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Geometry
Reference Packet
© Suzette Berry-Clark, Berry Pi Services, LLC, 2018
Topic Pg
Calculator tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Point, Line, Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Parallel , Perpendicular or Neither . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Equation of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Angles formed by Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Classifying Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Triangle Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Centers of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Steps to a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Never-Given-Givens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Proof Reasons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Coordinate Geometry Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
What To Do When You Need To Prove (Coord. Geo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Quadrilateral Family Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Right Triangle and Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Rigid Motion Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Two-Dimensional Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Three-Dimensional Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3
Calculator Tricks
Go to
Using 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 type c in Y1 and
C in Y2
Go to the table and look for the b value in the Y2
column.
Your factors are in the x and Y1 columns.
Resetting • Turn the calculator off and lay it flat
• Press and hold the left and right arrow keys
• Turn the calculator on
Fractions
• To enter a fraction
• To convert a fraction to a decimal or a decimal to a fraction
Simplifying
Radicals
Factoring
Trinomials
enter the number under the radical
Go to the table
Look in the y column for the last whole number
Answer is written in the form of 𝑥 𝑦
Y = ÷ x x2
Y =
÷ x
÷ x + x
ALPHA Y = 1
ALPHA Y = 4 ENTER
4
Points, Lines, Planes
Collinear Points on the same line
Congruent Equal
Coplanar Points on the same plane
Line A set of points with no thickness or width. Represented with a single script lower case letter or
𝐴𝐵 ⃡
Line segment A measureable part of a line consisting of two endpoints. Represented by 𝐴𝐵̅̅ ̅̅
Midpoint A point on a line segment that divides the segment into 2 congruent segments
Parallel Lines ( ∥ ) Two lines that never intersect
Perpendicular ( ⊥ ) Two lines that intersect to form right angles
Perpendicular Bisector
Two lines that intersect at a segments midpoint to form right angles
Plane A flat surface made up of points that extends in all directions without end. Represented by a single script capital letter or 3 non-collinear points
Point A location in space with no size or shape. Represented with a capital letter
Ray A line that extends in one direction without end.
Represented by 𝐴𝐵
Segment Addition piece + piece = whole 𝐴𝐵̅̅ ̅̅ + 𝐵𝐶̅̅ ̅̅ = 𝐴𝐶̅̅ ̅̅
Segment Bisector A line or part of a line that intersects a segment at its midpoint
5
Angles
Angle The intersection of two rays at an endpoint
Acute Angle An angle that measures less than 90°
Adjacent Angles Two angles that share a side
Angle Addition piece + piece = whole ∠𝐴𝐵𝐶 + ∠𝐶𝐵𝐷 = ∠𝐴𝐵𝐷
Angle Bisector A line part of a line that divides an angle into two congruent angles
Complementary Angles
Two angles that add up to 90°
Linear Pair Adjacent angles that are supplementary.
Obtuse Angle An angles that measures more than 90°, but less than 180°
Right Angle An angle that measures 90°
Straight Angle An angle that measures 180°
Supplementary Angles
Two angles that add up to 180°
Vertex The common end point of an angle
Vertical Angles
Two congruent angles across from each other on intersecting lines.
6
Coordinate Geometry
Parallel , Perpendicular or Neither
Δ𝑥 = 𝑥2 − 𝑥1
Δy = 𝑦2 − 𝑦1
Slope = ∆𝑦
∆𝑥
Distance
(∆𝑥)2 + (∆𝑦)2
Midpoint 𝑥2+𝑥1
2,𝑦2+𝑦1
2
If you have the midpoint, do not use
the formula, do the number line jump
𝐴(−5,7) 𝑀(−9,12) 𝐵( , )
-4 -4
+5 +5
Partitioning a line segment
𝑃 = (𝑥1 + 𝑘∆𝑥, 𝑦1 + 𝑘∆𝑦) 𝑘 =𝑎
𝑎+𝑏 where a:b
(𝑥1, 𝑦1) must be the first point named in the directed line segment
Horizontal Lines
y = # slope is zero (zero in the numerator)
Parallel Lines
Equal slopes
Perpendicular Lines
Negative Reciprocal Slopes
Ver
tica
l Lin
es
x =
#
s
lop
e is
zer
o
(zer
o i
n t
he
den
om
inat
or)
7
Equation of a Line
Point-intercept 𝑦 = 𝑚𝑥 + 𝑏 𝑚 = slope 𝑏 = y-intercept
Point-Slope Form (Geometry’s Favorite)
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 𝑚 = slope (𝑥1, 𝑦1) = a point on the line
Angles formed by Parallel Lines
Alternate Interior Angles – Congruent
Alternate Exterior Angles – Congruent
Corresponding Angles – Congruent
Consecutive Interior Angles –Supplementary
Classifying Triangles
By Sides By Angles
Scalene – no ≅ sides; no ≅ angles Acute – all ∠’s less than 90
Isosceles – 2 ≅ sides; 2 ≅ angles Obtuse – 1 ∠ greater than 90
Equilateral – All ≅ sides; all ≅ angles Right – 1 ∠ = 90
Equiangular – all ∠’s = 60
8
Triangle Theorems
Triangle Angle Sum Theorem The sum of the measures of a
triangle is 180°
𝑚∠𝐴 +𝑚∠𝐵 +𝑚∠𝐶 = 180
Exterior Angle Theorem An exterior angle of a triangle is always equal to the
sum of the two non-adjacent interior angles.
𝑚∠𝐴 = 𝑚∠𝐵 +𝑚∠𝐶
Isosceles Triangles If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
Midsegment If a segment joins the midpoints of two
sides of a triangle, then the segment is
parallel to the third side and half as long.
𝐷𝐸̅̅ ̅̅ ∥ 𝐴𝐶̅̅ ̅̅ and 2(DE) = AC
2(midsegments) = base
Ordering Sides & Angles
The longest side is opposite the largest angle.
The shortest side is opposite the smallest angle.
The largest angle is opposite the longest side.
The smallest angle is opposite the shortest side.
C
B
A
B
C A
C
B
A
A
B
C
D E
Largest Smallest
9
credit: All Things Algebra on Teachers Pay Teachers
Perpendicular Bisectors
Equidistance from the
vertices
Center of the
circumscribed circle
Located:
o In – acute
o On – right
o Outside - obtuse
Angle Bisectors Medians Altitudes
- Equidistance from the
sides
- Center of the inscribed
circle
- Located 2
3 of the way
from the vertex
- Forms a ratio of 2:1
o 𝐴𝑀̅̅̅̅̅ = long piece
o 𝑀𝑌̅̅̅̅̅ = short piece
- Located:
o In – acute
o On – right
o Outside – obtuse
10
Steps to a Proof
1) READ it what are you asked to prove? 2) WRITE it copy the given on the T chart 3) DEFINE it highlight the vocabulary word and
write the definition, theorem, or properties in the reason box on the diagonal
4) MARK it mark the picture with colored pencil 5) LABEL it is it an S or A
6) CHECK it check the given as done 7) REPEAT it steps 2-6 as necessary 8) N-G-G are there any never-given-givens? 9) PROVE it Triangles First! Then CPCTC
Never-Given-Givens
Vertical Angles look like
∠1 ≅ ∠2 and ∠3 ≅ ∠4
Reflexive Sides looks like Reflexive Angles look
like
2 1
1
2 4 3
11
Proof Reasons
Angle Bisector (bisects ∠)
An ∠ bisector creates 2 ≅ ∠’s
Midpoint A midpoint creates 2 ≅ segments
Median A median creates 2 ≅ segments
Parallel lines If 2 || lines are cut by a transversal, then _________ ∠’s are ≅.
Isosceles Triangle (≅ sides)
In a ∆, ∠’𝑠 opposite ≅ sides are ≅
Isosceles Triangle (≅ angles)
In a ∆, sides opposite ≅ ∠’s are ≅
Perpendicular Lines ⊥ lines form ≅ right ∠’s
Altitude An altitude forms ≅ right ∠’s
Segment Bisector (bisects 𝐴𝐵̅̅ ̅̅ )
A segment bisector creates 2 ≅ segments
Perpendicular Bisector (⊥ bisector)
1) ⊥ lines form ≅ right ∠’s 2) A segment bisector creates 2 ≅ segments
Supplements of Supplements
Supplements of ≅ ∠’s are ≅
Complements of Complements
Complements of ≅ ∠’s are ≅
Symmetric If 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , then 𝐶𝐷̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅ .
Reflexive 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅ and ∠𝐴 ≅ ∠𝐴
Transitive If 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅ , then 𝐴𝐵̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅
12
Congruent Triangles
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
Transformations
Translate – a shift or move (𝑥, 𝑦) → (𝑥 + 𝑎, 𝑦 + 𝑏)
Rotation – a turn 90° (𝑥, 𝑦) → (−𝑦, 𝑥)
(rules are for counterclockwise) 180° (𝑥, 𝑦) → (−𝑥,−𝑦)
270° (𝑥, 𝑦) → (𝑦−, 𝑥)
Reflection – a flip x-axis (𝑥, 𝑦) → (𝑥,−𝑦)
y-axis (𝑥, 𝑦) → (−𝑥, 𝑦)
Dilation – a stretch/shrink (𝑥, 𝑦) → (𝑘𝑥, 𝑘𝑦)
Line of symmetry – a line drawn through a shape such that both sides are
congruent figures
Point symmetry – a shape that is congruent when turned 180°
Rotational symmetry – a shape that is congruent when turned a specific
number of degrees
Degree of rotational symmetry – 360 divided by the number of rotations
Dilating a line Keep the slope and dilate the y-intercept
SSS SAS
ASA AAS
HL
13
Constructions
Perpendicular bisector
Perpendicular through a point on a line
Perpendicular through a point not on a line
Parallel line through a point
Angle bisector
Congruent angle
Equilateral triangle
Isosceles triangle
90° angle
45° angle 60° angle 30° angle
Inscribed hexagon
Inscribed equilateral
triangle
Inscribed square given the center
Inscribed square not given
the center
14
Polygons
Polygon – a 2D closed shape made of 3 or more straight lines
Regular Polygon – all sides and angles congruent
Sum of the Interior Angles 180(𝑛 − 2)
Sum of the Exterior Angles 360
One Interior Angle of a Regular Polygon 180(𝑛−2)
𝑛
One Exterior Angle of a Regular Polygon 360
𝑛
Interior and exterior angles are a linear pair
Lines of symmetry n
Minimal degree of rotation 360
𝑛
Coordinate Geometry Proofs
Use distance to prove congruent sides
Use midpoint to prove segments bisect
Use slope to prove parallelism – same slopes
Use slope to prove perpendicularity – negative reciprocals slopes
Midpoint Distance Slope Diagonal 1
Diagonal 2
Proves: Parallelogram (=) Rectangle (=) Rhombus (=) Square (=)
Rectangle (≅) Square (≅)
Rhombus (neg. rec.) Square (neg. rec.)
*** If your math does not support the shape, adjust your therefore statement
to say it should be what the questions wants, but your math is incorrect. ***
15
What To Do When You Need To Prove
The Conclusion Statement
Therefore _________________ __________ a ____________________ because (the name of the shape) (is/is not) (type of shape)
___________________________________________________________________.
(property you proved)
16
Quadrilateral Family Tree
17
Similar Triangles
Corresponding angles are congruent
Corresponding sides are proportional
Perimeters are proportional
Areas are proportional to the square of the scale factor
If a line intersects two sides of a triangle and is parallel to the third side, then it
divides those sides proportionally.
Left Right Base Perimeter Area2
Small ∆ Big ∆
Can be proven by:
o Angle-Angle (AA)
o Side-Side-Side (SSS) where the sides are proportional
o Side-Angle-Side (SAS) where the sides are proportional
Proofs can include
Statement Reason #) ∆ ABC ~ ∆DEF #) AA
#) 𝐴𝐵
𝐷𝐸=
𝐴𝐶
𝐷𝐹 #) CSSTP
Corresponding sides of similar triangles are proportional #) AB x DF = DE x AC
#) In a proportion, the product of the means equals the product of the extremes
Right triangles – The altitude to the hypotenuse of a right triangle forms two
triangles that are similar to each other and to the original triangle.
Little Leg
Medium Leg Hypotenuse
Small ∆
Medium ∆
Big ∆
18
Right Triangles & Trigonometry
Pythagorean Theorem 𝑎2 + 𝑏2 = 𝑐2
Converting Degrees to Radians multiply by 𝜋
180
Converting Radians to Degrees multiply by 180
𝜋
Trigonometric Rations for RIGHT Triangles
sin 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑠𝑒 cos 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑠𝑒 tan 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Law of Sines sin𝐴
𝑎=
sin𝐵
𝑏=
sin𝐶
𝑐 (works for all triangles)
Circles
Area = πr2 Circumference = 2πr
Arc Length 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒=
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑟𝑐
360 𝑜𝑟 2𝜋
Area of a Sector 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟
𝑎𝑟𝑒𝑎=
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑟𝑐
360 𝑜𝑟 2𝜋
Equation of a circle (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2
center-radius form center: (h,k) radius = r
Equation of a circle x2 + y2 + ab + by + c = 0
standard form
Use completing the square to go from standard form to center-radius form!
Segment Lengths
PP = PP Part x Part = Part x Part WE = WE Whole x External = Whole x External
19
Angles of a Circle
Location Formula Picture
In – center ∠ = arc
In – 2 chords 2∠ = 𝑓𝑟𝑜𝑛𝑡̂ + 𝑏𝑒ℎ𝑖𝑛𝑑̂
On 2∠ = arc
Outside 2∠ = 𝑓𝑎�̂� − 𝑐𝑙𝑜𝑠�̂�
Circle Theorems
-In a circle, if central angles are congruent, then their intercepted arcs are congruent. -In a circle, central angles are congruent if their intercepted arcs are congruent.
An angle inscribed in a semicircle is a right angle.
-In a circle, congruent central angles have congruent chords. -In a circle, congruent chords have congruent central angles.
-In a circle, congruent arcs have congruent chords. -In a circle, congruent chords have congruent arcs.
20
-In a circle, parallel lines create congruent arcs. -In a circle, congruent arcs create parallel lines.
A diameter perpendicular to a chord bisects the chord and its arc.
-If two chords of a circle are congruent, then they are equidistant from the center of the circle. -If the two chords of a circle are equidistant from the center of a circle, then the chords are congruent.
If two inscribed angles of a circle intercept the same arc, then they are congruent.
At a given point on a circle, one and only one line can be drawn that is tangent to the circle.
-If a line is perpendicular to a radius at a point on the circle, then the line is tangent to the circle. -If a line is tangent to a circle, then it is perpendicular to a radius at a point on the circle.
If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents.
Rigid Motion Conclusion
Rigid motions preserve side length and angle measure which makes the shapes congruent.
21
Two-Dimensional Geometry
Shape Picture Real Life Example Formulas
Parallelogram
Side of an eraser 𝐴 = 𝑏ℎ
Rectangle
Piece of paper 𝐴 = 𝑏ℎ or 𝐴 = 𝑙𝑤
Square
Rubik’s cube sticker 𝐴 = 𝑠2
Triangle
Yield sign 𝐴 = 1
2 𝑏ℎ
Trapezoid
Elementary table 𝐴 = 1
2(𝑏1 + 𝑏2)ℎ
Circle
Cookie 𝐶 = 2𝜋𝑟 𝐴 = 𝜋𝑟2
Rotating 2D
Three-dimensional geometric solid formed when 2D shapes are continuously rotated about an axis?
Shape Axis Solid Formed Rectangle/Square Horizontal & Vertical (line of symmetry or side) Cylinder
Right Triangle Either Leg Cone Circle Diamter Sphere
22
Three-Dimensional Geometry
Shape Picture Real Life Example Surface Area Volume
Rectangular Prism
Cereal box Find the area of 6 rectangles
and add together 𝑉 = 𝐵ℎ 𝑉 = 𝑙𝑤ℎ
Triangular Prism
Toblerone bar Find the area of 2 triangles and 3 rectangles and add together
𝑉 = 𝐵ℎ
𝑉 = (1
2𝑏ℎ)𝐻
H = height of prism
Cylinder
Can of soup 𝑆. 𝐴. = 2𝜋𝑟2 + 2𝜋𝑟ℎ 𝑉 = 𝜋𝑟2ℎ
Pyramid
Egyptian Pyramid Find the area of the triangles and 1 base and add together
𝑉 =1
3𝐵ℎ
𝑉 =1
3𝑙𝑤ℎ
Cone
Traffic cone 𝑆. 𝐴. = 𝜋𝑟2 + 𝜋𝑟𝑙 𝑉 =1
3𝜋𝑟2ℎ
Sphere
Earth 𝑆. 𝐴. = 4𝜋𝑟2 𝑉 =4
3𝜋𝑟3
23
What two-dimensional figure is formed when you slice each solid?
Shape Direction of Slice 2D shape
Rectangular Prism Parallel to base Rectangle
Perpendicular to base Rectangle
Triangular Prism Parallel to base Triangle
Perpendicular to base Rectangle
Cylinder Parallel to base Circle
Perpendicular to base Rectangle
Rectangular Pyramid Parallel to base Same as base
Perpendicular to base Triangle (through height)
Trapezoid
Cone
Sphere Any direction Circle
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Frustum - the portion of a cone or pyramid that remains after its upper part has been cut
off by a plane parallel to its base, or that is intercepted between two such planes.
Cavalieri’s Principle - If, in two solids of equal altitude, the sections made by planes parallel
to and at the same distance from their respective bases are always equal, then
the volumes of the two solids are equal.
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