The Ungerecht Family Tree

• Angles and Vectors
• Polygons
• Triangles
• Circles, Chords, etc
• Graphing curves, circles, etc.

The sum of angles in a polygon = 180 (n - 2), where n = number of sides. This can also be written as: 180 + 180 (n-3). (Basically you add in another triangle when adding a side to polygon)

Definitions:

• isosceles - 2 sides the same length
• equilateral - all 3 sides the same length
• similar triangles - triangles that have same shape (all 3 angles), but size can be different. The sides have same ratios
• congruent triangles - triangles that have the same shape and size
• CPCT - corresponding parts of congruent triangles (are equal)

Centroid of a triangle - center (where the lines bisecting each angle will meet)

• average the x corner values, and average of y corner values
• the bisecting line will have 2/3 of its length between the corner and the centroid and 1/3 from centroid to far side of triangle
• the six small triangles formed by bisecting lines all have equal area

Area of a triangle = base*height/2

Heron's formula - area of any triangle, with sides of length a,b,c, don't know height

semiperimeter (sp) = perimeter/2 = (a+b+c)/2
area = √(sp(sp-a)(sp-b)(sp-c))

Theorems to prove triangles are congruent:

• AAS - angle-angle-side
• ASA - angle-side-angle
• SAS - side-angle-side
• SSS - side-side-side
• RHS - right-angle, hypotenuse, side (basically ASS, which doesn't work for all angles, but does for right-angle). Using Pythagorean theorem can be converted to SSS.

45-45-90 triangle: hyp = side * √2 30-60-90 triangle: short side = a, long side = a*√3, hyp = 2a

area of a right triangle = 1/2*h*w

hypotenuse is the side opposite the right angle, opposite is the side opposite the given angle, adjacent is the side next to the given angle

SOHCAHTOA (Soak a toe-ah): Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj

law of sines:

sin(A)/a = sin(B)/b = sin(C)/c  (sometimes a/sin(A) = ...)

law of cosines - to find an angle when all the sides are known

cos(A) = (b² + c² - a²) / (2bc)
cos(C) = (a² + b² - c²) / (2ab), cos(B) is same pattern
  (side a is opposite angle A, etc)
rewriting it: c = √(a² + b² - 2abcos(C))

• chord has two endpoints on a circle
• secant is a line that contains a chord, but extends beyond the circle
• Intercepted arc is the part of the circle contained within the two lines
• Central angle is angle of two lines from the center of the circle
• Inscribed/Interior angle has two points and the vertex all on the circle itself
• central angle is same as the degrees of intercepted arc

if two chords AB, CD intersect at P, then AP * PB = CP * PD

Interior angle = 1/2 of intercepted arc

all inscribed angles going to the two same points on the circle have the same angle
angle of intersecting secants theorem: 
    angle formed by the secant intersection = (opposite arc - adjacent arc)/2

An angle outside the circle with two secants (or tangents) will have an angle

that is 1/2 * (difference of the intercepted arcs)

coterminal angle - the rest of the circle outside the angle.

eg. angle of 30', the coterminal is 330'
in radians, use the absolute value.  
    2pi - |angle|, or |angle| - 2pi for the negative angle

area of an arc:

a = rad*r²/2  .... comes from a = πr² for full circle, and the 
                    proportion of a circle in the arc is rad/2π
                    so a = (rad/2π)*(πr²)

• https://www2.clarku.edu/faculty/djoyce/trig/identities.html
• 11 pages of definitions, postulates and theorems: http://www.ouchihs.org/ourpages/auto/2013/7/26/52822673/Geo-PostulatesTheorems-List.pdf

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