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Geometry A Overview General Formulas Distance Formula: Given P=(x1,y1) and Q = (x2,y2), distance is d = ( x1 x2 ) 2 ( y1 y2 ) 2 x x2 y1 y 2 Midpoint Formula: Given P=(x1,y1) and Q = (x2,y2), midpoint is 1 , 2 2 y y2 Slope Formula: Given P=(x1,y1) and Q = (x2,y2), slope is m = 1 x1 x2 Symbols Symbol ≠ ≥ ≤ ¬, ~ Meaning Not equal Congruent Greater than or equal to Less than or equal to Not And Or Symbol ∆ ∩ Meaning Triangle Angle Perpendicular Union Intersection Implies If and only if Symbol ε ε Meaning Therefore Since There exists Such that For all or for every Is an element of Is not an element of Introduction to Geometry Vocabulary Angle ( ): formed by 2 rays that have same endpoint Bisect: to divide segment or angle into 2 congruent parts Bisector: point or ray that bisects segment or angle Collinear: points on same line Congruent ( ): same measure Line ( ): made up of points and is straight (symbol: arrowhead at each end) Line Segment/Segment ( ): has 2 endpoints, can be measured. (named after 2 endpoints) Midpoint: point that bisects a segment. Must be collinear Noncollinear: points that don’t lie on same line Ray( ): begins at an endpoint & then extends infinitely far in only 1 direction. Tick Marks: indicate congruency Trisect: to divide segment or angle into 3 congruent parts Trisectors: points or rays that trisect segment or angle Algebraic Phrases English Word Algebraic Translation Complement 90 –x Difference Equal = Greater than +, > Increased by + Less than -, < English Word Number Opposite of a number Product Sum Supplement Algebraic Translation N -N * + 180 – x Example: Angle Measures 1. Find the measures of 2 supplementary angels if the measure of 1 angle is 6 less than 5 times the measure of the other angle. 180 –x = 5x - 6; angles149, 31 1 Rev B Geometry A Overview Angle Relationships Vocabulary Acute Angle: between 0º and 90º Adjacent Angles: 2 that lie in same plane, have common vertex & side but no common interior point Complementary Angles: 2 angles whose measures have sum of 90º Linear Pair: pair of adjacent angels whose non-common sides are opposite rays. Obtuse Angle: between 90º and 180º Perpendicular ( ): lines that form 90º angles. Intersect to form congruent adjacent angles; rt angle symbol indicates lines are perpendicular Right Angle: 90º Straight Angle: 180º Supplementary Angles: 2 angles whose measures have sum of 180º Vertical Angles: 2 nonadjacent angles formed by 2 intersecting lines. Example: Angle Relationships Given the diagram to the right, identify the following: 1. Adjacent angles: AED & AEB, CED & CEB 2. Vertical Angles: DEA & CEB, AEB & CED 3. Linear Pair: DEA & BEA, DEC & BEC 4. Given the diagram to the right, identify all angle relationships. 1 and 5 are exterior angles 1 sup 2 , 1 + 2 =180, 1 & 2 are linear pair 4 sup 5 , 4 + 5 =180, 4 & 5 are linear pair 2 + 3 + 4 = 180 1 = 3 + 4, 1 > 3, 1 > 4 5 = 2 + 3 , 5 > 2, 5 > 3 Reasoning & Proofs Vocabulary Compound Statement: 2 or more statements joined together Conclusion: phrase immediately following then Conditional Statement/implication: if-then statement Conjecture: educated guess based on known info Conjunction ( ): compound statement formed by joining 2 or more statements with word and; true when both are true Contrapositive: if p then q if ~q then ~p Converse: statement associated with if p then q having form if q then p Disjunction ( ): compound statement formed by joining 2 or more statements with word or; true when at least 1 statement is true Hypothesis: if clause of conditional statement/implication Inverse: if p then q if ~p then ~q Negation (~ or ): opposite meaning as well as opposite truth value; represented by ~p Statement: sentence that is either true or false; represented by p 2 Rev B Geometry A Overview Assumptions: We can assume the following Straight lines & angles are as they appear Collinearity of points Betweenneess of points Relative positions of points Adjacent , linear pairs, supplementary We cannot assume the following Right angles Congruent segments Congruent angles Relative size of segments & angles Perpendicular Lines Examples: Assumptions Use the figure below, state whether you can make the following assumptions. If not, indicate the reason. 1. AOC = 90º no, cannot assume right angles 2. AO OB 3. FOB = 50º no, cannot assume congruency yes, given Logic Table Implication p q Converse qp Inverse Contrapositive ~p ~q ~q ~p Negation ~p (read “not p”), ~~p means p Example: Converse, Inverse & Contrapositive 1. Write the converse, inverse and contrapositive of the following true statement: “If 2 angles are right angles, then they are congruent.” Which of your new statements are true? Which are false? Implication: If 2 angles are right angles, then they are congruent. (True) Converse: If 2 angles are congruent, then they are right angles. (False) Inverse: If 2 angles are not right angles, then they are not congruent. (False) Contrapositive: If 2 angles are not congruent, then they are not right angles. (True) Examples: Truth Tables Construct a truth table for each compound statement. a. p ~q P Q ~q P ~q T T F T F=F T F T T T=T F T F FF = F F F T F T=F b. ~p ~q p q ~p T T F T F F F T T F F T ~q F T F T ~p ~q F F=F F T=T T F = T T T=T Properties of Equality Reflexive Property Symmetric Property Transitive Property Addition and Subtraction Properties Multiplication and Division Properties Distributive Property Substitution Property For every number a, a = a. For all numbers a & b, if a = b, then b = a If a = b and b = c then a = c If a = b then a + c = b + c and a – c = b – c a b If c 0 and a = b, then ac = bc and = c c A(b+c) = ab + ac If a = b, then a may be replaced by b in equation Properties of Inequalities: 3 Rev B Geometry A Overview a < b, a = b or a > b If a < b and b < c then a < c If a > b and b > c then a > c If a > b then a + c > b + c and a – c > b – c Addition and Subtraction Properties If a < b then a + c < b + c and a –c < b – c Multiplication and Division Properties a b If c > 0 and a < b, then ac < bc and < c c a b If c > 0 and a > b, then ac > bc and > c c a b If c < 0 and a < b, then ac > bc and > c c a b If c < 0 and a > b, then ac < bc and < c c Comparison Property Transitive Property Properties Reflexive: AB AB (any segment/angle is congruent to itself) Symmetric: if AB CD then CD AB Transitive: if AB CD and CD EF then AB EF Segment Addition Postulate: if B is between A and C, then AB + BC = AC Examples: Algebraic Properties State the property that justifies each statement: If x = y, then x + 8 = y + 8 Addition Property of Equality If x = y, then 8x = 8y Multiplication Property of Equality If a = 5 and 5 = b, then a = b Transitive Property of Equality Theorems and Postulates Angles Right Angle Theorem: If 2 angles are right angles, then they are congruent Straight Angle Theorem: If 2 angles are straight angles, then they are congruent Supplement Theorem: If 2 angles form a linear pair, then they are supplementary. Complement Theorem: if non-common sides of 2 adjacent s form right , then s are complementary If (or segments) are to same or (or segments) then they are to each other. (Transitive Property) Vertical Angle Theorem: if 2 angles are vertical angles, then they are congruent. Same Supplements: Angles supplementary to same or congruent angles are congruent. Same Complements: Angles complementary to same or congruent angles are congruent. Right Angles Perpendicular lines intersect to form 4 right angles All right angles are congruent Perpendicular lines form congruent adjacent angles Right Angle Theorem: If 2 angles are congruent & supplementary, then each angle is a right angle If 2 congruent angels form a linear pair, then they are right angles Points, Lines & Planes Through any 2 points, there is exactly 1 line. 4 Rev B Geometry A Overview Through any 3 noncollinear points, there is exactly 1 plane. A line contains at least 2 points A plane contains at least 3 noncollinear points If 2 points lie in plane then entire line containing those points lies in that plane. If 2 lines intersect, then their intersection is exactly 1 point. If 2 planes intersect, then their intersection is a line Midpoint Theorem: If M is midpoint of AB, then AM MB Segment Addition Postulate: if B is between A and C, then AB + BC = AC. Or if AB + BC = AC, then B is between A and C. General: Addition Property: If segments/angles are added to segments/angles, then the sums are . Subtraction Property: If segment /angle is subtracted from segments/angles, the differences are . Multiplication Property: If segments/angles are , their like multiples are . Division Property: If segments/angles are , their like divisions are All radii of a circle are congruent. Parallel, Perpendicular or Neither? When asked to determine whether a set of lines is parallel, perpendicular or neither, you need to find the slope. If the slopes are the same then the lines are parallel If the slopes are opposite reciprocal then the lines are perpendicular Parallel Lines & Transversal Vocabulary: Alternate Interior Angles: pair of angles in int of figure formed by 2 lines & transversal lying on alternate sides of transversal & having different vertices (forms a Z) Alternate Exterior Angles: pair of angles in ext of figure formed by 2 lines & transversal lying on alternate sides of transversal & having different vertices (forms opposite V) Corresponding Angles: in figure formed by 2 lines & transversal, pair of angles on same side of transversal, 1 in int & 1 in ext having different vertices (forms F) Exterior: outer part of figure Interior: inner part of figure Parallel Lines: coplanar lines that never intersect Transversal: line that intersects 2 coplanar lines in 2 distinct points Example: a. Which of the lines in the figure is a transversal? EF E b. Name all pairs of alternate interior angles. 3,6 and 4,5 c. Name all pairs of corresponding angles. 2,5 and 3,8 and 1,6 and 4,7 d. Name all pairs of alternate exterior angles. 2,7 and 1,7 F e. Name all pairs of interior angles on same side of transversal. 3,5 and 4,6 f. Name all pairs of exterior angles on the same side of the transversal. 2,8 and 1,7 Angles and Parallel Lines Theorems: 5 Rev B Geometry A Overview Alternate Interior Angles: If 2 parallel lines are cut by transversal then each pair of alternate interior angles are congruent. (║Alt int ) a if a ║ b 1 2 1 b 2 Alternate Exterior Angles: If 2 parallel lines are cut by transversal then each pair of alternate exterior angles are congruent. (║Alt ext ) a 1 if a ║ b 1 8 b 8 Corresponding Angles: If 2 parallel lines are cut by transversal then each pair of corresponding angles are congruent. (║corresponding ) 1 a if a ║ b 1 5 5 b Same Side Interior Angles: If 2 parallel lines are cut by transversal then each pair of interior angles on the same side of transversal are supplementary. 3 5 a b if a ║ b 3 5 180 Same Side Exterior Angles: If 2 parallel lines are cut by transversal then each pair of exterior angles on the same side of transversal are supplementary. a 1 if a ║ b 1 7 180 b 7 Perpendicular Transversal: In a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the other. a if a ║ b and ca cb b c Transitive Property of Parallel Lines: If 2 lines are parallel to a 3rd line, then they are parallel to each other. a if a ║ b and b ║ c a ║c b c Congruent/Supplementary: If 2 parallel lines are cut by transversal then any pair of angles a b x 180-x 180-x x x 180-x 180-x x || alt int , alt ext , corresponding , ssi = 180, sse = 180 Examples: Angles and Parallel Lines 1. If c || d, find 1 2. If a || b, find 1 6 Rev B Geometry A Overview c 1 d 2x + 10 a 3x + 5 100 b Since c ||d, alt int , 2x + 10 = 3x + 5 x=5 2 (5) + 10 = 20 1 c 40 This is a crook problem. 1 is determined by the following 1. 100 + part of 1 = 180 (SSI) = 80 2. 40 part of 1 (Alt Int) = 40 Therefore, 1 = 80 + 40 = 120 Since c ||d, corresponding , So 1 = 20 Proving Lines are Parallel Two lines can be proven to be parallel by: Alternate Interior Alternate Exterior Corresponding Same Side Interior (SSI) supplementary Same Side Exterior (SSE) supplementary Same slope Triangle Properties Vocabulary: Exterior Angle: formed by 1 side of ∆ & extension of another side Interior: inside Remote: far away Remote Interior Angle: interior of ∆ not adjacent to given exterior. Interior farthest from exterior Theorems: Angle Sum Theorem: Sum of measure of angles of a triangle is 180. No Choice Theorem/3rd Angle Theorem: if 2 angles of triangle are to 2 angles of another triangle, then the 3rd angle of the triangles are . Exterior Angle Theorem: The measure of the exterior angle of a triangle is equal to the sum of the measures of the 2 remote interior angles. Exterior Angle Inequality Theorem: if is exterior of a triangle then its measure is greater than the measure of its corresponding remote interior. Side Angle Theorem: longest side of a triangle is opposite the largest in triangle. Angle Side Theorem: largest of a triangle is opposite the longest side in triangle. Triangle Inequality Theorem: sum of lengths of any 2 sides of triangle is greater than the length of the 3rd side. (Hint: true if sum of smallest & middle > largest) Examples: Determining if it’s a triangle Determine whether the given measures can be the lengths of sides of a triangle. a. 2, 3, 4 b. 6, 8, 14 2+3>4 6 + 8 > 14 yes no Examples: Determining range of 3rd side of triangle 7 Rev B Geometry A Overview Find the range for the measure of the 3rd side of a triangle given the measures of 2 sides. a. 5 and 9 To determine the lower range, subtract the 2 numbers: 9-5 = 4 To determine the upper range, add the 2 numbers: 9 + 5 = 14 Therefore, the range is 4 < x < 14 Examples: Side Order Examples: Angle Order List the sides in order from least to greatest measure. 1st Order angles: V=28 U=70 W=82 List the angles in order from least to greatest measure. 1st Order sides: ST =7 RT = 8 RS = 13 2nd list associated with side R, S, T 2nd list sides associated with UW, VW, UV Classifying Triangles Vocabulary Acute Triangle: all 3 angels are acute Base: in isosceles triangle, the non congruent side Equiangular Triangle: all 3 angles are congruent Equilateral Triangle: all 3 sides are congruent Hypotenuse: side across from right angle in right triangle; longest side Isosceles Triangle: at least 2 sides are congruent Legs: 2 sides forming right angle in right triangle; or congruent sides of isosceles tri Obtuse Triangle: triangle with 1 obtuse angle & 2 acute angles Right Triangle: triangle with 1 right angle and 2 acute angles Scalene Triangle: all 3 sides are different lengths (no 2 sides are congruent) Vertex Angle: angle between 2 congruent legs of isosceles triangle Classifying Triangles by Angles Acute: all angles are acute Obtuse: 1 angle is obtuse Right: 1 angle is right Equiangular: all angles are congruent Classifying Triangles by Sides Scalene: no sides are congruent Isosceles: at least 2 sides are congruent Equilateral: all 3 sides are congruent Key Concepts: All equilateral triangles are isosceles but not all isosceles triangles are equilateral. If triangles are equilateral, then they are also equiangular and vice versa. If c is the length of the longest side of a triangle then o If a2 + b2 > c2, ∆ is acute o If a2 + b2 = c2, ∆ is right o If a2 + b2 < c2, ∆ is obtuse Congruent Triangles/Transformations 8 Rev B Geometry A Overview Vocabulary Congruent Polygons: same shape/size; all pairs of corresponding parts are congruent Congruent Triangles: all pairs of corresponding parts are congruent Reflection: mirror image of polygon/triangle. 2 congruent triangles can be reflections of each other Rotate: to turn polygon/triangle Translate: push/slide polygon/triangle Reflection: When done on the y-axis, the x-coordinate sign changes. When done on the x-axis, the y-coordinate sign changes. Translation: can slide up or down, right or left or diagonally. To right: Add number of units to x value Up: Add number of units to the y value To left: Subtract number of units from x value Down: Subtract number of units from y value Proving Triangle Congruency Included Angle: angle between 2 sides of ∆ Included Side: sides that compose an angle Methods for proving triangles are congruent SSS SAS ASA If there exists a correspondence If there exists a correspondence If there exists a correspondence between the vertices of 2 ∆s such between the vertices of 2 ∆s such between the vertices of 2 ∆s such that 3 sides of 1 ∆ are to 3 that 2 sides and the included that 2 s and the included side sides of other ∆, then the 2 ∆s of 1 ∆ are to corresponding of 1 ∆ are to the corresponding are . parts of the other ∆, then 2 ∆s parts of the other ∆, then 2 ∆s are . are . AAS If there exists a correspondence between the vertices of 2 ∆s such that 2 s and the non-included side of 1 ∆ are to the corresponding parts of the other triangle, then 2 ∆s are . HL If the hypotenuse and the leg of 1 right ∆ are to the hypotenuse and corresponding leg of another right ∆, then ∆s are . LL If legs of 1 right ∆ are to corresponding legs of another right ∆, then ∆s are . CPCTC What is CPCTC? Corresponding Parts of Congruent Triangles are Congruent When is it used? Only after 2 ∆ have been proven or stated to be . Cannot be used to prove ∆ Isosceles Triangles Angle-Side Theorems If sides then angles: If 2 sides of a triangle are , then the angles opposite those sides are . If angles then sides: If 2 angles of a triangle are , then the sides opposite those angles are . Parts of Triangles Vocabulary 9 Rev B Geometry A Overview Altitude: line/segment drawn from vertex to point on opposite side making them Median: line/segment drawn from 1 vertex of triangle to midpoint of opposite side. Altitude of ∆KMO: LO Altitude of ∆OLM: LN Median of ∆LKO: JL Median of ∆KMO: OL Equidistance Theorems Define: Equidistant: distance between 2 points is equal to distance between another set of points Perpendicular Bisector: line that is both perpendicular to and bisects a segment. (both altitude and median) TPEEEDPB: if 2 points are each equidistant from the endpoints of a segment, then the 2 points determine the perpendicular bisector of that segment BD is bis of AC POPBTEE: if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. If PQ is ┴ bisector of AB then PA PB Proofs Helpful hints to write a proof Make sure you know your previous definitions and theorems Use these definitions and theorems to expand the given Try to get to the prove statement Justify each step with a definition, postulate or theorem Mark your picture with your given and what you can observe Start your proof with given info. Then proceed to make conclusions from previous statements. Indirect Proofs Procedures for Indirect Proof 1. List the possibilities for the conclusion. a. Your conclusion is or is not true. 2. Assume that the negation of the desired conclusion is true. a. So the OPPOSITE of the conclusion 3. Write a “chain of reasons” until you reach an IMPOSSIBILITY or a CONTRADICTION. a. This will be a statement that either disputes a known theorem/definition/postulate or your given information. 4. State that what you assumed to start was WRONG and that the desired conclusion then must be true. Quadrilaterals 10 Rev B Geometry A Overview Parallelogram Opposite sides are parallel Opposite sides are Opposite angles are Consecutive angles are supplementary Diagonal bisect each other Kite Rectangle All properties of parallelogram All angles are right angles Diagonals are Diagonals divide rectangle into isosceles triangles. Trapezoid 2 disjoint pairs of consecutive sides are Diagonals are perpendicular 1 diagonal is perpendicular bisector of the other 1 of diagonals bisects a pair of opposite angles 1 pair of opposite angles is Rhombus 1 pair of parallel lines Isosceles Trapezoid all properties of parallelogram all properties of kite all sides are diagonals bisect angles diagonals are perpendicular bisectors of each other Diagonals divide rhombus into 4 right triangles. legs are congruent bases are parallel lower base angles are upper base angles are diagonals are congruent any lower base angle is supplementary to any upper base angle Square all properties of rectangle all properties of rhombus diagonals form 4 isosceles right triangles (45-45-90 triangles) 11 Rev B