Name____________________________________ Geometry Review

Unit 1 – Geometry Gallery STANDARD: LANGUAGE OF MATH ARGUMENT

Deductive Reasoning – Arriving at a conclusion based on given facts.

Inductive Reasoning – Arriving at a conclusion based on observations. Uncertain whether conclusion is actually true.

Conjecture – A hypothesis formed by reasoning.

Counterexample – A specific example that proves a statement false.

Indirect Proof – Based on process of elimination.

Conditional Statement – “If, then” statement where the “if” part is the hypothesis and the “then” part is the conclusion.

Converse – Switch the hypothesis and conclusion.

Inverse – Negate the hypothesis and conclusion.

Contrapositive – Switch and negate the hypothesis and conclusion.

1. It two angles have the same measure, then they are congruent. You know that . What can you conclude about these two angles? Is this inductive or deductive reasoning, why?

2. If you study hard, you will pass all your classes. If you pass all your classes, you will graduate. What can you conclude from this information? Is this inductive or deductive reasoning, why?

3. Using the pattern below, sketch the next figure in the set.

4. Using the set of numbers below, find the next two numbers and describe the pattern. Is this inductive or deductive reasoning, why?

5. Find a counterexample to the following

conjecture. All odd numbers are prime. 6. Ella factored the first five out of ten trinomials

on a test, and each one factored into a pair of binomials. She made this statement. “All of the trinomials on this test will factor into a pair of binomials.” Is this inductive or deductive reasoning, why?

7. Jesse is a girl who loves math class and her dog. Using this information, what can you conclude? Is this inductive or deductive reasoning, why? a. Jesse is a boy. b. Jesse dislikes math class. c. Jesse dislikes her dog. d. Jesse is a Justin Bieber fan.

8. Write the converse, inverse, and contrapositive of the following statement. “If a number is a natural number, then the number is greater than zero.”

Converse:

Inverse:

Contrapositive:

STANDARD: PROPERTIES OF POLYGONS

Supplementary – Add up to .

Complementary – Add up to .

Sum of Interior Angles - .

Sum of Exterior Angles - .

Triangle Inequality Theorem – The sum of any 2 sides must be greater than the third side.

Triangle Congruence Theorems – SSS, SAS, ASA, AAS, and HL.

Orthocenter – Intersection of 3 altitudes of a triangle.

Centroid – Intersection of 3 medians of a triangle.

Circumcenter – Intersection of 3 perpendicular bisectors of a triangle, circumscribed circle.

Incenter – Intersection of 3 angle bisectors of a triangle, inscribed circle.

Parallelogram -

Rhombus -

Rectangle –

Square –

Trapezoid –

Kite –

Special Quadrilaterals –

9. In , , , and . Order the sides and angles from smallest to largest.

10. Determine whether each of the following sets could be the lengths of the sides of a triangle.

a. 12, 13, 25

b. 2, 3, 4

c. 5, 1, 5

d. 49, 51, 99

11. The first four angles in a hexagon each have the same measure. The other two angles each measure more than each of the first four angles. What is the measure of one of the first four angles in the hexagon?

12. You are given that and

. Which side of the triangle has the shortest

length? Which side of the triangle has the longest

length?

13. The exterior angles of a pentagon have the measures

, , , , and . What is the measure of the smallest exterior angle?

14. What are the degree measures of the labeled

interior and exterior angles of this figure? Justify

your answer.

15. Determine the value of for the following

quadrilaterals.

16. In the given isosceles trapezoid, ,

, and .

What is the length of ?

17. If Joseph were to construct a circumscribed circle around a triangle, what point of concurrency would he need to locate first?

17. What are all the names for quadrilateral that

has its vertices plotted at , ,

, and .

18. In the given parallelogram, and

. What is the length of ?

19. In the given kite, , , and . What is the perimeter of kite ?

20. Find the values of and in the diagram.

21. Determine which triangle congruence theorem is used to prove that the following pairs of triangles are congruent.

Unit 2 – Coordinate Geometry STANDARD: PROPERTIES OF GEOMETRIC FIGURES ON THE

COORDINATE PLANE

Distance Formula – √

Midpoint Formula – (

)

Perimeter – Sum of the outside lengths of a figure.

22. How much further is point from point than point from point ?

23. Line segment has the endpoints and . Point is located at . What point

on would create a line segment with the shortest distance to point ?

24. Find the distance between points and .

25. Points and lie on the coordinate

plane. Find the midpoint of segment .

26. Determine the point on the graph that is equidistant from points , , and .

27. Parallelogram has the coordinates , , and . Find the coordinates of point , and also find the perimeter of the figure.

28. Line segment is the hypotenuse of a right isosceles triangle and has coordinate points and . What are the possible locations for vertex ?

29. Find the perimeter of a quadrilateral with vertices , , , and .

Class 1 Class 2

Mean 12 12

Median 12 11

Mode 10 and 11 13

Range 18 20

Standard Deviation 3.2 2.8

Class 1 Class 2

Mean 12 12

Median 12 11

Mode 10 and 11 13

Range 18 20

Standard Deviation 3.2 2.8

Winning 63 56 61 48 50

Losing 61 35 55 30 49

Unit 3 – Statistics

STANDARD: DATA ANALYSIS

Mean – Average of the numbers.

Median – Middle number when placed in order from smallest to largest.

Mode – Most repeated number.

Range – Difference between largest and smallest numbers.

Box Plot – Min, Q1, median, Q3, max.

Standard Deviation – Tells us how spread out the data is.

Normal Distribution -

Samples – Used when it is impossible collect data for an entire population. The means and standard deviations will vary from one sample to the next.

28. The winning margin for each game is the difference

between the winning score and the losing score. What is the mean and standard deviation of the winning margins for this data?

29. Compare the mean and standard deviations of the

two data sets.

Set A: 7, 3, 4, 9, 2 Set B: 5, 8, 7, 6, 4

30. The table below compares the age of students in Mrs. June’s 2 classes. Describe the similarities and differences. Which class has more variability in their age?

31. For a large population, the means is 4.8 and the standard deviation is 3.6. One random sample produces data values of 5, 1, 3, 4, 7, 6, 8, 2, 1, and 3. Another random sample produced data values of 8, 7, 5, 3, 4, 2, 2, 9, 7, and 3. Compare the means and standard deviations of the random samples to the population parameters.

32. Find the mean, median, mode, range, variance, and

standard deviation of the data set. If an outlier is defined as any value more than two standard deviations from the mean, which, if any, values in the data would be considered an outlier? 22, 18, 19, 25, 27, 21, 24, 18, 21, 21, 37

33. Describe a way in which a student could select people from their high school for a survey using least biased methods.

34. A data set describing the age of a population of 300 adults is normally distributed with a mean of 38 and a standard deviation of 4.

a. How many people would be expected to be older than 46?

b. How many people would be expected to be between 30 and 42?

Unit 4 – Right Triangle Trigonometry STANDARD: SPECIAL RIGHT TRIANGLES

Special Right Triangles –

35. Quadrilateral ABCD is an isosceles trapezoid with angles A and D measuring , segment BC

measuring , and segment CD measuring √ . What is the length of segment AD?

36. A square has a side length of 10 cm. What is the length of the diagonal of the square, rounded to the nearest whole number?

37. A 30 ft plank leaning against a building makes an angle of 60⁰ with the ground. How far from the base of the building is the plank?

38. The diagonal of a square is √ , what is the perimeter of the square?

39. What kind of triangles are formed when:

a. You construct the altitude of an equilateral triangle.

b. You construct a diagonal in a square?

40. Determine the lengths of the sides labeled and in the three diagrams below.

41. Find the length of the hypotenuse for the following triangles:

a. A right triangle that has legs with measures

of and √ .

b. A right triangle that has both legs with measures of 19.

42. A parallelogram has sides that are cm and cm

long. The measure of the acute angles of the parallelogram is . What is the area of the parallelogram?

43. An equilateral triangle has side lengths of 18. What is

the length of the attitude?

STANDARD: TRIGONOMETRIC RATIOS

Trig Ratios –

Sin =

Cos =

Tan =

Inverse Trig Ratios – Only used when finding the angle measure of a right triangle.

44. What does it mean for two angles to be complementary?

45. Angle and angle are complementary angles in a

right triangle. The value of is

. What is the

value of ?

46. Triangle is a right triangle with right angle , as shown. What is the area of triangle ?

47. A road ascends a hill at an angle of . For every 120

feet of road, how many feet does the road ascend?

48. Given triangle , what is ?

49. You are given that

. What is the measure

of angle ?

50. In a right triangle, if

, what is ?

51. In right triangle , if and are the acute

angles, and

, what is ?

52. Find the measure of angle . Round your answer to

the nearest degree.

53. Solve for .

54. Solve for .

55. A ladder is leaning against a house so that the top of the ladder is 18 feet above the ground. The angle

with the ground is 47. How far is the base of the ladder from the house?

2x

y 100

86

T

E

D

C

A

B

Unit 5 – Circles and Spheres STANDARD: CIRCLES

Area –

Circumference –

Parts of a Circle –

Properties of Tangent Lines – o Tangent and a radius form a right angle o You can use Pythagorean Theorem to find

the side lengths o Two tangents from a common external

point are congruent

Central Angles –

Inscribed Angles –

Angles Outside the Circle –

Intersecting Chords –

56. What is the value of in this diagram?

57. Given , with the inscribed quadrilateral, find the value of each variable.

58. is tangent to at point . measures 12

inches and measures 7 inches. What is the radius of the circle?

59. Given , the and the find the value of x.

60. If two tangents of meet at the

external point , find their congruent length.

61. The measure of is . What is the measure of ?

62. Isosceles triangle is inscribed in this circle.

and . What is the measure of ?

63. In this diagram, segment is tangent to circle at

point . The measure of minor arc is . What is

?

64. If and , find .

STANDARD: SPHERES

Surface Area –

Volume -

65. A sphere has a radius of 8 cm. What is the surface

area? Answer in both decimal and exact -form.

66. A sphere has a surface area of 50 m2. Find the

radius.

67. A soccer ball has a diameter of 10 in. Find its volume.

68. When comparing two different sized bouncy balls, by how much more is the volume of larger ball if its radius is 3 times larger than the smaller ball?

69. A hot air balloon is being deflated. Its full blown volume is about 80,000 ft

3.

After 30 minutes, the balloon’s radius

has decreased by

. What is the volume

of the balloon at this time?

70. A sphere has a volume of cubic inches. What is the surface area?

71. Find the volume of a hemisphere which

has a diameter of 7.

72. A sphere has a surface area of 81 m2. What must

its diameter be?

73. A balance ball has a surface area of 32 . What is its volume?

74. A sphere has a diameter of 32 cm.

a. What is the radius?

b. What is the circumference of the great circle?

c. What is the surface area?

d. What is the volume?

75. Find the volume of a hemisphere which has a radius of 5.

76. The radius of a sphere is changed from 20 cm to 5 cm. How has the surface area and volume of the sphere changes?

77. Find the diameter of a sphere which has a surface area of 200. Round to the nearest hundredth.

Unit 6 – Exponential and Inverses STANDARD: EXPONENTIAL FUNCTIONS

Properties of Exponents –

o Product of like bases:

o Quotient of like bases:

o Power to a power:

o Product to a power:

o Quotient to a power: (

)

o Zero exponent:

o Negative exponent:

or

Solving Exponential Equations – o Make the bases the same. o Set the exponents equal and solve for x.

Growth and Decay – o Growth: , Decay: o

Exponential Graphs – o Determine whether growth or decay o Horizontal Asymptote: o Y-intercept: Substitute 0 for

Translations – o Negative : reflects across x-axis o Negative : reflects across y-axis o Shifts horizontally by o Shifts vertically by

78. Solve the exponential equations.

a. b.

79. Jill invests $6000 at a rate of 4% interest compounded yearly. Approximately how much will Jill’s investment be worth in 5 years?

80. Mark buys a car new for $52,000. If the value of the car depreciates by 12% each year, how much will the car be worth in 4 years?

81. Identify a, b, h, k, domain, range, asymptote, y-intercept, end behavior, and translations. Then sketch the graphs.

(

)

82. A certain population changes according to the model

, where represents the time in years. What is the difference in the population between years 4 and 8?

83. Simplify the exponential expressions. a. b.

c.

d.

d. e.

STANDARD: INVERSES OF FUNCTIONS

Inverse of a Function – When x and y values switch. o Switch x and y variables. o Solve for y.

Composition – Verifies that two expressions are inverses of each other. Substitute one expression for x in the other expression, the resulting answer should be x if they are inverses.

Vertical Line Test – Determines whether a relation is a function. No more than 1 point of intersection.

One-to-One – There is exactly 1 y-value for each x-value (no x-values repeat and no y-values repeat). If a function is 1-to-1, then its inverse is also a function.

Horizontal Line Test – Determines whether a function is 1-to-1. No more than 1 point of intersection.

84. If , what is the domain and range of .

85. Find the inverse of the following functions.

a. b.

86. A table of ordered pairs for a function is shown.

a. Write the inverse table. b. Is a one-to-one function, why? c. Is the inverse table a function, why?

87. Using composition, verify that and

are inverse functions.

88. Using the graph of below, graph the inverse of the function. Is also a function?

89. Determine whether the functions below are one-to-one and determine whether the inverses of the graphed functions are also a function.

a.

b.

90. Using composition, and are inverses of

each other if and only if and equal

what?