Given: is a parallelogram.

Prove: ,

JKLM

JK LM KL MJ

Review: Semester 2 Final NAME __________________

Geometry

1. Find the SUM of the interior angle measures of a convex heptagon.

2. Find the measure of EACH interior angle of a regular 16-gon.

3. Find the measure of EACH interior angle of pentagon ABCDE.

4. Find the value of b in polygon FGHJKL.

5. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave

or convex.

6. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°.

Find CF, find m∠EFC, and find DF.

7. Use congruent triangles to prove the following:

_______________________________________________________________

J

K L

M

1

4

3

2

8. WXYZ is a parallelogram. Find YZ and m∠Z.

9. Use congruent triangles to prove the following:

GIVEN: ABCD is a parallelogram

PROVE: ∆AEB ≅ ∆CED

_________________________________________________________________

10. Show that JKLM is a parallelogram for a = 3 and b = 9.

11. Show that PQRS is a parallelogram for a = 2.4 and b = 9.

12. Show that JKLM is a parallelogram for a = 4 and b = 5.

13. Prove the following property:

Given: ABCD is a parallelogram, AB ≅ DA.

Prove: ABCD is a rhombus

______________________________________________________________________

14. Given that AB = BC = CD = DA, what additional information is needed to

conclude that ABCD is a square?

15. In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mBCD. Find mABC.

Find mFDA.

16. Find m∠F.

17. Find the value of a so that PQRS is isosceles.

18. m∠WZY = 61˚. Find m∠WXY. XV = 4.6 and WY = 14.2. Find VZ.

B

C D

A

19 - 20. Determine whether the polygons are similar. If so, write the similarity ratio and similarity statement.

19) 20)

21. A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model

to the nearest inch.

22. Explain why the triangles are similar and write a similarity statement.

23. Explain why the triangles are similar and write a similarity statement.

24. Explain why the triangles are similar and write a similarity statement.

25. Explain why the triangles are similar, then find BE and CD.

26. Verify that BE and CD are parallel.

45°

60°

27. Find the length of SR and RQ. 28. Newman wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?

29. The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:20 ft. 30. Find the geometric mean of 16 and 4. 31. Use the figure below, to find the values of x, y, AND z. 32. Use the figure below to complete the following statement:

“QS is the geometric mean of _________ and _________ .” 33. Find the following:

sin A cos B sin B

4 12

x y

z

R

S

P Q

34. Use special right triangles to write each trigonometric ratio as a simplified fraction. sin 30° cos 30° tan 45° 35. Solve for x. (round to the nearest hundredth) 36. Find the value of x AND y. (round to the nearest hundredth) 37. Find the value of x. (round to the nearest hundredth)

38. A tree 19 feet tall casts a shadow which forms an angle of 43 with the ground. How long is the shadow to the nearest hundredth? 39. A cameraman is standing on a tower above a racetrack. The angle of depression from the camera to the

track is 27. If the distance from the bottom of the tower to the track is 40 feet, how tall is the tower? 40. Find the measure of an acute angle that satisfies the given equation. Round answers to the nearest tenth of a degree.

6

11sin x

9

40tanY

60

3 x

y

x 10

43

19 ft

x

45° 3

41. Using the figure below, find the value of each missing angle. Round your answers to the nearest tenth of a degree. 42. Find the angle whose cosine is 0.866. 43. Classify each angle as an angle of elevation or an angle of depression. ∠1 = ∠2 = ∠3 = ∠4 = 44. Writing Sine in Cosine Terms and Cosine in Sine Terms

Write sin 52° in terms of the cosine.

Write cos 71° in terms of the sine.

45. An ice climber stands at the edge of a crevasse that is 115 ft wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52°. How deep is the crevasse at this point? Round to the nearest foot.

46. Find the two angles that satisfy the equation below.

sin (x + 5)° = cos (4x + 10)°

47. Find the two angles that satisfy the equation below.

sin(3x + 2)° = cos(x + 44)°

11

15

x

y

48. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot. 49. A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the transformation.

50. A figure has vertices at G(0, 0), H(-1, -2), I(-1.5, 0), and J(-2.5, 2). Find the coordinates for the image of GHIJ after the translation (x, y) (x - 2, y + 4).

G (0, 0) ( ) H(-1, -2) ( ) I (-1.5, 0) ( ) J (-2.5, 2) ( )

51. A figure has vertices at X(-1, 1), Y(-2, 3), and Z(0, 4). Draw the image of XYZ after the translation (x, y) (x +2, y) and a 180° rotation around the origin. X (-1, 1) ( ) ( ) Y (-2, 3) ( ) ( ) Z (0, 4) ( ) ( ) 52. Given points P(-2, -1) and Q(-1, 3), draw PQ and its reflection across the y-axis. P (-2, -1) Q (-1, 3)

x

y

A (-2, 3) ( )

B (1, 5 ) ( )

C (-3, -2) ( )

E (-3, -2) ( )

F (6, -4) ( )

G (-2, 1) ( )

57. What is the translation of the image (–6, 3) after the translation (x, y) (x – 3, y + 4) ?

(-6, 3) ( )

53. Tell whether the transformation appears to be a reflection.

54. Copy the figure and the line of reflection. Draw the reflection of the figure across the line.

For problems 55 and 56, reflect the figure with the given vertices across the given line.

55. A(–2, 3), B(1, 5), C(–3,-2); y = x

a) b) c)

a) b)

56. E(–3, –2), F(6, –4), G(–2, 1); x-axis

g

A

B

C

D

For questions 58 and 59, describe the translation using coordinate notation.

58. Every point moves right 3 units and down 4 units.

59. Every point moves left 8 units.

60. If figure WXYZ, W (2, 2), X (0, 4), Y (–1, 3), and Z (1, 1) were rotated 180, what are the NEW coordinates

of WXYZ ?

Graph WXYZ and its rotated image.

(2, 2) __________

(0, 4) __________

(–1, 3) __________

(1, 1) __________

61. AFTER the translation (a, b) (a + 5, b – 2), the coordinates of ABC are (0, 3), (2, 6) and (4, 1). What

were the original coordinates of the vertices of ABC ?

__________ (0, 3)

__________ (2, 6)

__________ (4, 1)

62. Graph the image of ABC after the given glide reflection:

Tranlation: (a, b) (a – 2, b + 3)

Reflection: over the x-axis.

Give the final coordinates:

A( ) A ( ) A ( )

B( ) B ( ) B ( )

C( ) C ( ) C ( )

x

y

A

B C

63. Graph MN with M (1, –2) and N (2, 3). Then graph its dilation with a scale factor of 1.5 centered at (0,0).

Give the new coordinates:

(1, –2) _________

(2, 3) _________

64. Draw the image of the given figure after a dilation of with scale factor -2 and centered at O .

(1, -2) ( )

(-3, -1) ( )

(-2, -4) ( )

65. Find the area of the following rhombus:

66. Find the area of the following isosceles trapezoid:

67. Find the area and/or circumference of the following circles:

Area 16

6 m

20 m 6 m

x

y

x

y

7m

55

10 ft

22 ft

h

68. Find the area of the regular pentagon.

69. Estimate the area of the irregular shape.

70. Find the area and perimeter the polygon with vertices A(-4, 1), B(2, 4), C(0, 4), and D(-2, -3).

71. A point is randomly chosen on AD . Find the probability of each event.

a. The point is on AC .

b. The point is not on AB .

c. The point is on AB or CD .

12

in

x

y

72. A skydiver jumps from an airplane and parachutes down to the 70 by 100 meter rectangular field shown.

What is the probability that he will miss all three targets?

73. Classify each figure. Name the vertices, edges, and bases.

a.

b.

74. Identify the three-dimensional figure from the given net.

a. b.

A B

D C

E F

G H

75. Find the volume of the figures.

9 cm