Geometry Cheat Sheet
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Parallel and Perpendicular Lines:
Notation:
Equation of a Line:
𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙:
Perpendicular: Take
≅ congruent
𝑦 = 𝑚𝑥 + 𝑏
Same Slope
negative reciprocal
~ similar
∆𝑦 𝑟𝑖𝑠𝑒
∆ triangle
𝑚=𝑚
𝑚 → − 1L𝑚
𝑚 = 𝑠𝑙𝑜𝑝𝑒 =
=
∆𝑥 𝑟𝑢𝑛
∢ angle
𝑏
=
𝑦
−
𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
∥ parallel
⊥ perpendicular
Point Slope Form:
2222
𝐴𝐵 line segment AB
𝑦 − 𝑦! = 𝑚(𝑥 − 𝑥! )
͡ arc AB
𝐴𝐵
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Distance Formula:
𝑑 = N(𝑥" − 𝑥! )" + (𝑦" − 𝑦! )"
Law of Sins:
𝑎
𝑏
𝑐
=
=
𝑆𝑖𝑛 𝐴 𝑆𝑖𝑛 𝐵 𝑆𝑖𝑛 𝐶
Law of Cosines:
𝑐 " = 𝑎" + 𝑏 " − 2𝑎𝑏𝑐𝑜𝑠𝐶
Converting Degrees to Radians:
π
𝜋
𝑒𝑥: 60° ×
=
180 3
𝑥°
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐿𝑖𝑛
𝐻𝑒𝑖𝑔ℎ𝑡
Midpoint Formula:
Converting Radians to Degrees:
𝑥" + 𝑥! 𝑦" + 𝑦!
𝜋 180
𝑀=(
,
)
𝑒𝑥: ×
= 60°
2
2
3
𝜋
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Angle of Elevation:
SOH CAH TOA:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑡
𝑒
ℎ
𝑠𝑖𝑛(𝑥)
=
𝑠
𝑔
𝑢
𝑆𝑖
𝑒𝑛
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑝𝑜𝑡
𝑒 𝑜𝑓
𝐻𝑦
𝑥°
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑐𝑜𝑠(𝑥) =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
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Inverse Trig. Functions:
Complimentary Angles:
1
𝑠𝑒𝑐(𝑥) =
°
𝑐𝑜𝑠(𝑥)
𝑠𝑖𝑛(90° − 𝜃) = 𝑐𝑜𝑠(𝜃) 𝑐𝑠𝑐(90 − 𝜃) = 𝑠𝑒𝑐(𝜃)
1
𝑐𝑠𝑐(𝑥) =
°
𝑠𝑖𝑛(𝑥)
𝑐𝑜𝑠(90° − 𝜃) = 𝑠𝑖𝑛(𝜃) 𝑠𝑒𝑐(90 − 𝜃) = 𝑐𝑠𝑐(𝜃)
1
𝑐𝑜𝑡(𝑥) =
°
𝑡𝑎𝑛(𝑥)
𝑡𝑎𝑛(90° − 𝜃) = 𝑐𝑜𝑡(𝜃) 𝑐𝑜𝑡(90 − 𝜃) = 𝑡𝑎𝑛(𝜃)
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𝑡𝑎𝑛(𝑥) =
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1
Probability:
Conditional: 𝑃(𝐵|𝐴) =
n=Total number of objects
r=Number of chosen objects
And:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵) (Independent)
Permutation:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × (𝐵|𝐴) (Dependent)
(Order matters)
Combinations:
%(' ∩*)
%(')
Or:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) (Not Mutually Exclusive)
(Order doesn’t matter)
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃 (𝐵) (Mutually Exclusive)
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Transversals: Given two lines are parallel and are cut by a transversal line.
Alternate Interior Angles:
∢𝑐 = ∢𝑓 𝑎𝑛𝑑 ∢𝑑 = ∢𝑒
Alternate Exterior Angles:
∢𝑎 = ∢ℎ 𝑎𝑛𝑑 ∢𝑏 = ∢𝑔
Corresponding Angles:
∢𝑎 = ∢𝑒, ∢𝑏 = ∢𝑓, ∢𝑐 = ∢𝑔, 𝑎𝑛𝑑 ∢𝑑 = ∢ℎ
Supplementary Angles:
∢𝑐 + ∢𝑒 = 180° , ∢𝑑 + ∢𝑓 = 180° , ∢𝑎 + ∢𝑏 = 180° ,
∢𝑐 + ∢𝑑 = 180° , ∢𝑒 + ∢𝑓 = 180° , ∢𝑔 + ∢ℎ = 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(𝑥, 𝑦) → 𝐴$ (−𝑦, 𝑥)
Reflection in the x-axis: 𝐴(𝑥, 𝑦) → 𝐴$ (𝑥, −𝑦)
Rotation of 180° : 𝐴(𝑥, 𝑦) → 𝐴$ (−𝑥, −𝑦)
Reflection in the y-axis: 𝐴(𝑥, 𝑦) → 𝐴$ (−𝑥, 𝑦)
Rotation of 270° : 𝐴(𝑥, 𝑦) → 𝐴$ (𝑦, −𝑥)
Reflection over the line y=x: 𝐴(𝑥, 𝑦) → 𝐴$ (𝑦, 𝑥)
Dilation of n: 𝐴(𝑥, 𝑦) → 𝐴$ (𝑥𝑛, 𝑦𝑛)
Reflection through the origin: 𝐴(𝑥, 𝑦) → 𝐴$ (−𝑥, −𝑦)
Transformation to the left m units and up n units: 𝐴(𝑥, 𝑦) → 𝐴$ (𝑥 − 𝑚, 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:
∆𝐴𝐵𝐶 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
𝐵𝐶 .
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 𝑥 and 𝑦.
1
𝑧 = (𝑥 + 𝑦)
2
𝑥
𝐴
𝐷
𝑥
𝐸
𝑧
2𝑥
𝑦
𝐶
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𝐵
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 = - 𝜋𝑟
Rectangle:𝑃 = 2𝑙 + 2𝑤
Trapezoid: 𝐴 = " (𝑏! + 𝑏" )ℎ
"
Square:𝑃 = 4𝑠
Cylinder: 𝑉 = 𝜋𝑟 ℎ
!
Triangle: 𝐴 = " 𝑏ℎ
!
Circle: Circumference = 𝜋𝑑
Pyramid: 𝑉 = - 𝑏ℎ
Rectangle:𝐴 = 𝑏ℎ
!
Cone: 𝑉 = - 𝜋𝑟 Pythagorean Theorem:
Square: 𝐴 = 𝑠 "
"
𝑎" + 𝑏" = 𝑐 "
Circle: 𝐴 = 𝜋𝑟
Prism: 𝑉 = 𝑏ℎ
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
͡
∢𝐴𝐶𝐵 = 𝐴𝐵
͡ = 90°
∢𝐴𝐶𝐵 = 90° and 𝐴𝐵
𝟏
𝟏
Inscribed Angle=𝟐Arc
Tangent/Chord Angle =𝟐 𝑨𝒓𝒄
𝟏
Angle formed by Two Intersecting Chords=𝟐 𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄𝒔
͡ = 160°
͡ = 50°
∢𝐴𝐵𝐶 = 80° and 𝐴𝐵
∢𝐴𝐶𝐵 = 25° and 𝐴𝐵
1
͡ + 𝐶𝐷
͡ )
∢𝐵𝐸𝐴 = ( 𝐴𝐵
2
1
∢𝐵𝐸𝐴 = (120° + 50° )
2
1
∢𝐵𝐸𝐴 = (170° )
2
∢𝐵𝐸𝐴 = 85°
𝟏
Tangents=𝟐 𝒕𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
͡ − 𝐵𝐶
͡ )
∢𝐵𝐴𝐶 = ( 𝐵𝐷𝐶
2
1
∢𝐵𝐴𝐶 = (200° − 40° )
2
1
∢𝐵𝐴𝐶 = (160° )
2
∢𝐵𝐴𝐶 = 80°
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𝟏
Angle formed by two Secants = 𝟐 𝒕𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
͡ − 𝐵𝐸
͡ )
∢𝐴𝐶𝐷 = (𝐴𝐷
2
1
∢𝐴𝐶𝐷 = (120° − 30° )
2
1
∢𝐴𝐶𝐷 = (90° )
2
∢𝐴𝐶𝐷 = 45°
𝟏
Angle formed by a Secant and Tangent =𝟐 𝒕𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒆𝒅 𝑨𝒓𝒄
1
͡ − 𝐵𝐷
͡ )
∢𝐴𝐶𝐷 = ( 𝐴𝐷
2
1
∢𝐴𝐶𝐷 = (180° − 70° )
2
1
∢𝐴𝐶𝐷 = (110° )
2
∢𝐴𝐶𝐷 = 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.
∢𝐴𝐶𝐵 = 90° and ∢𝐴𝐶𝐷 = 90°
∢𝐴 ≅ ∢𝐵
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 + ∢𝐶 = 180° and ∢𝐵 + ∢𝐷 = 180°
∢𝐵𝐴𝐶 ≅ 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)
͡ ≅ 𝐶𝐷
͡
∢𝐶𝑂𝐷 ≅ ∢𝐴𝑂𝐵 Therefore, 𝐴𝐵
͡ ≅ 𝐶𝐷
͡
∢𝐶𝑂𝐷 ≅ ∢𝐴𝑂𝐵 Therefore, 𝐴𝐵
Perimeter, Area and Volume:
Shape
Perimeter
Area
𝑃=𝑎+𝑏+𝑐
1
𝐴 = 𝑎𝑏
2
𝑃=4𝑠
𝐴 = 𝑠"
𝑃=2𝑙+2𝑤
𝐴 =𝑙×𝑤
𝑃=𝑎+𝑏+2𝑐
1
𝐴 = (𝑎 + 𝑏)ℎ
2
𝑃=2𝑙+2𝑤
𝐴 =𝑙×ℎ
𝑐
𝑎
Triangle
𝑏
s
Square
l
Rectangle
w
𝑎
Trapezoid
𝑐
ℎ
𝑏
𝑙
Parallelogram
𝑤
ℎ
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6
Volume
𝐴 = 𝜋𝑟 "
𝐶=𝜋𝑑
𝑑
Circle
𝑟
𝑑
Sphere
𝑆𝐴 = 4𝜋𝑟 "
4
𝑉 = 𝜋𝑟 #
3
𝑆𝐴 = 2𝜋𝑟 " + 2𝜋𝑟ℎ
𝑉 = 𝜋𝑟 " ℎ
𝑟
Cylinder
ℎ
Cone
ℎ
1
𝑉 = 𝜋𝑟 " ℎ
3
𝑟
ℎ
Pyramid
1
𝑉 = 𝑙𝑤 ℎ
3
𝑤
Rectangular Prism
𝑤
𝑙
𝑆𝐴 = 2(𝑙𝑤 + 𝑤ℎ + 𝑙ℎ)
𝑉 =𝑙×𝑤×ℎ
𝑆𝐴 = 6𝑠 "
𝑉 = 𝑠#
ℎ
Cube
𝑠
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