. What is the measure of one interior angle?

135°

c. What is the sum of the exterior angles?

360°

d. What is the measure of one exterior angle?

45°

2. 3. 4.

159° 69° 60°

5. 6. 7.

103° 145° z = 70°; y = 103°

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8. What is the sum of the interior angle measures of a polygon with…

a. 15 sides b. 50 sides

o2340180215 o8640180250

9. What is the sum of the measures of the exterior angles of a decagon?

ANSWER: 360°

10. 11.

a = 108° b = 3

145 °

12. Find the missing angles.

a = 67°

b = 58°

c = 125°

d = 23°

e = 90°

x = 38°, y = 36°, z = 90°

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13. Find the missing angles.

ANSWER: (all measurements in degrees). CHALLENGE?

49;131;46;49;96;46;84;48;33;57;57;123;57 srqpkjhgfecba

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XXI – Right Triangle Trigonometry

a) Write a true equation using sine, cosine, or tangent.

b) Solve the equation.

1.

x

672cos

x ≈ 19.4

2.

6

73cosx

x ≈ 1.8

3.

12

24tanx

x ≈ 5.3

4.

12

37tanx

; x ≈ 9.04

5.

?21

9tan;

21

9?tan 1

? = 23.2°

6.

?29

14tan;

29

14?tan 1

? = 25.8°

7.

?27

11tan;

27

11?sin 1

? = 24.0°

8.

?5

4cos;

5

4?cos 1

? = 36.9°

9.

?24

12tan;

24

12?sin 1

? = 30°

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10. Tamara drew the triangle pictured at right and measured

one side and one angle as shown in the diagram.

a. Which leg is opposite Which leg is adjacent to it?

Opposite: AT; Adjacent: TM

b. Write a trigonometry equation (use sine, cosine, or tangent) that you could use to solve for the

length of ̅̅̅̅̅.

AM

1028sin

c. Solve the equation you wrote in part b. Show your work.

3.2128sin

10

1028sin

AM

AM

d. Use the Pythagorean theorem and your results from part c to find the length of ̅̅̅̅̅

222 103.21 TM = 18.8

e. Ryan argues that he could have used trigonometry to find the length of ̅̅̅̅̅ without knowing the

length of ̅̅̅̅̅ Is he right? If so, explain how to do it. If not, explain why he is wrong.

Yes, he is right. You could do it like this:

8.1828tan

10

1028tan

TM

TM

28°

10 in

A

TM