Geometry 1-2 'Big Picture' Angles Area and Perimeter • Degrees, whole circle = 360 deg. • Area = space inside a shape • • Definitions / Terms Measuring (protractor) Perimeter = add up sides (length around) • • • Point, line, line segment, ray, angle, angle measure, vertex, triangle 'measure of'' an angle = how many degrees Rectangle: • o A= bi h • Union ∪ (combined), Intersection ∩ (overlap) Degrees-Minutes-Seconds, converting (multiply or divide by 60) o P=2 b + 2 h • Area (space), Perimeter (distance around) • 'Clock problems' (angle between hands) • Triangle: • Congruent (line seg = same length, angle = same measure, shape = all • Kinds of angles: acute, right, obtuse, 1 sides and angles equal.) o = i straight. A b h • Collinear, coplanar (same line, same plane) 2 o = • Parallel, perpendicular, midpoint, bisect P add the sides • Angles: complementary (add to 90), supplementary (add to 180) • Circle: • Transformations: slide, rotate o A= π r 2 • Triangle lines: median (to midpoint), altitude (perpendicular) Lines o P=circumference, C • Auxiliary line (a line added to a drawing) + + o C= 2π r • midpoint: x1 x 21 y y 2 • M = , Kinds of triangles: scalene, isosceles, equilateral, equiangular, right, 2 2 acute, obtuse • y− y • Kinds of polygons: quadrilateral, rhombus, kite, parallelogram, slope: m = 2 1 − rectangle, trapezoid, isosceles trapezoid, square, pentagon, hexagon, x2 x 1 heptagon, octagon, nonagon, decagon, dodecagon, pentadecagon. • parallel: slopes are the same Counting and Probability • Polygon terms: regular (equilateral/equiangular), diagonal (line • perpendicular: slopes are negative • Counting (simple cases): between corners not a side) reciprocals e.g. − 3 and 5 o List all possibilities and count • Transversal (line cutting 2 other lines) 5 3 o Use a tree diagram o Boxes method: one box for each selection, number in box is number of options for that selection. Proofs Logic • Counting 'choosing problems' – Order matters = • • 2 column proofs (statements – reasons) Conditional Statement: If p, then q, Permutations • Flowchart proof written ⇒ p q o n! • Indirect proof (prove something is not true by proving P = • Statement: p⇒ q n r − contrapositive) (n r )! • o • Counter-example: one example that proves a statement is not true. Converse: q⇒ p 'Boxes' method (easiest) • • General strategies: • Inverse: ∼p⇒ ∼ q Counting 'choosing problems' – Order doesn't o matter = Combinations Start with givens (in proof and on diagram) • Contrapositive: ∼q⇒ ∼ p o o n! What else do the givens tell you? Add lines to proof. • C = o Try working backwards from what you want to prove. If statement is true, only n r (n− r )!! r contrapositive is definitely true. o Overlapping triangles...draw separately. o =Pascal's Triangle (n=row, • nC r • Triangle congruency proofs: Chain of reasoning: If a⇒ b and r=column, both start counting from o Always include a line with b⇒ c , then a⇒ c zero.) ∆ ≅ ∆ and reason is a triangle something something o Boxes method: but remember to divide shortcut (SSS, SAS, ASA, HL, AAS) by boxes for how many ways those o Might use CPCTC after triangles are proved congruent. selected can be rearranged. • Parallel line proofs: • Probability, P is how likely an outcome is to o If given lines parallel can prove pairs of angle congruent. happen. o If given angles, can prove line parallel. o P=0, impossible • Make a proof problem from words: draw picture, write 'prove' o P=1, certain statement, then add given statements. o P=between 0 and 1 might happen o number desired outcomes P = number total outcomes Theorems Kinds of problems • Angles: • Clock problems (find angle between hands) o all right angles are congruent, all straight angles are • 'Crook problems' congruent. o angles supp./comp. to congruent angles are congruent • Parallel lines: given parallel lines, find angles o vertical angles are congruent • Parallel lines: given angles, say which lines are parallel o angles both supplementary and congruent are right angles • Give most descriptive name of polygon • Triangles: o interior angles add to 180 deg. • Lines cross, find all angles o exterior angle = sum of remote interior angles. o 2 shorter side lengths added > longest side length • Given area or perimeter, find sides o shortest side opposite smallest angle, largest side opposite • Given sides find area or perimeter largest angle. • Ratios: (usually just add an x...make a drawing) o Congruent means all 3 angles and all 3 sides congruent • Find midpoint given endpoints CPCTC. • o Find slope of a line from points Prove congruent by matching 3 things (shortcuts): SSS, • SAS, ASA, HL (right triangles only), AAS 'Reflect' shape over a line and find new coordinates ∆ ⇔ ∆ • Given an angle, find complement or supplement. o (only true within one triangle, not between • Given all angles in a polygon except one, find missing angle. triangles) • Make an equation from words problems (e.g. 'Complement is 10 more o Isosceles triangles: 2 sides congruent, base angle congruent. than 9 times the supplement'). • Parallel lines: • Put all givens on a picture, use theorems to solve for x (vertical angles o parallel lines = alt. int. angles congruent equal, isosceles sides equal, parallel same side angles add to 180, etc). o parallel lines = alt. ext. angles congruent Then plug in x to find an angle or length. 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 = − (for triangle: 3 angles add to 180 deg.) Si ( n 2)180 o = Se 360 o 360 E = E= one external angle. Only for regular polygons n o n( n − 3) d = d=number of diagonals 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. .