1

Geometry Level 2

Final Exam Review

2014- 2015

Due, with work, the day of your exam!!!!!!!!

In addition to reviewing all quizzes, tests, homework, and notes assigned throughout the second semester, students should always study by doing additional problems. The final exam is cumulative. The following packet consists of skill-based problems by chapter. In order to be successful on the Final Exam, students also need to know Geometry vocabulary, notation, theorems, and formulas. The following formulas will be given on the exam.

Distance Formula: d = √ Slope Formula: m=

Pythagorean Theorem: a2 + b2 = c2 Area of a Triangle: A =

In addition to this packet, the following pages in your book are additional problems that can be completed Chapter 5: p R5, #1-23

Chapter 6: p R6, #1-20

Chapter 7: p R7, #1-20

Chapter 8: p R8, #1-22

Unit 5- Properties of Triangles Unit 6- Quadrilaterals

Sec Topic New vocab, theorems

etc.

5-1 Bisectors of triangles

Perpendicular bisector, perpendicular bisector

theorem, converse of the perpendicular bisector

theorem, concurrent lines, point of concurrency,

circumcenter, circumcenter

theorem, angle bisector theorem, converse of the

angle bisector theorem, incenter, incenter theorem

5-2 Medians and

altitudes of triangles

Median, centroid, centroid

theorem, altitude, orthocenter,

Midsegment

5-3 Inequalities in one triangle

Definition of inequality, properties of inequality for

real numbers, exterior angle

inequality, angle side inequalities,

5-4 Indirect proof Indirect reasoning, indirect proof, proof by

contradiction,

5-5 Triangle inequality Triangle inequality theorem,

5-6 Inequality in two triangles

Hinge theorem, converse of the hinge theorem

Convers to pythag

Sec Topic New vocab, theorems etc.

6-1 Angles of polygons Diagonal, polygon interior

angles sum, polygon exterior angles sum,

6-2 Parallelograms Parallelogram, properties of

parallelogram, diagonals of parallelograms,

6-3 Test for

parallelograms

Conditions for

parallelograms,

6-4 Rectangles Rectangle, diagonals of

rectangle,

6-5 Rhombi and squares Rhombus, square, diagonals of rhombus, conditions for

rhombi and squares,

6-6 Trapezoid and kites Trapezoid, bases, legs of a trapezoid, base angles,

isosceles trapezoid, midsegment of a trapezoid,

kite, trapezoid midsegment

theorem,

2

Unit 7- Similarity

Unit 8- Right Triangles and Trigonometry

Sec Topic New vocab, theorems etc.

7-1 Ratios and

proportions

Ratio, extended ratios,

proportions, extremes, means, cross products,

cross product property,

equivalent proportions,

7-2 Similar polygons Similar polygons, scale

factor, perimeters of similar

polygons,

7-3 Similar triangles AA similarity, SSS similarity,

SAS similarity, reflexive, symmetric, transitive

properties of similarity,

7-4 Parallel lines and proportional parts

Midsegment of triangle, triangle midsegment

theorem, proportional parts

of parallel lines, congruent parts of parallel lines

7-5 Parts of similar triangles

Special segments of similar triangles, triangle angle

bisector

7-6 Similarity transformations

Dilation, similarity transformation, center of

dilation, scale factor of a

dilation, enlargement, reduction,

7-7 Scale drawings and models

Scale model, scale drawing, scale,

Sec Topic New vocab, theorems etc.

8-1 Geometric Mean Geometric mean, theorem 8.1 (altitude of right

triangles), right triangle

geometric mean theorems,

8-2 Pythagorean

Theorem and

Converse

Pythagorean triple,

converse to Pythagorean

theorem, Pythagorean inequality theorems,

8-3 Special right

Triangle

45-45-90 theorem, 30-60-

90 theorem

8-4 Trigonometry Trigonometry, trimetric

ratio, sine, cosine, tangent, inverse sine, inverse cosine,

inverse tangent,

8-5 Angle of elevation and depression

Angle of elevation, angle of depression,

3

CHAPTER 5- PROPERTIES OF TRIANGLES

Find the value of x. Then classify the triangle by its angles. 1) _______ 2) _______

3) _______

4) _______

5) _______

6) _______

7) _______

8) _______

9) _______ 10) _______

11) _______

12) _______

13) _______

14) _______

15) _______ 16) _______

Use Δ DEF, where J, K, and L are midpoints of the sides

1) If DE = 8x + 12 and KL = 10x – 9, what is DE?

2) If JL = 7x – 6 and EF = 9x + 8, what is EK?

3) If DF = 18x – 6 and JK = 3x + 15, what is JK?

4) Use you answer for JK. if JL=20, and EJ=17, find the perimeter of Δ DEF.

A triangle has the given vertices. A(-1,1), B(4,4), C(-1,4). 5) Graph and label the triangle.

6) State the slope formula.

7) Find the slope of the three sides,

, and AB BC AC

8) Classify the triangle by its sides. Explain your choice.

Find the unknown side length x.

9) 10)

Tell whether the given side lengths of a triangle can represent a right triangle.

11) 9, 15, and 3 34 12) 12, 14, and 18

A triangle has the given vertices. A(-1, 1), B(3, 1), C(3, 4).

13) Graph and label the triangle.

14) Explain, in detail, how you would ‘prove’

that ABC is a right triangle.

15) Please state any formulas used and use

correct notation.

16) Find the coordinates of the endpoints of the midsegment that is parallel to

BC .

4

Find each measure

17) _______

18) _______

19) _______ 20) _______ 21) _______ 22) _______ 23) _______

24) _______

25) _______

26) _______

27) _______

28) _______

29) _______

30) _______

31) _______

32) _______

17) XW

18) BF

Point P is the circumcenter of EMK. List any segment(s) congruent to each

segment below.

19) ̅̅̅̅̅

20) ̅̅ ̅̅

21) ̅̅ ̅̅ ̅̅

22) ̅̅ ̅̅

Point U is the incenter of GHY. Find each measure

23) MU

24) m∠UGM

25) m∠PHU

26) HU

In ABC, AU = 16, BU = 12, and CF = 18. Find each measure.

27) UD

28) EU

29) CU

30) AD

31) UF

32) BE

5

In CDE, U is the centroid, UK = 12, EM = 21, and UD = 9. Find each measure.

33) _______

34) _______

35) _______

36) _______

37) _______

38) _______ 39) _______ 40) _______

41) _______ 42) _______

43) _______

44) _______

45) _______ 46) _______

33) CU

34) MU

35) CK

36) JU

37) EU

38) JD

COORDINATE GEOMETRY Find the coordinates of the orthocenter of the triangle with the given vertices. 39) J(1, 0), H(6, 0), I(3, 6)

40) S(1, 0), T(4, 7), U(8, −3)

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

41) measures are less than m∠1

42) measures are greater than m∠1

43) measures are less than m∠7

44) measures are greater than m∠2

45) measures are greater than m∠5

46) measures are less than m∠4

6

List the angles and sides in order from smallest to largest.

47) _______

48) _______

49) _______ 50) _______

51) _______

52) _______

53) _______ 54) _______

47) 48)

Is it possible to form a triangle with the given side lengths? If not, explain why not.

49) 6, 9, 15 50) 4, 8, 16

Find the range for the measure of the third side of a triangle given the measures of two sides.

51) 1 cm and 6 cm 52) 1.5 ft and 5.5 ft

Write an inequality for the range of values of x.

53) 54)

7

Chapter 6: Special Quadrilaterals and Polygons

Give the most specific name for the quadrilateral

1) _________

2) _________

3) _________

4) _________

5) _________

6) _________

7) _________

8) _________

9) _________

10) ________

11) ________

12) ________

13) ________

1) 2)

3) 4)

Three of the vertices of ABCD are given. Find the coordinates of point D.

5) A(3, 6), B(6, 7), C(6, 3), D(x, y)

Find the value of x

6) 7)

8) 9)

10) 11)

12)

13)

8

14) ________

15) ________

16) ________

17) ________

18) ________

19) ________

20) ________

21) ________

22) ________

23) ________

24) ________

25) ________

26) ________

27) ________

28) ________

29) ________

30) ________

31) ________

32) ________

33) ________

34) ________

14) The measure of one interior angle of a parallelogram is 42 degrees more than twice the measure of another angle. Find the measure of each angle.

The diagonals of rhombus STUV intersect at W. Given that mUVT = 31.8°, TU = 20, and TW = 17, find the indicated measure.

15) mUVS

16) mTUV

17) SU

18) mTWU

19) UW

20) VT

Find the sum of the measures of the interior angles of the indicated convex polygon.

21) Decagon 22) Heptagon

23) 18-gon 24) 30-gon

The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides.

25) 1080° 26) 1800°

27) 2520° 28) 3960°

Find the value of x and y in each parallelogram.

29) 30)

31) 32)

33) 34)

9

For what value of x is the quadrilateral a parallelogram?

35) ________

36) ________

37) ________

38) ________

39) ________

40) ________

41) ________

42) ________

43) ________

44) ________

45) ________

46) ________

47) ________

48) ________

35)

Find the value of n for each regular n-gon described.

36) Each interior angle of the regular n-gon has a measure of 162°.

37) Each exterior angle of the regular n-gon has a measure of 5°.

38) The measure of one interior angle of a parallelogram is 30 degrees less than 9 times the measure of another angle. Find the measure of each angle.

39) What is the measure of one exterior angle for a regular 40-gon

Tell whether the statement is always, sometimes, or never true.

40) The diagonal of a polygon connects two adjacent vertices.

41) A quadrilateral is convex.

42) A hexagon is equiangular but not equilateral.

43) A pentagon is a plane figure.

44) A hexagon has six congruent sides. 45) A quadrilateral is equiangular but not

equilateral.

46) A triangle is concave. 47) The exterior angle sum for a convex

heptagon is 360°.

48) A line containing a convex polygon’s side shares interior points with the polygon.

10

Quadrilateral ABCD is a rectangle.

49) ________

50) ________

51) ________

52) ________

53) ________

54) ________

55) ________

56) ________

57) ________

58) ________

59) ________

60) ________

61) ________

62) ________

63) ________

64) ________

49) If AE = 36 and CE = 2x – 4, find x. 50) If BE = 6y + 2 and CE = 4y + 6,

find y.

51) If BC = 24 and AD = 5y – 1, find y. 52) If m∠BEA = 62, find m∠BAC.

53) If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.

54) If BD = 8y – 4 and AC = 7y + 3, find BD.

55) If m∠DBC = 10x and m∠ACB =

– 6, find m∠ ACB.

56) If AB = 6y and BC = 8y, find BD in terms of y.

Quadrilateral ABCD is a rhombus. Find each value or measure.

57) If m∠ABD = 60, find m∠BDC. 58) If AB = 26 and BD = 20, find AE.

59) Find m∠CEB. 60) If m∠CBD = 58, find m∠ACB.

61) If AE = 3x – 1 and AC = 16, find x. 62) If AE = 8, find AC.

63) If m∠CDB = 6y and m∠ACB = 2y +

10, find y.

64) If AD = 2x + 4 and CD = 4x – 4, find x.

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Chapter 7: Similarity

List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality.

1) _________ 2) _________ 3) _________ 4) _________ 5) _________

6) _________

7) _________

8) _________

9) _________

10) _______

1) ABC DFE

2) WXYZ ~ MNOP

Triangles ABC and DEF are similar. 3) Which statement is not correct?

,

Determine whether the polygons are similar. If they are, write a similarity staement and find the scale factor.

4) 5)

In the diagram, WXYZ MNOP. 6) Find the scale factor of WXYZ to MNOP.

7) Find the values of x, y, and z.

8) Find the perimeter of WXYZ. 9) Find the perimeter of MNOP.

10) Find the ratio of the perimeter of MNOP to the perimeter of WXYZ.

CA FD

AB DE

AB DE

BC EF

12

The two triangles are similar. Find the values of the variables. 11) _______

12) ________

13) ________

14) ________

15) ________ 16) ________

17) ________

18) ________

19) _______

20) ________

21) ________

22) ________

11) 12)

Triangles RST and WXY are similar. The side lengths of RST are 10

inches, 14 inches, and 20 inches, and the length of an altitude is 6.5 inches. The shortest side of WXY is 15 inches long. Sketch each of

the two triangles.

13) Find the lengths of the other two sides of WXY.

14) Find the length of the corresponding altitude in

WXY. 15) The ratio of one side of ∆ ABC to the

corresponding side of a similar DEF is

4:3. The perimeter of ∆DEF is 24 inches. What is the perimeter of ABC?

18 inches 24 inches 32 inches

In the diagram, ∆XYZ ∆MNP

16) Find the scale factor of XYZ to MNP.

17) Find the unknown side lengths of both triangles.

18) Find the length of the altitude shown

in XYZ.

19) Find and compare the areas of both triangles.

The community park has a rectangular swimming pool enclosed by a rectangular fence for sunbathing. The shape of the pool is similar to the shape of the fence. The pool is 30 feet wide. The fence is 50 feet wide and 100 feet long.

20) What is the scale factor of the pool to the fence?

21) Find the area reserved strictly for sunbathing.

22) What is the length of the pool?

13

Use the diagram to complete the statements. 23) ________

24) ________

25) ________

26) ________

27) ________

28) ________

29) ________ 30) ________ 31) ________ 32) ________ 33) ________ 34) ________

23) ABC ~ __?__

24)

?

? ?

AB CA

EF

25) B __?__

26)

? 8

12 ?

27) x = __?__

28) y = __?__

Determine whether the triangles are similar. If they are, write a similarity statement.

29)

30)

31)

32)

33)

34)

14

35) In the diagram at the right, find the

length of BC

35) ________ 36) ________ 37) ________ 38) ________ 39) ________ 40) ________

41) ________

42) ________

43) ________

44) ________

45) ________

36) List three pairs of congruent angles.

37) Name two pairs of similar triangles and write a similarity statement for each.

38) Is ACD ~ BCE?

39) ls AED EAB?

40) Triangles ABC and DEF are right

triangles that are similar. AB and BC

are the legs of the first triangle. DE and

EF are the legs of the second triangle. Which of the following is false?

A. A D

B. AC = DF

C. =

In order to estimate the height h of a flag pole, a 5 foot tall male student stands so that the tip of his shadow coincides with the tip of the flag pole’s shadow. This scenario results in two similar triangles as shown in the diagram. 41) Why are the two overlapping triangles

similar?

42) Using the similar triangles, write a proportion that models the situation.

43) What is the height h (in feet) of the

flag pole?

44) Is either ΔLMN or ΔRST similar to ΔABC?

45) Is either ΔLMN or ΔRST similar to ΔABC?

AC DF

AB DE

15

Show that the triangles are similar and write a similarity statement. Explain your reasoning.

46) ________ 47) ________ 48) ________ 49) ________

50) ________

51) ________

52) ________

53) ________ 54) ________

46) 47)

48) In the diagram at the right, ΔACE ~ ΔDCB. Find the length of AB.

Use the diagram at the right to copy and complete the statement

49) ΔABC ~ __?__

50) mDCE = __?__

51) AB = __?__

52) mCAB + mABC = __?_ In order to estimate the height h of a tall pine tree, a student places a mirror on the ground and stands where she can see the top of the tree, as shown. The student is 6 feet tall and stands 3 feet from the mirror which is 11 feet from the base of the tree.

53) What is the height h (in feet) of the pine tree?

54) Another student also wants to see the top of the tree. The other student is 5.5 feet tall. If the mirror is to remain 3 feet from the student's feet, how far from the base of the tree should the mirror be placed?

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Use the figure to complete the proportion 55) ________

56) ________

57) ________

58) ________

59) ________

60) ________

61) ________ 62) ________ 63) ________ 64) ________ 65) ________

66) __________

67) __________

68) ________

55)

56)

57)

58)

59)

60) Use the given information to determine whether ║

61)

62)

63)

64)

Determine the length of each segment

65)

66)

67)

68)

DB CF GC ?

BD ?

FC AF

? GD

FB CD

? GE

CD AE

AE ?

GE GD

? FB

AG FG

GB

BC CD FC

.

AE BD

17

69) ________ 70) ________ 71) ________ 72) ________ 73) ________ 74) ________ 75) ________ 76) ________ 77) ________ 78) ________ 79) ________

find the value of x.

69) 70)

71)

72) Find the value of the variable.

73) x 74) m

75) a

Determine whether the given information implies . Explain.

76) 77)

78) 79)

DE BC

18

Find the value of the variable. 80) ________ 81) ________ 82) _______

80) 81)

82)

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Chapter 8: Right Triangles 1) Which equation is not correct? Ⓐ t2 – r2 = s2 Ⓑ t2 + r2 = s2

Ⓒ s2 – t2 = –r2 Ⓓ t2 – s2 = r2

1) _______ 2) _______ 3) _______ 4) _______ 5) _______ 6) _______ 7) _______ 8) _______ 9) _______

2) A 25-foot ladder leans against a wall 7 feet from the base of the wall. How high up the wall does the ladder touch?

3) Find the area of the rectangle.

4) Classify ΔABC if the vertices are A(–12,

5), B(12, 5), and C(10, 17).

5) Find the approximate area of the triangle.

6) Find sin F and sin G.

7) Which equation could be used to find

the value of x in the diagram?

Ⓐ cos 55° =

12

x Ⓑ cos 35° = 12

x

Ⓒ cos 35° =

12

x Ⓓ cos 55° = 12

x

8) Which is not enough given information needed to solve a right triangle?

Ⓐ two acute angles and one side

length Ⓑ measure of the hypotenuse

Ⓒ two side lengths

Ⓓ one side length and the measure

of one acute angle

9) Find mA.

20

10) What is the value of x? Round your answer to the nearest tenth.

10) _______ 11) _______ 12) _______ 13) _______ 14) _______

15) _______

16) _______

17) _______

11) What is the value of x? Round your answer to the nearest tenth.

12) What is the value of y? Round your

answer to the nearest tenth.

13) A shipping dock has a mobile ramp

that is used to help load and unload cargo from trucks. The ramp is 125 inches long and has a base that is 120 inches long. What is the height h of the ramp?

Real Estate An investor owns a triangular plot of land as shown in the diagram. 14) Find the perimeter of the plot of land.

15) Find the area of the triangular plot. 16) One acre of land is equivalent to

43,560 square feet. 17) How many acres are in this plot of

land? Round to two decimal places. The investor is planning on selling the land. The market rate in this area is $5000 per acre. How much should the investor ask for the land?

21

Garden You have a garden that is in the shape of a right triangle with the dimensions shown.

18) _______ 19) _______ 20) _______ 21) _______ 22) _______ 23) _______ 24) _______

18) Find the perimeter of the garden. 19) You are going to plant a post every 15

inches around the garden's perimeter. How many posts do you need?

20) You plan to attach fencing to the posts

to enclose the garden.

21) If each post costs $1.25 and each foot

of fencing costs $.70, how much will it

cost to enclose the garden? Explain.

Maps The distances between three towns are given in the diagram.

22) Is the triangle (∆ABC) formed by the three towns a right triangle?

23) Town B is directly west of town C. Is town A directly north of town C?

24) Complete the table. Give exact answers.

X 5 2 9

Y 4 2 24

22

25) Complete the table. Give exact answers.

25) _______ 26) _______ 27) _______ 28) _______ 29) _______ 30) _______ 31) _______ 32) _______ 33) _______

a 9 11

b 9

5 3

c 16

You are replacing the roof on the house shown, and you want to know the total area of the roof. The roof has a 1-1 pitch on both sides, which means that it slopes upward at a rate of 1 vertical unit for each 1 horizontal unit. 26) Find the values of x and y in the

diagram.

27) Find the total area of the roof to the nearest square foot.

Find the value of each variable. Round decimals to the nearest tenth.

28)

29)

30) 31)

32) 33)

23

34) Highway You are traveling along a stretch of highway that has a slight grade with an angle of inclination of 5°. After traveling for 4 miles, what is the vertical v and horizontal h change in feet? (1 mi = 5280 ft) Round your answer to the nearest foot.

34) _______

35) _______ 36) _______ 37) _______

35) Skyscraper You are a block away from a skyscraper that is 780 feet tall. Your friend is between the skyscraper and yourself. The angle of elevation from your position to the top of the skyscraper is 42°. The angle of elevation from your friend’s position to the top of the skyscraper is 71°. To the nearest foot, how far are you from your friend?

36) Ladder You lean a 16 foot ladder against the wall. If the ladder makes an angle of 70° with the ground, how far away from the wall is the base of the ladder? Round your answer to the nearest tenth of a foot

37) Skyscraper You are standing 350 feet

away from a skyscraper that is 750 feet tall. What is the angle of elevation from you to the top of the building?