Unit 11 Objective 0 1. The point (3, 1) lies on a circle whose equation is (x + 1) 2 + (y + 2) 2 = r 2. Find the radius of the circle.

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Unit 11 Objective 0

1. The point (3, 1) lies on a circle whose equation is (x + 1)² + (y + 2)² = r². Find the radius of the circle.

2. Find the value of x in the parallelogram shown.

3. Find the length of AC and AB.

4. Find sin L.

5. Find the value of z.

6. Two sides of a triangle are 5 and 9. What is the maximum integer length that the third side could be?

7. ABC has the angle measures shown in the diagram. Which side of the triangle is the shortest?

8. Find the measure of R.

9. Which property below is demonstrated by the statement: If a = b and b = c, then a = c ? A. Reflexive Property

B. Symmetric Property C. Transitive Property D. Substitution Property

10. An angle is 4 times as large as its complement. Find both the angle and the complement.

11x–14 5x+2

22 C

B A

30°

N

L M

20

21 29

z°

98°

S

R P

42°

119°

C B

A

59°

60°

61°

(2)

Geometry Unit 11 1. Areas of Triangles and Parallelograms (Section 11.1) 2. Areas of Trapezoids, Rhombuses, and Kites (Section 11.2) 3. Perimeter and Area of Similar Figures (Section 11.3) 4. Circumference and Arc Length (Section 11.4)

5. Areas of Circles and Sectors (Section 11.5) 6. Areas of Regular Polygons (Section 11.6) Review

(3)

Unit 11 Worksheet 6

For problems 1 and 2 use this information: O is the center of a regular n-sided polygon with

consecutive vertices A and B.

1. If AOB has the given measure, find the value of n.

a. m AOB = 45°, n = _______ b. m AOB = 30°, n = _______

2. Find the measure of AOB for the given value of n.

a. n = 10, m AOB = _________ b. n = 15, m AOB = _________

For problems 3 – 5 find the apothem of each regular polygon.

3. Hexagon with radius 8.

4. Square with side 10.

5. Equilateral triangle with radius 4 3.

In problems 6 and 7 find the radius of each regular polygon.

6. Square with area 64.

7. Triangle with apothem 12 3.

In problems 8 and 9 find the perimeter of each regular polygon.

8. Triangle with radius 4 3. 9. Hexagon with radius 8.

In problems 10 – 19 find the area of each polygon described.

10. A square with perimeter 44.

11. A square with apothem 4.

12. A square with radius 6.

13. A regular pentagon with perimeter 60 and apothem x.

14. A regular 12-sided polygon with side s and apothem a.

15. A regular hexagon with sides 12.

16. A regular hexagon with radius 8.

17. An equilateral triangle with radius 6.

18. An equiangular triangle with perimeter 36.

19. An equilateral triangle with apothem 2.

(4)

Unit 11 Review

In problems 1 – 12, choose the correct multiple choice response.

NOTE: Diagrams are not drawn to scale.

1. Find the area of this figure.

A. 42 units² B. 144 units² C. 208 units² D. 272 units²

2. Find the area of this figure.

3. Find the area of the trapezoid below.

4. The figure below shows a circle inscribed in a square whose sides are 8 feet.

Find the area of the shaded region.

A. (64 – 8π) ft.² B. (64 – 4π) ft.² C. (64 – 16π) ft.² D. (64 – 64π) ft.²

5. A regular pentagon has sides of length 6 in. Another regular pentagon has sides of length 8 in. Find the ratio of the area of the smaller pentagon to the area of the larger pentagon.

A. 9: 16 B. 27: 64 C. 3: 4 D. 4: 3

6. The area of a square is 36 cm². Find the perimeter of the square.

A. 6 cm.

B. 12 cm.

C. 24 cm.

D. 72 cm.

(cont) 8

8 8

10 8

9 8 11

7

8 feet

(5)

Unit 11 Review (cont)

7. Find the length of AB in O if diameter BC = 20 in. and m AOC = 120°

A. ^5π

3 in

B. 40π

3 in

C. 20π

3 in

D. ^10π

3 in

8. Find the area of the figure shown.

9. Find the area of the shaded region.

A. 4π mm² B. 8π mm² C. 16π mm² D. 64π mm²

A. 7π units²

B. ^11π

2 units²

C. 99π

2 units²

D. 63π

2 units²

11. In O the measure of AB is ^16π

3 in. Find the length of the radius.

A. 6 in.

B. 12 in.

C. 24 in.

D. 36 in. (cont)

B A C

O

3 5

3

8 mm ^O

A

B

9

220°

A

C

O 80°

(6)

Unit 11 Review (cont)

12. Find the area of the kite shown.

Solve the following problems. Show all work.

13. Find the area of the figure below.

14. The area of the trapezoid shown is 30 cm². Find h.

15. Find the area of the parallelogram. Express answer in simplest radical form.

16. Find the area of the figure shown. Express answer in simplest radical form.

(cont)

6 in

30 in

10 in

17 in 21 in

6 cm

9 cm

5 cm h

45o

14

26

12 8

28

18

30°

45°

(7)

14

6

10

Unit 11 Review (cont) 17. In P, the area of the shaded region is 27π

2 units², find m MPN .

In problems 18 – 20, each shape shown is a regular polygon. Find the following:

A. apothem B. radius C. area of each polygon

18. 19. 20.

21. The figure shown has two tangent circles inscribed in a rectangle with width of 12 cm and length of 24cm. Find the area of the shaded region.

22. Find the area of the polygon with vertices A (–3, –3), B (–1, 5), C (3, 5), and D (1, –3).

6

N M

P

24 12

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Geometry Unit 12

1. Exploring polyhedron: Faces, vertices, edges. (Section 12.1) 2. Surface areas of prisms. (Section 12.2)

3. Surface areas of cylinders and prisms. (Section 12.3) 4. Volumes of prisms and cylinders. (Section 12.4) 5. Volumes of pyramids and cones. (Section 12.5) 6. Surface area and volume of spheres. (Section 12.6)

7. Similar figures: r: r²: r³ theorem for similar solids. (Section 12.7) Review

(9)

Unit 12 Worksheet 2A

Prisms: L.A.

(lateral area)

& T.A.

(total surface area)

[1-3]: Rectangular (based) Prisms.

1. 6, 4, 2;

& .

l w h

find LA TA 2. ⁶ ^, ³ ^, ⁵⁴ ²^;

& .

l cm w cm LA cm

find h TA 3. ^{9 ,} ^{8 ,} ³¹⁴ ²^;

& .

l ft w ft TA ft find h LA

[4-5]: Cubes.

4. edge 3m;

find TA. 5.

TA 600in ;2

find edge length.

[6-8]: Triangular (based) Prisms. Leave all answers in simplest radical form.

6. Isosceles based prism with sides 13, 13, 10, and prism height 7;

find LA & TA.

7. Equilateral based prism with sides of 8 and prism height 10;

find LA & TA.

8. based prism with sides 9, 12, 15 and prism height 10;

find LA & TA.

[9-11]: Other (based) Prisms.

9.

Isosceles trapezoidal based prism with bases 10 & 4, legs of 5,

& prism height 20;

find LA & TA.

10. R hom bus based prism with diagonals 6 & 8, and prism height 9; find LA & TA.

11. Re gular hexagonal based prism with sides of 8, and prism height 12;

find LA & TA.

l

w h

l

w h l

w h

(10)

Unit 12 Worksheet 2B

Prisms & Cylinders: L.A.

(lateral area)

& T.A.

(total surface area) Reminder: LA(prismorcylinder) (perimeterof base)(heightof prismorcylinder) ph

TA(prismorcylinder) LA 2B, whereB areaof thebase.

[1-5]: Cylinders. Leave answers in terms of .

1.radius 5, height 4;

find LA & TA. 2. diameter 6cm, height 8cm;

find LA & TA. 3. ^radius ^{7ft, LA} ^{70 ft ;}² find h & TA.

4.

height 8ft, LA 96 ft ;2

find r & TA. 5.

circumference 13 m, TA 188.5 m ;2

find r, h, & LA.

[6-10]: Prisms - review. Leave all answers in simplest radical form.

6. ^Isosceles based prism with sides 17, 17, 16, and prism height 12;

find LA & TA.

7. Equilateral based prism with sides of 12 and prism height 15;

find LA & TA.

8. based prism with sides 7, 24, 25 and prism height 30;

find LA & TA.

9. ^{r hom bus} based prism with diagonals 10 & 24, and prism height 20; find LA & TA.

10. ^hexagonal based prism with sides of 14, and prism height 17;

find LA & TA.

(11)

Unit 12 Worksheet 3

Pyramids & Cones: L.A.

(lateral area)

& T.A.

(total surface area) Reminder

pl cone

or pyramid of

height slant

base of perimeter cone

or pyramid

LA 2

) 1 )(

2( ) 1 (

TA(pyramidorcone) LA B, whereB areaof thebase.

[1-5]: Pyramids. Leave answers in simplest radical form and/or in terms of when appropriate.

1. Square-based pyramid 2. Square-based pyramid 3. Square-based pyramid

base edge 20, slant height 15;

find LA & TA.

base edge 6, pyramid height 4;

find LA & TA.

base edge 16, lateral edge 17;

find LA & TA.

4. Square-based pyramid with 5. Square-based pyramid with

pyramid height 24, slant height 25;

find LA & TA.

lateral edge 15, slant height 12;

find LA & TA.

[6-10]: Cones. Leave answers in simplest radical form and/or in terms of when appropriate.

6.

A right Cone with radius 10, and slant height 19;

find LA & TA.

7.

A right Cone with diameter 14, and cone height 24;

find LA & TA.

8.

A right Cone with diameter 16, and cone height 8 3;

find LA & TA.

9.

A right Cone with

circumference of base 12 , and cone height 8;

find LA & TA

10.

A right Cone with LA 136 , and slant height 17;

find radius & TA.

(12)

Unit 12 Worksheet 4 Volume of Prisms & Cylinders

Bh cylinder or

prism of

height base

the of area cylinder

or prism any of

Volume ( )( )

Reminder: LA(prismorcylinder) (perimeterof base)(heightof prismorcylinder) ph .

, 2 )

(prismorcylinder LA B whereB areaof thebase TA

Solve for the following. Show work! Leave answers in terms of and/or in simplest radical form when appropriate.

1. Rectangular solid with 2. Rectangular solid with

8 , 5 , 2 ;

&

l m w m h m

find V TA 10 , 4 , 3 ;

&

l ft w ft h ft

find V TA

3. ^Isosceles based prism with sides 15cm, 15cm, 18cm, and prism height 25cm;

find V & TA.

4. based prism with sides 6yd., 8yd., 10yd. and prism height 14yd.;

find V & TA.

5. Equilateral based prism with base edges of 10m and prism height 20m;

find V & TA.

6. Cube with edge 3m;

find V. 7.

Cube with TA 150ft ;2

find V.

8. ^{R hom bus} based prism with diagonals 16cm & 30cm, and prism height 15cm;

find V & TA.

9. Re gular hexagonal based prism with base edges of 8m, and prism height 15m;

find V & TA.

l

w l

w

(13)

Unit 12 Worksheet 5 Volume of Pyramids & Cones

Bh cone

or pyramid of

height base

the of area cone

or pyramid any

for Volume

3 ) 1 )(

3( 1

Reminder: LA(prismorcylinder) (perimeterof base)(heightof prismorcylinder) ph TA(prismorcylinder) LA 2B, whereB areaof thebase.

LA pyramid orcone perimeter of base slant heightof pyramid orcone pl 2 ) 1 )(

2( ) 1 (

. ,

)

(pyramidorcone LA B whereB areaof thebase TA

Bh cylinder or

prism of

height base

the of area cylinder

or prism

Volume( ) ( )( )

1. Right square-based pyramid with 2. Right square-based pyramid with ^height 8 & slant height 17;

find LA, TA, and V. ^{base edge} 6 & height 4;

find LA, TA, and V.

3. Right square-based pyramid with 4. Right regular hexagonal-based pyramid with

base edge 16 & slant height 10;

find LA, TA, and V. ^{base edge} 8 & height 12;

find V.

5. Right regular triangle-based pyramid with 6. A right square-based pyramid has base edge = 24 & height = 12; base area 16 cm² & volume 32 cm³ Find V & slant height. Find its height & TA

7. Right square-based pyramid with lateral edge 25 & base perimeter 120;

find LA, TA, and V.

(cont)

(14)

Unit 12 Worksheet 5 (cont)

8.

A right Cone with radius 9, and height 12;

find LA, TA, & V.

9.

A right Cone with diameter 12, and height 6 3;

find LA, TA, & V.

10.

A right Cone with radius 15, and volume 600 ;

find height, LA & TA.

11.

A right Cone with radius 21, and LA 609 ;

find TA & V.

12.

A right Cone with volume 432 , and height 9;

find slant height, LA, & TA.

13.

Isosceles based prism with sides 20m, 20m, 24m, and prism height 40m;

find LA, TA, & V.

(15)

Unit 12 Worksheet 6 Sphere: Volume & Surface Area

3

3 ) 4

(sphere r

V & S.A.(sphere) 4 r²

[1-5]: Find the volume and the surface area for the following spheres.

1. radius 6m 2. radius 19feet 3. diameter 22 inches

4. radius 15 yards 5. diameter 42m

[6-7]: Find the radius and the volume of the spheres with the given surface area.

6. S.A. 324 cm² 7. S.A. 4 ft²

[8-9]: Find the radius and the surface area of the spheres with the given volume.

8. V 2304 yd³ 9. V 36 in³

10. Find the volume for the figure (cylinder w/ hemisphere on top) to the right.

18 feet

(16)

Geometry Spring Practice Final Select the correct multiple choice response.

Note: Diagrams are not drawn to scale.

1. If the length of one leg of a right triangle is 4 and the length of the hypotenuse is 12, what is the length of the other leg?

A. 8 2

B. 4 10

C. 8

D. 4 2

2. Determine the length of x.

A. 73

B. 55

C. 5

D. 3 3

3. Which classification applies to a triangle with sides of lengths 16, 30, and 34?

A. right B. obtuse C. acute

D. not a possible triangle

4. Which triangle formed by the side lengths given is a right triangle?

A. 6, 8, 9 B. , 4, 6 C. 3, 3,

D. 2, 4, 5

5. Determine the length of x .

A. ⁹

5

B. 25

3

C. 4

D. 15

6. What is the value of x?

A. ¹⁵

2

B. ⁹

2

C. 3 2

D. 4 3

(cont)

x 6

8 x

3 5

8 x 3

(17)

Geometry Spring Practice Final (cont) 7. Determine the values of x and y.

A. x = 5 y = 5 3

B. x = 5 y = 5

C. x =10 y =10 D. x = 5 y = 5 2 8. Determine the values for x and y.

A. x = 12 y = 6 B. x = 12 3 y = 6 C. x = 6 y = 12 D. x = 8 3 y = 8 9. What is cos B?

A. ⁸

15

B. ¹⁵

8

C. ⁸

17

D. ¹⁵

17

10. What equation could you use to find x?

A. sin 20° = 50

x

B. cos 70° = 100

x

C. tan 20° = 50

x

D. cos 20° = 50

x

11. Which equation could you use to find the measure of angle A?

A. sin A = B. cos A = C. tan A =

D. sin A = (cont)

17

C

8

15

B

A

20°

x 50

y 60°

30°

x

6 3

12

7 C

B A

10 _y

x

(18)

Geometry Spring Practice Final (cont) 12. Which equation could be used to determine the value of y?

A. sin y = ¹⁰ 8

B. cos y = 8 10 C. tan y = ⁶

8

D. tan y = ⁸ 6

13. What is the value of x in the pentagon shown?

A. 58°

B. 77°

C. 139°

D. 122°

14. Determine the measure of the sum of the interior angles of a polygon with 13 sides.

A. 1980°

B. 2700°

C. 360°

D. 2340°

15. Determine the measure of one interior angle in a regular hexagon?

A. 45°

B. 60°

C. 180°

D. 120°

16. Determine the measure of the sum of the exterior angles of a polygon with octagon.

A. 1440°

B. 900°

C. 1080°

D. 360°

17. Determine the measure of one exterior angle of a regular hexagon.

A. 108°

B. 360°

C. 60°

D. 150°

18. Solve for x.

A. 156°

B. 68°

C. 102°

D. 90°

(cont)

86°

70°

x°

2x°

10

8 6

y°

103°

x°

122°

87°

89°

(19)

7x - 6

3x + 2

B C

A D

x + 8 2x + 4

Geometry Spring Practice Final (cont)

19. The points (1, -1), (4, -1) and (0, -4) are plotted below. Which of the following points would complete the parallelogram?

A. (5, 2) B. (2, -4) C. (-4, -4) D. (5, -4)

20. For what value of x is the quadrilateral a parallelogram?

A. 4

B. 56

C. 26

D. 11

21. ABCD is a parallelogram. What is the measure of x.

A. 110°

B. 37°

C. 70°

D. 33°

22. Solve for 1 and 2 in the rhombus to the right.

A. 1 = 34 & 2 = 90°

B. 1 = 56 & 2 = 90°

C. 1 = 90 & 2 = 90°

D. 1 = 56 & 2 = 34°

23. If ABCD is a rhombus, calculate AB

A. 8

B. 2

C. 6

D. 14

24. ABCD is a rectangle, determine the measure of x.

A. 90°

B. 124°

C. 62°

D. 28°

(cont)

28°

°

A B

D C

x

110° x°

37°

D B A

C

O

N

M

L 34° 2

1

(20)

A D B C

(x + 11)°

(2x + 1)°

(3x – 3)°

25. Determine measure of A in the following trapezoid.

A. 90°

B. 87°

C. 86°

D. 60°

26. Find the length of MN of the trapezoid

A. 22

B. 11

C. 6

D. 3

27. Given trapezoid ABCD. Determine the length of DC.

A. 6

B. 12

C. 14

D. 10

28. A segment whose endpoints lie on a circle is defined to be a _______, and it is a portion of a _________.

A. chord, secant B. diameter, radius C. chord, diameter D. secant, chord

29. In circle P, which of the following segments are guaranteed to be congruent.

I. BP and PE II. BC and DE III. AB and AF IV. BC and CD A. II and IV B. III and IV C. I, II, and III

D. II, III, and IV 30. Determine m G given circle O.

A. 50 B. 40 C. 30 D. 20

31. Determine m 2 given circle O.

A. 15°

B. 65°

C. 70°

D. 95°

(cont)

F

D C

A 2x – 4 B E 10

x – 3

G

A B

C

D 70 80

O

O 110

100

2 8

M N

14

C B

D

E

A

P

90

90 F

(21)

32. In circle P, if QS = 20, US = 4, and QS TR, determine the measure of TR.

A. 16

B. 12

C. 10

D. 8

33. Determine the length of arc AB in circle P.

A. 8

B. 12 C. 16 D. 48

34. Determine the value of x in circle P.

A. 96°

B. 90°

C. 48°

D. 24°

35. Determine m 1 in circle M.

A. 140 B. 80 C. 40 D. 20

36. is the ratio of the _______of a circle to its ______.

A. circumference, area B. circumference, chord C. circumference, diameter D. circumference, radius 37. Determine m 1 given circle O.

A. 60 B. 65 C. 90 D. 130

38. Determine the measure of EF if AF = 10, AD = 13, and CF = 5.

A. 6

B. 11

C. 15

D. 26

(cont)

T R

S Q

P U

1 40

M

O A

B C

D E 100

30°

80 1

O C A

F D E A 120° 1

2

B P

A

B 12

120°

48 P x° 48°

(22)

Geometry Spring Practice Final (cont) 39. Determine the measure of GD if GC = 12, GB = 4, and GA = 3.

A. 9

B. 112 3 C. 142

3

D. 16

40. Find the arc length of DE in  O with radius of 9 in.

A. ^207π

4 in

B. ^117π

4 in

C. ^13π

2 in

D. ^23π

2 in

A. ^16π

3 units² B. 16π units²

C. 4π

3 units²

D. ^2π

3 units²

42. Circle P has a radius of 2 in. If the area of the shaded region is ^14π

9 in², determine m APE .

A. 100°

B. 120°

C. 140°

D. 160°

43. A rhombus has diagonals of 16 m & 30 m. Determine the perimeter of the rhombus.

A. 92m

B. 76m

C. 68m

D. 46m

(cont)

A ^C

D B

O A

G

E C D

O 130°

B 30°

A

O 8

P A

E

(23)

44. Calculate the area of a rhombus with perimeter 40 feet and one diagonal with length 12 feet.

A. 480 ft² B. 192 ft² C. 100 ft² D. 96 ft²

45. Calculate the area of an equilateral triangle with side length 24 cm.

A. 576 cm² B. 288 cm² C. 144 3cm² D. 72 3cm²

46. What is the perimeter of an equilateral triangle with radius 8 3 in. ?

A. 24 in.

B. 72 in.

C. 12 in.

D. 48 in.

47. Determine the area of the trapezoid below.

A. 135 units² B. 30 units² C. 108 units² D. 37.5 units²

48. A square is inscribed in a circle with radius 5. What is the area of the square?

A. 25

B. 10 C. 50 2 D. 50

49. Determine the area of a regular hexagon with perimeter 72.

A. 216 B. 432 3 C. 432 D. 216 3

(cont)

5

8 3

5

9 6

(24)

12 ft 16 ft

20 ft

15 ft Geometry Spring Practice Final (cont)

50. If the area of triangle ABC is 56 m² and the base of the triangle is 7 m, find the height.

A. 16m

B. 8m

C. 4 m

D. 9 m

51. If the circumference of a circle is 24 m, then find the area of the circle.

A. 144 m² B. 576 m²

C. 24 m²

D. 12 m²

52. If the ratio of the lengths of the corresponding sides of two similar polygons is 4:9, then determine the ratio of their areas.

A. 2 : 3 B. 4:9 C. 8 :18 D. 16 :81

53. If Figures I & II are similar and the area of Figure I is 35 cm², find the area of Figure II.

A. 70 cm² B. 140 cm² C. 36 cm² D. 35 cm²

54. If RSTW is a rhombus, what is the area of WXT ? A. 18 3 units²

B. ^{36 3} units

2

C. 36 units² D. 48 units²

55. Calculate the total surface area for the

right triangular-based prism shown to the right.

A. 960 ft² B. 912 ft² C. 1440 ft² D. 720 ft²

I

3 cm

II

6 cm

~

R W

S T

X 60°

12

(25)

15 m

16 m Geometry Spring Practice Final (cont)

56. Calculate the total surface area for the

right square-based pyramid shown to the right.

A. 800 m² B. 1280 m² C. 736 m² D. 1344 m²

57. Calculate the total surface area for a right cylinder with diameter 8 yards and height 6 yards.

A. 96 yd² B. 112 yd² C. 64 yd²

D. 80 yd²

58. Calculate the total surface area for a sphere with diameter 30 meters.

A. 900 m² B. 4500 m² C. 3600 m²

D. 300 m²

59. Calculate the total surface area for the right triangular-based prism with base edges 15 in., 15 in., & 18 in., and prism height is 30 in.

A. 3240 in² B. 1440 in² C. 1656 in² D. 1710 in²

60. Calculate the lateral surface area for the right cone with base circumference 10 meters and cone height 12meters. A. 100 m²

B. 130 m²

C. 90 m²

D. 65 m²

61. Determine the volume for the right regular hexagonal prism with base edge 14 feet and prism height 20 feet.

A) 5880 3 10184.459 ft ³ B) 11760 3 20368.918 ft ³ C) 294 3 509.223 ft³ D) 1680 588 3 2698.446 ft ³

(cont)

(26)

15

9 20

24 25

Geometry Spring Practice Final (cont) 62. Determine the volume for the right triangular-based prism

shown to the right.

A. 1350

B. 1080

C. 828

D. 2160

63. Determine the volume for the right cone with slant height 18 and radius for the base 9.

A. 729 3 1262.665 B. 27 3 46.765

C. 243 3 420.888 D. 54 3 93.531

64. Determine the volume for the right square-based pyramid with slant height 25 and pyramid height 24.

A. 1596

B. 1568 C. 4704 D. 896

65. The perimeters of the bases for two similar solids are 20yards and 32yards respectively. Determine the ratio of the volumes for these two similar solids.

A) 25:64 B) 5:8

C) 125:512 D) None of these

66. The volumes for two similar solids are 192 5m and ³ 81 5 m³ respectively.

Determine the ratio of the areas for these two similar solids.

A) 64:27 B) 4:3 C) 192:81 D) 16:9

67. The base areas for two similar solids are 9 ft² and 25 ft².

Determine the volume of the larger solid if the volume of the smaller is 36 ft³. A. 166.667 ³

3

500 ft

B. 60 ft³ C. 100 ft³

D. 133.333 ³ 3

400 ft

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1.0 Identify and give examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Rule

Deductive Reasoning:

Conclusions are based on accepted, true statements like definitions, postulates, and theorems.

It is logically valid.

Two-column proofs use deductive reasoning.

Inductive Reasoning

: Conclusions are based on several past observations.

Conclusions are probably true, but not necessarily true. They tend to follow patterns.

Example of Inductive Reasoning

If all the people you‟ve ever met from a particular town have been very strange, you might then say “all the residents of this town are strange”. This is inductive reasoning. It is not logically valid. Just because all the people you happen to have met were strange is no guarantee that all the people there are strange.

Homework

For each of the following state whether the approach represents deductive reasoning or inductive reasoning.

1. Conclusions are based on corollaries.

2. Conclusions are based on predictions from previous patterns.

3. Conclusions are based on observing physical objects.

4. Conclusions are based on definitions.

5. Conclusions are based on logical, accepted statements.

6. Conclusions are based on very strong suspicions.

7. Conclusions are based on constructing rigorous logical arguments using proven facts and accepted statements.

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2.0 Be able to write geometric proofs, including proofs by contradiction.

Notes:

When trying to prove a statement is true, it may be beneficial to ask yourself, “What if this statement was not true?” and examine what happens. This is called Proof by Contradiction or Indirect Proof.

In an indirect proof we start by assuming that what we are trying to prove true is actually false.

We then continue the proof until we reach a contradiction (something we know must be true).

This tells us that our original assumption was false and therefore what we‟re trying to prove must be true.

Rule

1. State your „Givens‟ in Step 1.

2. In step 2, assume temporarily that the conclusion (the ‘Prove’ part) is not true.

3. Work the proof like a normal proof until you reach a contradiction

4. Point out that since you got a contradiction to step 2, what you assumed in step 2 must be false. Since it‟s false, it must then be true.

Example 1

Given: Sue walks home from work and enters her house with a dry umbrella.

Prove: It‟s not raining.

Solution:

Assume temporarily that it is raining. Then Sue‟s umbrella would be wet. But it was given that the umbrella was dry and this contradicts that. Therefore our assumption of it raining must be false. So, it is not raining.

Example 2

Given: In ABC AB = 7, BC = 7 , AC = 7 Prove: ABC is equilateral

Solution:

Statements Reasons

1. AB = 7, BC = 7 , AC = 7 1. Given 2. Assume ABC is not equilateral 2. Assumption (Remember, this is the opposite of

the PROVE)

3. AB BC AC 3. Definition equilateral

4. But, AB = 7, BC = 7 , AC = 7 4. Re-stating the Given and and since 7 = 7 = 7, then substitution

AB = BC = AC

5. Line 4 contradicts line 3, so what we 5. Logic of Indirect Proof assumed in Line 2 must be incorrect,

therefore ABC is equilateral (cont)

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2.0 Be able to write geometric proofs, including proofs by contradiction. (cont)

Example 3

Given: XYZ is a right triangle Prove: m Y < 90º

Solution:

Statements Reasons

1. XYZ is a right triangle 1. Given

2. Assume m Y 90º 2. Assumption

(Remember, this is the opposite of the PROVE)

3. m X = 90º 3. Defn. right angle

4. m X + m Y + m Z = 180º 4. Sum of angles in = 180º 5. 90º + (90º or greater) + m Z > 180º 5. Substitution from lines 2 and 3 6. Line 5 contradicts Line 4, so what we 6. Logic of Indirect Proof

assumed in Line 2 must be incorrect, therefore m Y < 90º

Homework:

1. Select the correct multiple choice response for the following statement:

After stating the given, the next step in an indirect proof is ___________?

a. Assuming the given is false b. Creating a table

c. Assuming what you are trying to prove is true d. Assuming what you are trying to prove is false

Write the sentence you are going to try to contradict in an indirect proof for each of the following.

2. Given: m A m B m C Prove: ABC is equilateral

3. Given: AX = 5, BX = 5, X is between AB Prove: X is the midpoint of AB

4. Given: a = b Prove: a – b = 0

5. Use this diagram and given information

If you assume m 1 m 4 then p n. This contradicts p  n.

What conclusion can you make? (Select one of the multiple choices)

For this problem, remember that what you conclude is always the opposite of what you assume.

a. p ║ n b. p n c. ^m ¹ ^m ⁴ d. m 1 m 4

(cont)

Z

X Y

p

n ³

2

4 1

p ║ n

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2.0 Be able to write geometric proofs, including proofs by contradiction. (cont) 6. Fill in the missing steps in an indirect proof for the following:

Given: LMN is an isosceles triangle, L M Prove: L is not a right angle

Statements Reasons

1. LMN is an isosceles triangle, L M 1. Given 2. _________________________________ 2. Assumption

3. m L = 90º 3. Defn. right angle

4. since L M , then m M = _____ 4. Substitution using line 3

5. m L m M m N = _____ 5. Sum of angles in = 180º

6. 90º + 90º + m N = _____ 6. Substitution into line 5 7. 180º + m N = _____ 7. Addition

8. m N = _____ 8. Subtraction Prop. Of Equality 9. If m N = 0º, then LMN is not a 9. _________________

triangle, and this contradicts the given, so what we assumed in Line 2 must be incorrect, therefore L is not a right angle

7. AY bisects XYZ

If ^m ¹ ^m ², then AY is not the bisector of XYZ This contradicts the given that AY bisects XYZ What conclusion can you make?

(For this problem, remember that what you conclude is always the opposite of what you assume. The word if can be replaced with the word assume.)

Select one of the multiple choices.

a. m 1 m 2 b. m 1 m 2

c. AY bisects XYZ

d. AY does not bisect XYZ

8. In ∆LMN, m N > m M

If we assume that LM LN , it follows that either LM = LN or LM < LN.

If LM = LN then m N = m M If LM < LN then m N < m M

These both contradict the given statement that m N > m M .

What conclusion can be drawn from this contradiction? Select one of the multiple choices.

a. m N = m M b. LM > LN c. LM LN d. LM = LN

L M

N

Z A X

Y

1 2

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3.0 Construct and judge the validity of a logical argument and give counterexamples to disprove a statement.

Rule:

A counterexample is an if-then statement for which the hypothesis is true and the conclusion is false.

Example:

Given the statement: If a number is divisible by 4, then it is divisible by 6.

Provide a counterexample for this statement.

Solution:

A counterexample would be the number 16 . 16 is divisible by 4, but is not divisible by 6.

Homework:

1. Which multiple choice is a counterexample to the statement:

If point K is on AB

then K is on BA

a. b.

A K B A B K 2. Which multiple choice is a counterexample to the statement:

If a four-sided figure has four right angles, then it has four congruent sides.

a. a square b. a rectangle

The midpoint of a segment is the point that divides the segment into two congruent segments.

a. b. c.

If m AOB = 75º and m BOC = 15 º, then m AOC = 90º

a. b.

An angle is an obtuse angle if its measure is greater than 90º.

a. 141º b. 180º c. 100º d. 91º

W

X Y

Z T

A

B

O C

A

C

O B

Z X Y

X

Y

Z

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4.0 Prove basic theorems involving congruence and similarity.

Definition:

Congruent figures have the same size and shape.

Similar figures have the same shape.

Rule:

To prove Congruency in Triangles Show SSS, SAS, ASA, AAS, or HL To prove Similarity in Triangles

Show angles are congruent and sides proportional using AA, SAS, SSS

Example:

Two triangles, ∆XYZ and ∆LMN have side lengths such that ⁵ 8 XY

LM and

XY YZ

LM MN

If ⁵

8 XZ

LM , then ∆XYZ ~ ∆LMN

Which multiple choice property justifies the similarity?

a. AA b. SSS c. SAS Solution: Draw 2 triangles.

Since ⁵

8 XY

LM we can mark the sides XY and LM of the triangles.

(We will use 5 and 8, but they might really be doubled like 10 and 16 or tripled like 15 and 24, etc.)

Since XY YZ

LM MN

we can mark the sides YZ and MN. We will use 5m and 8m. (This just means „a multiple of 5‟ and „a multiple of 8‟

5 8 XZ

LM means we can mark sides XZ and LM with 5p and 8p

Notice that the 3 sides of each triangle are marked therefore the property that is used for Similarity is SSS which is choice b

(cont)

X Y

Z

L M

N

X Y

Z

5 L M

N 8

Z X ₅ Y

5m

5p L M

N

8 m

8m 8p

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4.0 Prove basic theorems involving congruence and similarity. (cont)

Homework:

1. Two triangles, ∆ABC and ∆GFH have side lengths such that BC

FH = AC GH If C H, then ∆ABC  ∆GFH.

a. AA b. SSS c. SAS

2. In the diagram A E , then ∆ABC  ∆EDC Which multiple choice property justifies the similarity?

a. AA b. SSS c. SAS

3. In the diagram if BC

CE = AC

CD, then ∆BCA ~ ∆ECD.

a. AA b. SSS c. SAS

4. Using the diagram, which of the following facts would be sufficient to prove that ∆ABC and ∆ADE are similar?

a. AB and AC are congruent b. m B m ADE

c. AD and AE are congruent d. B and C are congruent

B A

C

D

E

B

E A

C D

B A

C

D

E

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5.0 Prove triangles are congruent or similar and be able to use the concept of corresponding parts of congruent triangles.

Definition

If two triangles are congruent then their vertices can be matched up so that the

corresponding angles and sides of the triangles are congruent. This is known as CPCT.

Rule

To prove Congruency in Triangles Show SSS, SAS, ASA, AAS, or HL .

Example

Given: AB CD, D is midpoint of AB Prove: AC BC

Solution:

Statement Reason

1. AB CD, D is midpoint of AB 1. Given

2. AD _____ 2. Defn. midpoint

3. ADC and BDC are right angles 3. Defn.

4. ADC BDC 4. All right angles are

5. CD CD 5. ___________________

6. ∆ADC ∆BDC 6. S A S

7. AC BC 7. CPCT

Homework

1. According to the ASA Postulate If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent.

Which example below is not using this postulate correctly?

a. b.

c. d.

(cont)

C

A D B

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5.0 Prove triangles are congruent or similar and be able to use the concept of corresponding parts of congruent triangles. (cont)

Homework

2. According to the SSS Postulate If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent.

Which is a counterexample?

a. b.

c. d.

.

3. Supply the missing reasons for the proof below.

Given: WZ bisects XZY , XZ YZ Prove: XW YW

Statements Reasons

1. WZ bisects XZY , XZ YZ 1._______________________

2. 1 2 2._______________________

3. WZ WZ 3._______________________

4. XWZ YWZ 4._______________________

5. XW YW 5._______________________

1 Z

X W Y

2

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6.0 Be able to use the triangle inequality theorem.

Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example 1

In the given triangle which multiple choice is possible for the length of side „x‟ ?

a. 4

b. 5

c. 7

d. 31

Solution: Try each multiple choice.

Add the 2 smaller sides and their sum must be greater than the third side.

a. The 3 sides would be 4, 11, and 16. Since 4 + 11 16, choice „a‟ doesn‟t work b. The 3 sides would be 5, 11, and 16. Since 5 + 11 16, choice „b‟ doesn‟t work c. The 3 sides would be 7, 11, and 16. Since 7 + 11 > 16, choice „c‟ does work d. The 3 sides would 11, 16, and 31. Since 11 + 16 31, choice „d‟ doesn‟t work

Example 2

In the given triangle which multiple choice is the smallest possible value for x ?

a. 2

b. 1

c. 4

d. 3

a. The 3 sides would be 2, 5, and 7. Since 2 + 5 7, choice „a‟ doesn‟t work b. The 3 sides would be 1, 5, and 7. Since 1 + 5 7, choice „b‟ doesn‟t work c. The 3 sides would be 4, 5, and 7. Since 4 + 5 > 7, choice „c‟ does work d. The 3 sides would 3, 5, and 7. Since 3 + 5 > 7, choice „d‟ does work Both choice „c‟ of „4‟ and choice „d‟ of „3‟ work. The question asks for the smallest value.

Therefore, you would choose choice „d‟.

(cont)

16 11

x

7 x

5

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6.0 Be able to use the triangle inequality theorem. (cont)

Example 3

In the given triangle which multiple choice is the smallest possible value for x ? a. 8

b. 9 c. 10 d. 31

a. The 3 sides would be 8, 8, and 18. Since 8 + 8 18, choice „a‟ doesn‟t work b. The 3 sides would be 9, 9, and 18. Since 9 + 9 18, choice „b‟ doesn‟t work c. The 3 sides would be 10, 10, and 18. Since 10 + 10 > 18, choice „c‟ does work d. The 3 sides would 18, 31, and 31. Since 18 + 31 > 31, choice „d‟ does work

Both choice „c‟ of „10‟ and choice „d‟ of „31‟ work. The question asks for the smallest value.

Therefore, you would choose choice „c‟.

Homework

In problems 1-4 state whether it is possible for a triangle to have the indicated side lengths.

1. 1, 2, 3 2. 4, 9, 4 3. 8, 8, 8 4. 5, 12, 9 5. In the given triangle which multiple choice is the smallest possible value for x ?

a. 8 b. 11 c. 15 d. 33

6. In the given triangle which multiple choice is the largest possible value for x ? a. 2

b. 9 c. 10 d. 20

x x

18

7 x

18

x 5

11

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p n

4 6 1

3 2

7 8 5 p  n

7.0 a. Prove and use theorems involving the properties of parallel lines cut by a transversal.

Theorems involving parallel lines cut by a transversal.

If two parallel lines are cut by a transversal then:

corresponding angles are congruent 1 5

2 6 3 7 4 8

alternate interior angles are congruent 3 6

4 5

alternate exterior angles are congruent 1 8

2 7

same-side interior angles are supplementary m 3 + m 5 = 180°

m 4 + m 6 = 180°

Example 1

If m 1 = 20º, find m 3

Solution: 1 and 3 are corresponding angles.

Corresponding angles are congruent, so m 3 = 20º.

Example 2

If m 6 = (4x + 8)° and m 7 = 100°, find the value of x.

Solution: 6 and 7 are same-side interior angles and they are supplementary (total 180º)

So m 6 + m 7 = 180°

(4x + 8)° + (100°) = 180°

4x + 108° = 180°

4x = 72°

x = 18°

(cont)

1 2 3 4 6

5 7 8

a c a _c

Use this diagram for Ex.1 and Ex.2

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p n

4 6 1

3 2

7 8 5 p  n

Use this diagram for Problems 1-5

p n

4 6 1

3 2

7 8 5

p  n Use this diagram for Problems 6-7

13 15 11 12 9 10

16 14

c a

a  c

k n

4 7 1

2 3

6 8 5 k  n

7.0 a. Prove and use theorems involving the properties of parallel lines cut by a transversal.

(cont)

Homework

1. Which multiple choice is true?

a. m 1 + m 4 = 180°

b. m 2 m 8 c. m 4 m 8 d. m 3 m 5 2. m 1 = 98º, find m 3 3. m 4 = 17º, find m 5 4. m 5 = 59º, find m 3 5. m 7 = 71º, find m 4

6. m 4 = 62º find m 14 7. m 1 = 28º find m 16

8. Find the value of x a

c 9. Find the value of w

10. Analyze each statement as true or false.

a. 1 and 8 are supplementary b. 2 and 5 are congruent c. 3 and 7 are congruent d. 6 and 4 are supplementary e. 1 and 4 are supplementary f. 4 and 5 are congruent

80°

(5 x– 10)°

a c

(3w+1)°

79°

m n

m

n

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7.0b. Prove and use theorems involving the properties of quadrilaterals

Theorems involving quadrilaterals

A quadrilateral is a 4-sided polygon. The sum of the 4 angles is 360º.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel and both pairs of opposite sides congruent.

A rectangle is a special parallelogram.

It has all of the parallelogram properties and four right angles.

A rhombus is a special parallelogram.

It has all of the parallelogram properties and all sides congruent.

A square is a special parallelogram.

It has all of the parallelogram properties, all the rectangle properties, all the rhombus properties.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

An isosceles trapezoid has congruent legs and congruent base angles.

Property Parallelogram Rectangle Rhombus Square

1. Opposite sides are parallel X X X X

2. Opposite sides are congruent X X X X

3. Opposite angles are congruent X X X X

4. Consecutive angles are supplementary X X X X

5. Diagonals bisect each other X X X X

6. Diagonals are congruent X X

7. Diagonals are perpendicular X X

8. Each diagonal bisects two angles X X

9. Has 4 right angles X X

10. All 4 sides are congruent X X

(cont)

leg leg

base base