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

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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

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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 ̅̅

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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

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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

24

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.