Download Geometry 1-2 `Big Picture`

Geometry 1-2 'Big Picture'
Definitions / Terms
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Point, line, line segment, ray, angle, angle measure, vertex, triangle
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Union ∪ (combined), Intersection ∩ (overlap)
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Area (space), Perimeter (distance around)
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Congruent (line seg = same length, angle = same measure, shape = all
sides and angles equal.)
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Collinear, coplanar (same line, same plane)
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Parallel, perpendicular, midpoint, bisect
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Angles: complementary (add to 90), supplementary (add to 180)
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Transformations: slide, rotate
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Triangle lines: median (to midpoint), altitude (perpendicular)
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Auxiliary line (a line added to a drawing)
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Kinds of triangles: scalene, isosceles, equilateral, equiangular, right,
acute, obtuse
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Kinds of polygons: quadrilateral, rhombus, kite, parallelogram,
rectangle, trapezoid, isosceles trapezoid, square, pentagon, hexagon,
heptagon, octagon, nonagon, decagon, dodecagon, pentadecagon.
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Polygon terms: regular (equilateral/equiangular), diagonal (line
between corners not a side)
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Transversal (line cutting 2 other lines)
Proofs
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2 column proofs (statements – reasons)
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Flowchart proof
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Indirect proof (prove something is not true by proving
contrapositive)
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Counter-example: one example that proves a statement is not true.
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General strategies:
o Start with givens (in proof and on diagram)
o What else do the givens tell you? Add lines to proof.
o Try working backwards from what you want to prove.
o Overlapping triangles...draw separately.
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Triangle congruency proofs:
o Always include a line with
∆ something ≅ ∆ something and reason is a triangle
shortcut (SSS, SAS, ASA, HL, AAS)
o Might use CPCTC after triangles are proved congruent.
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Parallel line proofs:
o If given lines parallel can prove pairs of angle congruent.
o If given angles, can prove line parallel.
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Make a proof problem from words: draw picture, write 'prove'
statement, then add given statements.
Angles
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Degrees, whole circle = 360 deg.
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Measuring (protractor)
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'measure of'' an angle = how many degrees
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Degrees-Minutes-Seconds, converting
(multiply or divide by 60)
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'Clock problems' (angle between hands)
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Kinds of angles: acute, right, obtuse,
straight.
Area and Perimeter
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Area = space inside a shape
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Perimeter = add up sides (length around)
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Rectangle:
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A = bi h
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P = 2b + 2h
• Triangle:
1
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A = b ih
2
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P = add the sides
• Circle:
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A = π r2
o P=circumference, C
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C = 2π r
Lines
midpoint: M =  x1 + x2 , y1 + y2 


2 
 2
slope: m = y2 − y1
x2 − x1
parallel: slopes are the same
perpendicular: slopes are negative
reciprocals e.g. − 3 and 5
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Logic
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Conditional Statement: If p, then q,
written p ⇒ q
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Statement: p ⇒ q
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Converse: q ⇒ p
Inverse: ∼ p ⇒∼ q
Contrapositive: ∼ q ⇒∼ p
If statement is true, only
contrapositive is definitely true.
Chain of reasoning: If a ⇒ b and
b ⇒ c , then a ⇒ c
Counting and Probability
• Counting (simple cases):
o List all possibilities and count
o Use a tree diagram
o Boxes method: one box for each
selection, number in box is number of
options for that selection.
• Counting 'choosing problems' – Order matters =
Permutations
n!
o
n Pr =
(n − r )!
o 'Boxes' method (easiest)
• Counting 'choosing problems' – Order doesn't
matter = Combinations
n!
o
n Cr =
(n − r )!r !
o
=Pascal's
Triangle (n=row,
C
n r
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r=column, both start counting from
zero.)
o Boxes method: but remember to divide
by boxes for how many ways those
selected can be rearranged.
Probability, P is how likely an outcome is to
happen.
o P=0, impossible
o P=1, certain
o P=between 0 and 1 might happen
number desired outcomes
o
P=
number total outcomes
Theorems
• Angles:
o all right angles are congruent, all straight angles are
congruent.
o angles supp./comp. to congruent angles are congruent
o vertical angles are congruent
o angles both supplementary and congruent are right angles
• Triangles:
o interior angles add to 180 deg.
o exterior angle = sum of remote interior angles.
o 2 shorter side lengths added > longest side length
o shortest side opposite smallest angle, largest side opposite
largest angle.
o Congruent means all 3 angles and all 3 sides congruent
CPCTC.
o Prove congruent by matching 3 things (shortcuts): SSS,
SAS, ASA, HL (right triangles only), AAS
∆ ⇔ ∆ (only true within one triangle, not between
triangles)
o Isosceles triangles: 2 sides congruent, base angle congruent.
Parallel lines:
o parallel lines = alt. int. angles congruent
o parallel lines = alt. ext. angles congruent
o parallel lines = corr. angles congruent
o parallel lines = same side int. angles supplementary
o parallel lines = same side ext. angles supplementary
Parallelograms:
o opp. sides congruent
o opp. sides parallel
o opp. angles congruent
o consecutive angles congruent
Polygons:
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Si = (n − 2)180 (for triangle: 3 angles add to 180 deg.)
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Se = 360
o
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360 E= one external angle. Only for regular polygons
n
n
(
n − 3) d=number of diagonals
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d=
2
o n is the number of sides in a polygon
Addition/Subtraction/Multiplication/Division properties
Transitive/Substitution properties
Circle: radii are congruent
2 pts equidistant from endpts of a line segment make a perpendicular
bisector to the line segment.
Pts on a perpendicular bisector of a line segment are equidistant from
segment endpoints.
o
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E=
Kinds of problems
• Clock problems (find angle between hands)
• 'Crook problems'
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Parallel lines: given parallel lines, find angles
Parallel lines: given angles, say which lines are parallel
Give most descriptive name of polygon
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Lines cross, find all angles
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Given area or perimeter, find sides
Given sides find area or perimeter
Ratios: (usually just add an x...make a drawing)
Find midpoint given endpoints
Find slope of a line from points
'Reflect' shape over a line and find new coordinates
Given an angle, find complement or supplement.
Given all angles in a polygon except one, find missing angle.
Make an equation from words problems (e.g. 'Complement is 10 more
than 9 times the supplement').
Put all givens on a picture, use theorems to solve for x (vertical angles
equal, isosceles sides equal, parallel same side angles add to 180, etc).
Then plug in x to find an angle or length.
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