Geometry AIR Study Guide
Table of Contents Topic Slide Formulas 3 5 Angles 6 Lines and Slope 7 Transformations 8 Constructions 9 10 Triangles 11 Congruency and Similarity 12 Right Triangles Only 13 Other Triangles 14 Quadrilaterals 15 Proofs 16 Polygons 17 Circles 18 3D Solids 19 Probability 20 Combinations and Permutations 21
Formulas (not on provided formula sheet) distance = x 2 x 1 2 + y 2 y 1 2 midpoint = x 2 + x 1 2 slope = y 2 y 1 x 2 x 1, y 2 + y 1 2 Sum of Interior Angles: n 2 180 One: n 2 180 n One Exterior/Central Angle: 360 n x 45 x x 2 45 n 3 30 n 60 2n sin = opp hyp cos = adj hyp tan = opp adj SOHCAHTOA a x b Geometric Mean a x = x b
Formulas, continued A A A A Equation of a Circle x h 2 + y k 2 = r 2 Center: (h, k) Radius: r Angle = arc Angle = arc 2 Angle = arc 2 Angle = (sum of arcs) 2 Length of Arc = n 360 2πr A A D B a d Area of Sector = n 360 πr2 Angle = far near 2 B D A + B B = C C + D D A A + B B = D 2 b c a c = b d
Surface Area Formulas: Formulas, continued
Angles 1 2 3 4 5 6 7 8 l m Alternate Interior Angles: Alternate Exterior Angles: 4, 5; 3, 6; congruent when l m 1, 8; 2, 7; congruent when l m Corresponding Angles: Consecutive Interior Angles: 1, 5; 2, 6; 3, 7; 4, 8; congruent when l m 4, 6; 3, 5; supplementary when l m Complementary Angles: Supplementary Angles: Vertical Angles: add to 90 add to 180 opposites are equal
Lines and Slope Parallel Lines never touch coplanar (same plane) slopes equal Perpendicular Lines intersect at 90 coplanar slopes opposite reciprocals (multiply to - 1)
Transformations Translation slide Reflections flip Rotation turn Dilation size *Scale factor/scalar: how much object enlarges/shrinks (multiply) Line Symmetry Rotational Symmetry
Key: First step Second step Third step Fourth step Congruent Line Segment use compass to mark length Constructions Congruent Angles A B A B Angle Bisector Parallel Lines
Constructions, continued Perpendicular Bisector Perpendicular from point on line Perpendicular from point not on line Key: First step Second step Third step
Triangles Does a Triangle Exist? -given x, y, z x + y > z x + z > y y + z > x Acute, Right, or Obtuse? *Apply Pythagorean Theorem a 2 + b 2 > c 2 acute a 2 + b 2 = c 2 right a 2 + b 2 < c 2 obtuse [example] 4, 9, 13 4 2 + 9 2 13 2 16 + 81 169 97 < 169 Obtuse If yes to all three, makes a triangle! Range of a Missing Side -given a, b (missing side x) Case 1: a or b largest side a + x > b, solve for x Case 2: x largest side a + b > x, solve for x Order Sides and/or Angles -from smallest to largest or vice versa -length of side corresponds to measurement of opposite angle.
Congruency and Similarity Congruent *same size AND shape *CPCTC Corresponding Parts of Congruent Triangles are Congruent Similar *same shape, DIFFERENT SIZE *proportional SSS Congruency *all sides equal SAS Congruency ASA Congruency AAS Congruency HL * Congruency *right triangle SSS Similarity * All sides proportional AA(A) Similarity *all angles equal SAS similarity *corresponding sides proportional, the angle between equal
Right Triangles Only *Find the missing side, given: leg TWO SIDES Pythagorean Theorem a 2 + b 2 = c 2 a c leg must be hypotenuse b ONE SIDE, ONE ANGLE Pythagorean Theorem Shortcuts Trig SOHCAHTOA x 45 x x 2 45 n 3 30 n 60 2n sin = opp hyp h o cos = adj hyp tan = opp adj *in relation to angle given a
Other Triangles **need a side length or angle and triangle is not a right angle** B a c C b A Law of Sines *useful with two sides and one angle (need another angle), or one side and two angles (need another side) sin A a sin B = b sin C = c Law of Cosines *useful with three sides (need one angle), or one angle and two sides (need third side) c 2 = a 2 + b 2 2ab cos C b 2 = a 2 + c 2 2ac cos B a 2 = b 2 + c 2 2bc cos A
Quadrilaterals > > Parallelogram Opposite sides Diagonals bisect each other Rectangle Parallelogram Diagonals bisect, > Trapezoid Square Parallelogram, rectangle, and rhombus Diagonals bisect,,, bisect opposite angles Rhombus Parallelogram Diagonals bisect,, bisect opposite angles > > > Isosceles Diagonals Avg. bases to find median Kite
Proofs Reflexive Property a = a Symmetric Property a + b = b + a ab = ba Transitive Property If a = b and b = c, then a = c First line or two normally what is given in the problem, the last line is what you are trying to prove. General process: 1. Mark any given information in picture (if no picture, might be helpful to make one!) 2. Follow proof, mark additional information in proof to picture 3. Read through proof line by line. What are they doing? How did they get each line? Know vocabulary and properties these will provide you with most of your answers! Need hints? Look at your answer options the correct answer is there somewhere!
Polygons *Circles are not polygons! *polygons do not have curves! *2-D Concave vs. Convex Regular: equilateral, equiangular Interior Angle Central Angle Exterior Angle Sum of Interior Angles: n 2 180 n 2 180 one interior angle: n Sum of Exterior/Central Angles: 360 one exterior/central angle: 360 n
radius Circles diameter chord Sector Arc common internal tangent common external tangent secant Central angle Co-centric circle *share same center tangent Inscribed angle Chord length: a c = b d
3D Solids Volume: how much space on the inside Surface Area: areas of all the outside faces added together *face: flat side/surface Lateral Area: area of vertical surfaces (does not include bases) Read questions carefully! Similar Solids: have three ratios for use in proportions original/scale factor: surface area: volume: a b a 2 : b 2 a 3 : b 3 *must use correct ratio in proportion!
Probability *The likelihood that an event will occur. number of what is wanted total More than 1 event? More than one fraction! >multiply your fractions< A jar has 5 red marbles, 3 green marbles, 2 yellow marbles, and 10 blue marbles. Independent Events: A and B are separate. What is the probability of randomly drawing a marble and getting blue, replacing it, then selecting another blue marble? 10 20 10 20 = 100 400 = 1 4 Dependent Events: B changes based on A. What is the probability of randomly drawing a marble and getting blue, then immediately selecting another blue marble? 10 20 9 19 = 90 380 = 9 38 More Independent Events: P A and B = P A B = P A P B P A or B = P A B = P A + P B P A B Conditional Probability of Independent Events Probability of B, given A P B A = P(A and B) P(A)
Combinations and Permutations how many ways can you pick! Factorial: multiply number by every number below until you get to 1 5! = 5 4 3 2 1 10! = 10 9 8 7 6 5 4 3 2 1 Permutation: order is important *a combination lock is actually a permutation Pick 1 st, 2 nd, 3 rd place Arrange [objects, letters, etc.] Notation: n P r Formula: n P r = (n r)! *easiest to use calculator n=total number r=how many wanted n! Combination: order is not important Toppings on a pizza Pick a number of items/people from a group Notation: n C r Formula: n C r = n! r!(n r)! *easiest to use calculator