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

Congruent

H

Endpoint

E

B. An angle with a measure greater than 90° and less than 180° C. Angle with a measure of greater than 0° and less than 90° D. An angle with a measure of 90°

Intersecting

A

Line

J

Line segment

F

Midpoint of a segment

G

Obtuse angle

B

Parallel

P

Perpendicular

I

Point

L

Right angle

D

Side

Q

Supplementary angles

N

E. The point at the beginning or end of a segment F. Two points on a line and all the points in between them G. A point on a segment that is equidistant from both endpoints H. Having the same size and shape I. Two lines that intersect and form a right angle at the point of intersection J. A one dimensional figure extending in two directions forever K. Two angles that share a vertex and no sides and that are formed by intersecting lines L. A location in space (zero dimensions) M. A line, segment or ray that cuts an angle into two congruent angles. N. Angles with a sum of 180°

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

1

II – Drawing – for each description or symbolic statement, draw a figure. 1. Acute angle RED with angle bisector ⃗⃗⃗⃗⃗ R

2. ̅̅̅̅ with midpoint M

T

F

O

M

E D

3. ̅̅̅̅

⃡⃗⃗⃗⃗

⃡⃗⃗⃗⃗

4. ⃡⃗⃗⃗⃗⃗ F

M D

C

P

N Q

G

5. Vertical angles

6. Supplementary angles J

A

D B C

K G

H

E

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,

11 )   1 , 5.5 2

2

IV -- For each of the following symbolic or written statement, draw a labeled diagram to match. 1. Parallelogram FROG

with base ̅̅̅̅̅

2. Isosceles

R

O

F

G 4. Right

3. Trapezoid HGFT with ̅̅̅̅ ̅̅̅̅

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.

2.

3.

L

M

O Q 114°

K

N

Rectangle KNML

Isosceles Triangle 4.

5.

V

R

P

Obtuse isosceles triangle OPQ 6. Y Z

60°

T

S

Right Triangle SRT 7.

B

U

60°

60°

A

X

W

Equilateral Triangle UWV 8.

Parallelogram XAZY 9.

C I

A

D

Square ADCB

K

J

Hexagon

L Parallelogram IJKL

3

VI -- Determine whether the three side lengths given could form a triangle. Write yes or no. 1. 4 miles, 5 miles, 6 miles YES

3. 2 in, 2 in, 2 in YES

2. 8 km, 5 km, 1 km NO

4. 6 cm, 5 cm, 2 cm YES

VII – Algebra – Solve for x 1.

2.

N (2x + 7) cm

M

P (3x - 1.5) cm

4x

(4x + 10)°

O

Q

(8x-22)°

x=8.5 R

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.

4

IX -- Use the angle measures to determine whether or not each pair of parallel, or cannot be determined. 1.

2.

LINES is parallel, not

3.

89°

66°

89°

74°

91°

PARALLEL

91° NOT PARALLELcorresponding angles not 4.

Both given angles are supplementary to the same angle

NOT PARALLEL AEA not  5.

6.

89°

89°

89°

91° 89°

89°

CANNOT BE DETERMINED

PARALLEL AIA are 

PARALLEL

Corresponding angles are 

X – Pythagorean Theorem 1.

2. ? 10 cm

5 cm 8 cm

7 cm ?

?  74  8.6

?=6

3.

4.

X

35

28 120

50

X

x = 130

x=21 5

XI – Proportions – Solve these problems. 1.

2.

12  7  5 x  4

75 x  15  45 x

84  20 x

15  45 9 75

x  4.2

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

7.5 x  5 2 x  $3.00

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 6 x

Shadow length 8 27

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

6 8  x 27 8 x  162 x  20.25

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 480 120

$$ 80 x

Set up the Proportion:

480 80  120 x 480 x  9600 x  $20 6

XII – Transformations 1.

2.

3.

4.

Identify the transformation performed below: 5.

6.

x, y   x  4, y  4 or Right 4, Down4

Reflection over the x-axis

7

7.

8.

Translate: down 1, right 3 or 180° Rotation (either direction)

x, y   x  3, y  1

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). J   1,1

K   2,3 L   3,1

M   2,1

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.

8

XIII – Similar Figures – Find the missing measures of these similar figures. 1.

Work Space

AL = 6 RA = 10 RG = 4 KN= 6 2.

Work Space

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

9

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

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

b = 8 yd

P =40 yd2

Find the area of the triangle.

Find the area and perimeter of the parallelogram.

A =96 in2

P =42 in. Area = 39 cm2, find the height of the triangle.

39=(1/2)*13*h 2508=44b

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

h =6 cm

b = 57 cm

P =57*2+48*2=210 cm

10

Find the area of the trapezoid.

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

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

12  2r

A=

1 6  14  6  60 cm2 2

r =6

180 

1 24  b  9 2

A =pi*r2 = 36*pi = 113

b = 16 m

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.

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 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,

sin 60  Area=



V =160*pi cm3= 502.7

3 h  ; therefore, h = 7 3 and 2 14



1 14  7 3 =84.9 cm2 2

Area of shaded region = 84.9 – 51.2 = 33.7 cm2

11

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

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

LA = 4  2.5  2  3  2.5  2  35 ft 2 SA = LA + 2(3)(4) = 59 ft2 SA = 36*pi = 113.1

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

1  52 12  100 3 Volume of cylinder:  5 2 6  75 Volume of cone:

 

Total volume = 175*pi ft3= 549.8

12

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.)

=

6 2  7 2  85  9.22

LA =2 *pi*(6)(7)=84*pi≈263.9

SA = 6  85  84  437.7

1 V =   6 2  7  84  263.9 3

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.)

= 4 2  9 2  9.85

LA =2(8)(9.85)=157.6

SA =157.6+64=221.6

V = (1/3)(64)9=192

13

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: 2 3 y  x  4 and y   x  2 Therefore, the lines are perpendicular because their slopes are 3 2 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. 1 3. Graph the line on the coordinate plane: y   x  1 3

14

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

y = –2x +2

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

Many possible answers: y 

1 x  2 (any line with slope = ½) 2

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

y

1 x 3 3

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:

y=4

b. Write an equation for a line parallel to the line above that goes through the point (2,2) y=2 15

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.

YES: HL

c.

YES: AAS

NO: AAA

16

d.

e.

f.

YES: SSS

YES: ASA

g.

NO: SA

h.

YES: SSS

i.

YES: SAS

YES: ASA

3. ∆CGI  ∆MPR. a. Draw a picture of this situation.

G

P

I

R M

C

P

b. If mC = 27 and mG = 63, what is mR? 90° c. If PR = 6 cm, what do you know about ∆CGI ? GI = 6

O

Q R

XVII – Proof Directions: Fill in each of the missing steps (statements or reasons) in the proofs below. 1. Given: ⃗⃗⃗⃗⃗⃗ ; Prove: Statement ⃗⃗⃗⃗⃗⃗ 1. 3. POQ  ROQ

Reason 2. Definition of bisects

4.

3.GIVEN

5. OQ  OQ

4. Same Side (Reflexive property)

6.

5. AAS

2. GIVEN

17

2.

Given: Prove: ̅̅̅̅

; Q is the midpoint of ̅̅̅̅ ̅̅̅̅;

S

R Q P

Statement 1.

T

Reason 1. Given ̅̅̅̅

2. 3. SQ  QP 4. RQS  TQP

2. Given 3. Definition of midpoint

4. Vertical angles are congruent

5. ASA 5.

6.

̅̅̅̅

̅̅̅̅

6. CPCTC

3.

18

Directions: Find all of the mistakes in the proofs below. 4.

CORRECTED PROOF: CHECK CAREFULLY! Statement 1. ̅̅̅̅

Reason

̅̅̅̅;

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 ̅̅̅̅ 1. GIVEN 1. ̅̅̅̅ ̅̅̅̅

2. 3. ̅̅̅̅

̅̅̅̅

4. ̅̅̅̅

̅̅̅̅ ̅̅̅̅

Reason

2. Given 3. DEFINITION OF MIDPOINT

̅̅̅̅

4. Given

5.

5. Definition perpendicular lines

6.

6. All right angles are congruent

7.

7. SAS 19

6.

Statement 1. ̅̅̅̅

Reason

̅̅̅̅

2. ̅̅̅̅

1. Given ̅̅̅̅

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: M , L, K

ANSWER: A, B, C

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: 2 x  10  3x  20  70  180 That gives x = 20. The largest angle is angle C (80 degrees). Therefore the longest side is opposite. ANSWER: side AB. 20

XIX – Algebra disguised as geometry- Set up an equation, then find the missing variable. 1. AC = 44 2. HJ = 6x - 7 3x+4

C

H 2x

11 I

B

J

A

ANSWER: x = 8

3. ÐDEF =111

ANSWER: x = 5

4. ÐJKM =117 Find the measure of ÐLKM .

ANSWER: x = 8

5.

ANSWER: x = 5

12

ANSWER: x = 11

6.

ANSWER: x = 60

21

8. Suppose RQ bisects ÐPQS and ÐPQS = 98 .

7. B is the midpoint of AC. Find x. ANSWER: x = 3

ANSWER: x = 8; y = 15

16

3x+7 A

C B

9.

FOR THIS PROBLEM YOU NEED TO ASSUME THE LINES ARE PARALLEL OR THERE IS NO WAY TO DO IT. IF YOU MAKE THAT ASSUMPTION, AIA are congruent and: x=5

10.

SAME HERE: x = 8 (angles supplementary if lines are parallel)

22

XX -- Polygon Sum 1. Regular Octagon (8-gon) a. What is the sum of the interior angles?

8  2 180  1080 ° 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°

23

8. What is the sum of the interior angle measures of a polygon with… a. 15 sides

15  2 180  2340 o

b. 50 sides

50  2 180  8640o

9. What is the sum of the measures of the exterior angles of a decagon? ANSWER: 360°

10.

a = 108°

11.

1 b = 45 ° 3

12. Find the missing angles.

a = 67° b = 58° c = 125° d = 23° e = 90°

x = 38°, y = 36°, z = 90°

24

13. Find the missing angles.

ANSWER: (all measurements in degrees). CHALLENGE?

a  57; b  123; c  57; e  57; f  33; g  48; h  84; j  46; k  96; p  49; q  46; r  131; s  49

25

XXI – Right Triangle Trigonometry a) Write a true equation using sine, cosine, or tangent. b) Solve the equation. 1.

2.

6 cos72  x

3.

cos73 

x 6

tan24 

x ≈ 1.8

x ≈ 5.3

4.

5.

6.

x ≈ 19.4

x 12

x ; x ≈ 9.04 tan37   12

tan? 

9 9 ; tan 1    ? 21  21  ? = 23.2°

14  14  ; tan 1    ? 29  29  ? = 25.8°

7.

8.

9.

11  11  ; tan 1    ? 27  27  ? = 24.0° sin ? 

4 4 cos?  ; cos 1    ? 5 5 ? = 36.9°

tan? 

sin ? 

12  12  ; tan 1    ? 24  24 

? = 30°

26

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?

A

10 in

Opposite: AT; Adjacent: TM

28° T

M

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

sin 28 

10 AM

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

AM sin 28  10 AM 

10  21.3 sin 28

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

TM 2  21.32  10 2 = 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:

10 TM 10 TM   18.8 tan 28 tan 28 

27