Math Geometry

1. Table of Contents
1
I. Number & Quantity
II. Algebra
III. Functions
IV. Geometry
V. Statistics & Probability
VI. Integrating Essential Skills

2. IV. Geometry
2
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

3. 1. Intersecting Lines
 When two lines intersect…
 Adjacent angles are supplementary (add up to 180
degrees)
 1 and 2
 2 and 3
 3 and 4
 4 and 1
 Vertical angles are equal
 1 and 3
 2 and 4
3

4. 2. Parallel Lines and Transversals
 Transversal: a line that crosses two other lines
 Facts
 A transversal across parallel lines forms four congruent
acute angles and four congruent obtuse angles.
 Only exception? If all angles are right angles
 Any of the acute angles are supplementary to any of the
obtuse angles
4

5. #1
5
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6. #1
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Answer: C
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7. 3. Interior and Exterior Angles
 Interior Angles: angles contained inside a polygon
 Ex: 1, 2, 3, 4, 5
 Exterior Angles: angles supplementary to adjacent
interior angles
 Ex: 6, 7, 8, 9, 10
6

8. 3. Interior and Exterior Angles
Important Facts
 Sum of all interior angles in a triangle 180°
 Sum of all exterior angles in every convex polygon is
360°
7

9. 3. Interior and Exterior Angles
Sum of Interior Angles in Any Convex
Polygon
 How to find sum of angles in any regular polygon
 Choose one vertex
 Draw as many triangles as possible by connecting this
vertex to other vertices
 Count the number of triangles
 Multiply this number by 180°
8

10. 3. Interior and Exterior Angles
Sum of Interior Angles in Any Convex
Polygon
 How to find sum of angles in any regular polygon
 Choose one vertex
 Draw as many triangles as possible by connecting this
vertex to other vertices
 Count the number of triangles
 Multiply this number by 180°
8

11. 3. Interior and Exterior Angles
Sum of Interior Angles in Any Convex
Polygon
 How to find sum of angles in any regular polygon
 Choose one vertex
 Draw as many triangles as possible by connecting this
vertex to other vertices
 Count the number of triangles
 Multiply this number by 180°
8

12. 3. Interior and Exterior Angles
Sum of Interior Angles in Any Convex
Polygon
 How to find sum of angles in any regular polygon
 Choose one vertex
 Draw as many triangles as possible by connecting this
vertex to other vertices
 Count the number of triangles
 Multiply this number by 180°
8
3(180)=540o

13. 3. Interior and Exterior Angles
Sum of Interior Angles in Any Convex
Polygon
 How to find sum of angles in any regular polygon
 Choose one vertex
 Draw as many triangles as possible by connecting this
vertex to other vertices
 Count the number of triangles
 Multiply this number by 180°
8
3(180)=540o
Formula for a Polygon with n sides

14. 3. Interior and Exterior Angles
Regular Polygons
 Regular polygon: a polygon in which all angles have the
same measure and all sides have the same length
 How to find the measure of an angle in a regular polygon
with n sides
 Find the sum of all interior angles
 Divide by n
9

15. 4. Similar Figures
 Similar Figures: corresponding angles have same measure
and corresponding sides are proportionally related by a specific
factor
 X, Y, Z have same measure
 Side lengths are proportionally related
 Perimeter is proportionally related
 Areas are proportionally related by the square of the scale
10

16. 4. Similar Figures
Special Right Triangles
3-4-5 5-12-13
11

17. 4. Similar Figures
Special Right Triangles
30-60-90
30o
60o
2
1
45-45-90
1
45o
45o
1
12

18. IV. Geometry
13
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

19. B. Distance
 Distance: refers to the length of
a line (straight or curved)
connecting two points
 Types of “distances” that will be
covered
 Distance between two points
 Midpoint
 Circumference on a Circle
 Arc Length
 Perimeter of a Polygon
14

20. 1. Find the Distance Between 2 Points
Method #1 – with Pythagorean
Theorem
Pythagorean Theorem
 Sketch the points on an x-y grid
 Connect them with a line
(representing the distance)
 This becomes the hypotenuse
 The difference in x-values is one
leg, and the difference in y-
values is the other leg
 Use the Pythagorean Theorem
to find the distance!
Example
Find the distance between the
points (1,2) and (3,6).
15a2 + b2 = c2

21. 1. Find the Distance Between 2 Points
Method #2 – with Distance Formula
Distance Formula
 Given two points (x1, y1) and (x2,
y2), the distance between them
is…
Example
Find the distance between the
points (1,2) and (3,6).
16

22. #2
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23. #2
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Answer: E
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24. 2. Find the Midpoint of 2 Points
 Midpoint: a point located
exactly halfway (distance-wise)
between two given points
 Use the following formula to find
the midpoint (xm, ym):
 Possible question types
1. Find midpoint given two
points
2. Find the other point given a
point and the midpoint
Example
Find the midpoint of the points
(1,2) and (3,6).
18
Think of these as the “averages”
of the x- and y-values

25. #3
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26. #3
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Answer: D
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27. 3. Circumference and Arc Length
Circumference of Circle
 Circumference: distance
traveled around the edge of a
circle (i.e. perimeter of a circle)
 Formula: C = 2πr where r is
the radius of the circle
20
r

28. 3. Circumference and Arc Length
Circumference of Circle
 Circumference: distance
traveled around the edge of a
circle (i.e. perimeter of a circle)
 Formula: C = 2πr where r is
the radius of the circle
Arc Length
 Arc: a piece of the
circumference
 Central angle: the angle at the
center of the circle that the arc
covers (see figure)
 Formula:
where n is the central angle
20
r
r
n

29. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
21

30. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
Example
Find the perimeter of the figure below.
21

31. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
Example
Find the perimeter of the figure below.
This appears to be a problem since we only
are given two sides. But…
21

32. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
Example
Find the perimeter of the figure below.
This appears to be a problem since we only
are given two sides. But…
21

33. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
Example
Find the perimeter of the figure below.
This appears to be a problem since we only
are given two sides. But…
21

34. 4. Perimeter of a Polygon
 Perimeter: the distance around
a 2-dimensional figure
 No consistent formula
 Instead, you should just find
the sum of all side lengths!
Example
Find the perimeter.
Example
Find the perimeter of the figure below.
This appears to be a problem since we only
are given two sides. But…
Now we just have a rectangle and can find its
perimeter!
21

35. #4
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36. #4
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Answer: C
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37. IV. Geometry
23
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

38. C. Area
24
 Area: a two-dimensional quantity
that describes the number of unit
squares that can fit inside a figure
 Units are always “units” squared,
square “units,” or u2
 Types of areas that will be
discussed
 Area of a Triangle
 Area of a Circle
 Area of a Sector
 Area of Quadrilaterals
 Area of Other Polygons
 Surface Area of Rectangular
Solids

39. 1. Area of a Triangle
Right Triangle
 Formula:
Oblique (Non-Right) Triangle
 Formula:
 What’s the difference?
 Height in an oblique triangle is
not a side length.
 Find the perpendicular
distance between the side
that’s chosen as the base and
the opposite vertex.
25

40. 2. Area of a Circle and Sector Area
Area of a Circle
 Formula: A = πr2 where r is
the radius of the circle
26
r
r
n

41. 2. Area of a Circle and Sector Area
Area of a Circle
 Formula: A = πr2 where r is
the radius of the circle
Sector Area
 Sector: a piece of the area
referred to as S
 Formula:
where n is the central angle
26
r
r
n

42. #5
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43. #5
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Answer: B
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44. 3. Area of Quadrilaterals
Rectangles and Parallelograms
Rectangles
 Rectangle: four-sided figure with
four right angles
 Opposite sides are equal in
length
 Diagonals are equal in length
 Area formula:
28
h
b

45. 3. Area of Quadrilaterals
Rectangles and Parallelograms
Rectangles
 Rectangle: four-sided figure with
four right angles
 Opposite sides are equal in
length
 Diagonals are equal in length
 Area formula:
Parallelogram
 Parallelogram: four-sided figure
with two pairs of parallel sides
 Opposite sides are equal in
length
 Opposite angles are equal is
measure
 Consecutive angles are
supplementary
28
h
b

46. 3. Area of Quadrilaterals
Rectangles and Parallelograms
Rectangles
 Rectangle: four-sided figure with
four right angles
 Opposite sides are equal in
length
 Diagonals are equal in length
 Area formula:
Parallelogram
 Parallelogram: four-sided figure
with two pairs of parallel sides
 Opposite sides are equal in
length
 Opposite angles are equal is
measure
 Consecutive angles are
supplementary
 Area formula:
28
h
b

47. 3. Area of Quadrilaterals
Squares and Trapezoids
Squares
 Square: rectangle with four equal
sides
 Area formula:
29

48. 3. Area of Quadrilaterals
Squares and Trapezoids
Squares
 Square: rectangle with four equal
sides
 Area formula:
Trapezoids
 Trapezoid: four-sided figure with
one pair of parallel sides and one
pair of nonparallel sides
29

49. 3. Area of Quadrilaterals
Squares and Trapezoids
Squares
 Square: rectangle with four equal
sides
 Area formula:
Trapezoids
 Trapezoid: four-sided figure with
one pair of parallel sides and one
pair of nonparallel sides
 Area formula:
29

50. 3. Area of Quadrilaterals
Squares and Trapezoids
Squares
 Square: rectangle with four equal
sides
 Area formula:
Trapezoids
 Trapezoid: four-sided figure with
one pair of parallel sides and one
pair of nonparallel sides
 Area formula:
 b1 and b2 must be the parallel
sides
29

51. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30

52. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30

53. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30

54. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30

55. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30

56. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
30
Total area:
15 + 50 = 65 u2

57. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
Example
Find the shaded area in the figure
below.
30
r=6
Total area:
15 + 50 = 65 u2

58. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
Example
Find the shaded area in the figure
below.
Area = Square Area – Circle Area
Square Area = 122 = 144
Circle Area = π62 = 36π
30
r=6
Total area:
15 + 50 = 65 u2

59. 4. Areas of other Polygons and Figures
 Strategies
 Count the number of squares of
area if possible
 Break the figure into small
pieces
Example
Find the area of the figure below.
Example
Find the shaded area in the figure
below.
Area = Square Area – Circle Area
Square Area = 122 = 144
Circle Area = π62 = 36π
Area = 144 – 36π ≈ 30.9 u2
30
r=6
Total area:
15 + 50 = 65 u2

60. #6
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61. #6
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Answer: D
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62. 5. Surface Area of Rectangular Solids
b
w
h
 Surface Area: the total area
covered by all faces of a given
three-dimensional object
32

63. 5. Surface Area of Rectangular Solids
b
w
h
 Surface Area: the total area
covered by all faces of a given
three-dimensional object
 Rectangular prism:
 SA = 2bw + 2hw + 2bh
32

64. IV. Geometry
33
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

65. D. Volume
34
 Volume: a three-dimensional
quantity that describes the
amount of space occupied by a
particular figure
 Units are always “units” cubed,
cubic “units,” or u3

66. 1. Volume of Rectangular Solids and Cubes
Rectangular Solids
 Volume formula:
Cubes
 Volume formula:
35
b
w
h

67. #7
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68. #7
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Answer: A
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69. 2. Volume of Cylinders and Cones
Cylinders
 Volume formula:
Cones
 Volume formula:
37
r
h
r
h

70. 3. Volume of Spheres and Pyramids
Spheres
 Volume formula:
Pyramids
 Volume formula:
where B is the area of the base
38
r
h

71. IV. Geometry
39
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

72. 1. Triangles
Important Facts
 Sum of interior angles is 180°
 Sum of any two side lengths on
a triangle must be larger than
the third side length
 Largest side length is across
from the largest angle, smallest
side length across from the
smallest angle
 Isosceles triangle has two
identical side lengths and two
identical angles
40

73. 2. Right Triangles
Pythagorean Theorem
 Pythagorean Theorem allows you to find an unknown side
length in a right triangle
a2 + b2 = c2
 a and b are interchangeable
 c must be the hypotenuse
41

74. 2. Right Triangles
Basic Trigonometry
 Trigonometric functions give relationships between angles
and ratios of side lengths in right triangles only
 Function definitions
 Consistent structure: function(angle)=side/side
42

75. 2. Right Triangles
Basic Trigonometry – Example #1
Example #1: Unknown Side Length
Find the unknown side length using trigonometry.
43

76. 2. Right Triangles
Basic Trigonometry – Example #1
Example #1: Unknown Side Length
Find the unknown side length using trigonometry.
15: opposite
x: hypotenuse
43

77. 2. Right Triangles
Basic Trigonometry – Example #1
Example #1: Unknown Side Length
Find the unknown side length using trigonometry.
15: opposite
x: hypotenuse
So we’ll use sin(45°).
43

78. 2. Right Triangles
Basic Trigonometry – Example #1
Example #1: Unknown Side Length
Find the unknown side length using trigonometry.
15: opposite
x: hypotenuse
So we’ll use sin(45°).
43

79. 2. Right Triangles
Basic Trigonometry – Example #1
Example #1: Unknown Side Length
Find the unknown side length using trigonometry.
15: opposite
x: hypotenuse
So we’ll use sin(45°).
43
Make sure calculator
is in “DEGREE” mode
by pressing “MODE”

80. 2. Right Triangles
Basic Trigonometry – Example #2
Example #2: Unknown Angle
Find the unknown angle using trigonometry.
44

81. 2. Right Triangles
Basic Trigonometry – Example #2
Example #2: Unknown Angle
Find the unknown angle using trigonometry.
8: opposite
14: adjacent
44

82. 2. Right Triangles
Basic Trigonometry – Example #2
Example #2: Unknown Angle
Find the unknown angle using trigonometry.
8: opposite
14: adjacent
So we’ll use tan(x).
44

83. 2. Right Triangles
Basic Trigonometry – Example #2
Example #2: Unknown Angle
Find the unknown angle using trigonometry.
8: opposite
14: adjacent
So we’ll use tan(x).
44

84. 2. Right Triangles
Basic Trigonometry – Example #2
Example #2: Unknown Angle
Find the unknown angle using trigonometry.
8: opposite
14: adjacent
So we’ll use tan(x).
44
Inverse trig function
gives an angle

85. #8
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86. #8
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Answer: C
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87. #9
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88. #9
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Answer: C
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89. 3. Oblique (Non-Right) Triangles
Law of Sines and Law of Cosines
Law of Sines
 Used to solve the
following types of
triangles:
 SSA, ASA, SAA
Law of Cosines
 Used to solve the following
types of triangles:
 SSS, SAS
47

90. 3. Oblique (Non-Right) Triangles
Law of Sines Example
Example #4:
Find the unknown side length.
48

91. 3. Oblique (Non-Right) Triangles
Law of Cosines Example
Example #5:
Find the unknown side length.
49
x
105o
12
7

92. IV. Geometry
50
A. Angles
B. Distance
C. Area
D. Volume
E. Triangles and Trigonometry
F. Transformations

93. Transformations
51

94. Transformations
 Translation: “sliding” an image
in the x and/or y directions
51

95. Transformations
 Translation: “sliding” an image
in the x and/or y directions
51

96. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
51

97. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
51

98. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
 Reflection: “flipping” an image
across an axis
51

99. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
 Reflection: “flipping” an image
across an axis
51

100. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
 Reflection: “flipping” an image
across an axis
 Dilation: “scaling” an image to
make it proportionally larger or
smaller
51

101. Transformations
 Translation: “sliding” an image
in the x and/or y directions
 Rotation: ”turning” an image by
a certain degree measure
 Reflection: “flipping” an image
across an axis
 Dilation: “scaling” an image to
make it proportionally larger or
smaller
51

102. #10
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103. #10
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Answer: D
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