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Geometry 1-2 'Big Picture' Definitions / Terms • Point, line, line segment, ray, angle, angle measure, vertex, triangle • Union ∪ (combined), Intersection ∩ (overlap) • Area (space), Perimeter (distance around) • Congruent (line seg = same length, angle = same measure, shape = all sides and angles equal.) • Collinear, coplanar (same line, same plane) • Parallel, perpendicular, midpoint, bisect • Angles: complementary (add to 90), supplementary (add to 180) • Transformations: slide, rotate • Triangle lines: median (to midpoint), altitude (perpendicular) • Auxiliary line (a line added to a drawing) • Kinds of triangles: scalene, isosceles, equilateral, equiangular, right, acute, obtuse • Kinds of polygons: quadrilateral, rhombus, kite, parallelogram, rectangle, trapezoid, isosceles trapezoid, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, dodecagon, pentadecagon. • Polygon terms: regular (equilateral/equiangular), diagonal (line between corners not a side) • Transversal (line cutting 2 other lines) Proofs • 2 column proofs (statements – reasons) • Flowchart proof • Indirect proof (prove something is not true by proving contrapositive) • Counter-example: one example that proves a statement is not true. • 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. • 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. • Parallel line proofs: o If given lines parallel can prove pairs of angle congruent. o If given angles, can prove line parallel. • Make a proof problem from words: draw picture, write 'prove' statement, then add given statements. Angles • Degrees, whole circle = 360 deg. • Measuring (protractor) • 'measure of'' an angle = how many degrees • Degrees-Minutes-Seconds, converting (multiply or divide by 60) • 'Clock problems' (angle between hands) • Kinds of angles: acute, right, obtuse, straight. Area and Perimeter • Area = space inside a shape • Perimeter = add up sides (length around) • Rectangle: o A = bi h o P = 2b + 2h • Triangle: 1 o A = b ih 2 o P = add the sides • Circle: o A = π r2 o P=circumference, C o 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 • • • • 5 3 Logic • Conditional Statement: If p, then q, written p ⇒ q • Statement: p ⇒ q • • • • • 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 • 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: o Si = (n − 2)180 (for triangle: 3 angles add to 180 deg.) o Se = 360 o • • • 360 E= one external angle. Only for regular polygons n n ( n − 3) d=number of diagonals o 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 • • • • • E= Kinds of problems • Clock problems (find angle between hands) • 'Crook problems' • • • Parallel lines: given parallel lines, find angles Parallel lines: given angles, say which lines are parallel Give most descriptive name of polygon • Lines cross, find all angles • • • • • • • • • 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. •