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Geometry Cheat Sheet

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Geometry Cheat Sheet
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Parallel and Perpendicular Lines:
Notation:
Equation of a Line:
π‘ƒπ‘Žπ‘Ÿπ‘Žπ‘™π‘™π‘’π‘™:
Perpendicular: Take
≅ congruent
𝑦 = π‘šπ‘₯ + 𝑏
Same Slope
negative reciprocal
~ similar
βˆ†π‘¦ π‘Ÿπ‘–π‘ π‘’
βˆ† triangle
π‘š=π‘š
π‘š → − 1Lπ‘š
π‘š = π‘ π‘™π‘œπ‘π‘’ =
=
βˆ†π‘₯ π‘Ÿπ‘’π‘›
− angle
𝑏
=
𝑦
−
π‘–π‘›π‘‘π‘’π‘Ÿπ‘π‘’π‘π‘‘
βˆ₯ parallel
⊥ perpendicular
Point Slope Form:
2222
𝐴𝐡 line segment AB
𝑦 − 𝑦! = π‘š(π‘₯ − π‘₯! )
Ν‘ arc AB
𝐴𝐡
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Distance Formula:
𝑑 = N(π‘₯" − π‘₯! )" + (𝑦" − 𝑦! )"
Law of Sins:
π‘Ž
𝑏
𝑐
=
=
𝑆𝑖𝑛 𝐴 𝑆𝑖𝑛 𝐡 𝑆𝑖𝑛 𝐢
Law of Cosines:
𝑐 " = π‘Ž" + 𝑏 " − 2π‘Žπ‘π‘π‘œπ‘ πΆ
Converting Degrees to Radians:
π
πœ‹
𝑒π‘₯: 60° ×
=
180 3
π‘₯°
π·π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’
π‘‚π‘π‘π‘œπ‘ π‘–π‘‘π‘’
𝐿𝑖𝑛
π»π‘’π‘–π‘”β„Žπ‘‘
Midpoint Formula:
Converting Radians to Degrees:
π‘₯" + π‘₯! 𝑦" + 𝑦!
πœ‹ 180
𝑀=(
,
)
𝑒π‘₯: ×
= 60°
2
2
3
πœ‹
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Angle of Elevation:
SOH CAH TOA:
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
𝑑
𝑒
β„Ž
𝑠𝑖𝑛(π‘₯)
=
𝑠
𝑔
𝑒
𝑆𝑖
𝑒𝑛
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
π‘π‘œπ‘‘
𝑒 π‘œπ‘“
𝐻𝑦
π‘₯°
π΄π‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
π‘π‘œπ‘ (π‘₯) =
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
β„Žπ‘¦π‘π‘‘π‘’π‘›π‘’π‘ π‘’
π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
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Inverse Trig. Functions:
Complimentary Angles:
1
𝑠𝑒𝑐(π‘₯) =
°
π‘π‘œπ‘ (π‘₯)
𝑠𝑖𝑛(90° − πœƒ) = π‘π‘œπ‘ (πœƒ) 𝑐𝑠𝑐(90 − πœƒ) = 𝑠𝑒𝑐(πœƒ)
1
𝑐𝑠𝑐(π‘₯) =
°
𝑠𝑖𝑛(π‘₯)
π‘π‘œπ‘ (90° − πœƒ) = 𝑠𝑖𝑛(πœƒ) 𝑠𝑒𝑐(90 − πœƒ) = 𝑐𝑠𝑐(πœƒ)
1
π‘π‘œπ‘‘(π‘₯) =
°
π‘‘π‘Žπ‘›(π‘₯)
π‘‘π‘Žπ‘›(90° − πœƒ) = π‘π‘œπ‘‘(πœƒ) π‘π‘œπ‘‘(90 − πœƒ) = π‘‘π‘Žπ‘›(πœƒ)
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π‘‘π‘Žπ‘›(π‘₯) =
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1
Probability:
Conditional: 𝑃(𝐡|𝐴) =
n=Total number of objects
r=Number of chosen objects
And:
𝑃(𝐴 ∩ 𝐡) = 𝑃(𝐴) × π‘ƒ(𝐡) (Independent)
Permutation:
𝑃(𝐴 ∩ 𝐡) = 𝑃(𝐴) × (𝐡|𝐴) (Dependent)
(Order matters)
Combinations:
%(' ∩*)
%(')
Or:
𝑃(𝐴 ∪ 𝐡) = 𝑃(𝐴) + 𝑃(𝐡) − 𝑃(𝐴 ∩ 𝐡) (Not Mutually Exclusive)
(Order doesn’t matter)
𝑃(𝐴 ∪ 𝐡) = 𝑃(𝐴) + 𝑃 (𝐡) (Mutually Exclusive)
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Transversals: Given two lines are parallel and are cut by a transversal line.
Alternate Interior Angles:
βˆ’π‘ = βˆ’π‘“ π‘Žπ‘›π‘‘ βˆ’π‘‘ = βˆ’π‘’
Alternate Exterior Angles:
βˆ’π‘Ž = βˆ’β„Ž π‘Žπ‘›π‘‘ βˆ’π‘ = βˆ’π‘”
Corresponding Angles:
βˆ’π‘Ž = βˆ’π‘’, βˆ’π‘ = βˆ’π‘“, βˆ’π‘ = βˆ’π‘”, π‘Žπ‘›π‘‘ βˆ’π‘‘ = βˆ’β„Ž
Supplementary Angles:
βˆ’π‘ + βˆ’π‘’ = 180° , βˆ’π‘‘ + βˆ’π‘“ = 180° , βˆ’π‘Ž + βˆ’π‘ = 180° ,
βˆ’π‘ + βˆ’π‘‘ = 180° , βˆ’π‘’ + βˆ’π‘“ = 180° , βˆ’π‘” + βˆ’β„Ž = 180°
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Properties of a Parallelogram:
The following shapes are all
1) Opposite sides are parallel.
Parallelograms:
2) Pairs of opposite sides are congruent.
1) Square (also a rhombus and a rectangle)
3) Pairs of opposite angles are congruent.
2) Rhombus
4) Diagonals bisect each other.
5) Diagonals separate parallelogram into 2
3) Rectangle
congruent triangles.
6) Interior angles add up to 360° .
Transformations:
Rotation of 90° : 𝐴(π‘₯, 𝑦) → 𝐴$ (−𝑦, π‘₯)
Reflection in the x-axis: 𝐴(π‘₯, 𝑦) → 𝐴$ (π‘₯, −𝑦)
Rotation of 180° : 𝐴(π‘₯, 𝑦) → 𝐴$ (−π‘₯, −𝑦)
Reflection in the y-axis: 𝐴(π‘₯, 𝑦) → 𝐴$ (−π‘₯, 𝑦)
Rotation of 270° : 𝐴(π‘₯, 𝑦) → 𝐴$ (𝑦, −π‘₯)
Reflection over the line y=x: 𝐴(π‘₯, 𝑦) → 𝐴$ (𝑦, π‘₯)
Dilation of n: 𝐴(π‘₯, 𝑦) → 𝐴$ (π‘₯𝑛, 𝑦𝑛)
Reflection through the origin: 𝐴(π‘₯, 𝑦) → 𝐴$ (−π‘₯, −𝑦)
Transformation to the left m units and up n units: 𝐴(π‘₯, 𝑦) → 𝐴$ (π‘₯ − π‘š, 𝑦 + 𝑛)
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2
Congruent Triangles ≅:
SAS
SSS
AAS
HL –(only for right triangles)
ASA
When proven use: Corresponding parts of
congruent triangles are congruent (CPCTC)
Midpoint Triangles Theorem:
βˆ†π΄π΅πΆ has midpoints at point D
and point E. When points D and E are
connected, the length of 2222
𝐷𝐸 is half the
length of base 2222
𝐡𝐢 .
Similar Triangles ~:
AA
SSS
SAS
When proven use: Corresponding sides of
similar triangles are in proportion.
Medians of a Trapezoid:
In a trapezoid, the length of median z
is equal to half the length of the sum
of both bases π‘₯ and 𝑦.
1
𝑧 = (π‘₯ + 𝑦)
2
π‘₯
𝐴
𝐷
π‘₯
𝐸
𝑧
2π‘₯
𝑦
𝐢
_____________________________________________________________________________
𝐡
Types of Triangles:
Scalene: No sides are equal.
Equilateral: All sides are equal.
Isosceles: Two sides are equal.
External Angle Triangles Theorem:
When any side of a triangle is extended the
value of its angle is supplementary to the
angle next to it (adding to 180° ). ex:
Acute: All angles are < 90° .
Obtuse: There is an angle > 90° .
Right: There is an angle = 90° .
40° + 140° = 180°
40° 140°
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Volume:
Perimeter:
Area:
,
!
Sphere: 𝑉 = - πœ‹π‘Ÿ
Rectangle:𝑃 = 2𝑙 + 2𝑀
Trapezoid: 𝐴 = " (𝑏! + 𝑏" )β„Ž
"
Square:𝑃 = 4𝑠
Cylinder: 𝑉 = πœ‹π‘Ÿ β„Ž
!
Triangle: 𝐴 = " π‘β„Ž
!
Circle: Circumference = πœ‹π‘‘
Pyramid: 𝑉 = - π‘β„Ž
Rectangle:𝐴 = π‘β„Ž
!
Cone: 𝑉 = - πœ‹π‘Ÿ Pythagorean Theorem:
Square: 𝐴 = 𝑠 "
"
π‘Ž" + 𝑏" = 𝑐 "
Circle: 𝐴 = πœ‹π‘Ÿ
Prism: 𝑉 = π‘β„Ž
Polygon Angle Formulas:
How to Prove Circles Congruent ≅:
n=number of sides
Circles are equal if they have congruent
!./(01")
radii, diameters, circumference, and/or area.
Value of each Interior Angle:
0
Sum of Interior Angles: 180(𝑛 − 2)
-2/
Value of each Exterior Angle: 0
Sum of Exterior Angles:360°
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3
Parts of a Circle:
Central Angles=Measure of Arc
Ν‘
−𝐴𝐢𝐡 = 𝐴𝐡
Ν‘ = 90°
−𝐴𝐢𝐡 = 90° and 𝐴𝐡
𝟏
𝟏
Inscribed Angle=𝟐Arc
Tangent/Chord Angle =𝟐 𝑨𝒓𝒄
𝟏
Angle formed by Two Intersecting Chords=𝟐 𝒕𝒉𝒆 π’”π’–π’Ž 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄𝒔
Ν‘ = 160°
Ν‘ = 50°
−𝐴𝐡𝐢 = 80° and 𝐴𝐡
−𝐴𝐢𝐡 = 25° and 𝐴𝐡
1
͑ + 𝐢𝐷
Ν‘ )
−𝐡𝐸𝐴 = ( 𝐴𝐡
2
1
−𝐡𝐸𝐴 = (120° + 50° )
2
1
−𝐡𝐸𝐴 = (170° )
2
−𝐡𝐸𝐴 = 85°
𝟏
Tangents=𝟐 𝒕𝒉𝒆 π’…π’Šπ’‡π’‡π’†π’“π’†π’π’„π’† 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
Ν‘ − 𝐡𝐢
Ν‘ )
−𝐡𝐴𝐢 = ( 𝐡𝐷𝐢
2
1
−𝐡𝐴𝐢 = (200° − 40° )
2
1
−𝐡𝐴𝐢 = (160° )
2
−𝐡𝐴𝐢 = 80°
______________________________________________________________________________
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4
𝟏
Angle formed by two Secants = 𝟐 𝒕𝒉𝒆 π’…π’Šπ’‡π’‡π’†π’“π’†π’π’„π’† 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
Ν‘ − 𝐡𝐸
Ν‘ )
−𝐴𝐢𝐷 = (𝐴𝐷
2
1
−𝐴𝐢𝐷 = (120° − 30° )
2
1
−𝐴𝐢𝐷 = (90° )
2
−𝐴𝐢𝐷 = 45°
𝟏
Angle formed by a Secant and Tangent =𝟐 𝒕𝒉𝒆 π’…π’Šπ’‡π’‡π’†π’“π’†π’π’„π’† 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
Ν‘ − 𝐡𝐷
Ν‘ )
−𝐴𝐢𝐷 = ( 𝐴𝐷
2
1
−𝐴𝐢𝐷 = (180° − 70° )
2
1
−𝐴𝐢𝐷 = (110° )
2
−𝐴𝐢𝐷 = 55°
Circle Theorems:
In a circle when
two inscribed
angles
intercept the
same arc, the
angles are
congruent.
In a circle
when a
tangent and
radius come
to touch, the
form a 90°
angle.
−𝐴𝐢𝐡 = 90° and −𝐴𝐢𝐷 = 90°
−𝐴 ≅ −𝐡
When a
quadrilateral is
inscribed in a
circle, opposite
angles are
supplementary.
In a circle
when an
angle is
inscribed by
a semicircle,
it forms a
90° angle.
−𝐴 + −𝐢 = 180° and −𝐡 + −𝐷 = 180°
−𝐡𝐴𝐢 ≅ 90°
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5
In a circle when
central angles
are congruent,
arcs are also
congruent. (and
vice versa)
In a circle when
central angles
are congruent,
chords are also
congruent. (and
vice versa)
Ν‘ ≅ 𝐢𝐷
Ν‘
βˆ’πΆπ‘‚π· ≅ βˆ’π΄π‘‚π΅ Therefore, 𝐴𝐡
Ν‘ ≅ 𝐢𝐷
Ν‘
βˆ’πΆπ‘‚π· ≅ βˆ’π΄π‘‚π΅ Therefore, 𝐴𝐡
Perimeter, Area and Volume:
Shape
Perimeter
Area
𝑃=π‘Ž+𝑏+𝑐
1
𝐴 = π‘Žπ‘
2
𝑃=4𝑠
𝐴 = 𝑠"
𝑃=2𝑙+2𝑀
𝐴 =𝑙×𝑀
𝑃=π‘Ž+𝑏+2𝑐
1
𝐴 = (π‘Ž + 𝑏)β„Ž
2
𝑃=2𝑙+2𝑀
𝐴 =𝑙×β„Ž
𝑐
π‘Ž
Triangle
𝑏
s
Square
l
Rectangle
w
π‘Ž
Trapezoid
𝑐
β„Ž
𝑏
𝑙
Parallelogram
𝑀
β„Ž
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6
Volume
𝐴 = πœ‹π‘Ÿ "
𝐢=πœ‹π‘‘
𝑑
Circle
π‘Ÿ
𝑑
Sphere
𝑆𝐴 = 4πœ‹π‘Ÿ "
4
𝑉 = πœ‹π‘Ÿ #
3
𝑆𝐴 = 2πœ‹π‘Ÿ " + 2πœ‹π‘Ÿβ„Ž
𝑉 = πœ‹π‘Ÿ " β„Ž
π‘Ÿ
Cylinder
β„Ž
Cone
β„Ž
1
𝑉 = πœ‹π‘Ÿ " β„Ž
3
π‘Ÿ
β„Ž
Pyramid
1
𝑉 = 𝑙𝑀 β„Ž
3
𝑀
Rectangular Prism
𝑀
𝑙
𝑆𝐴 = 2(𝑙𝑀 + π‘€β„Ž + π‘™β„Ž)
𝑉 =𝑙×𝑀×β„Ž
𝑆𝐴 = 6𝑠 "
𝑉 = 𝑠#
β„Ž
Cube
𝑠
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7
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