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Geometry A Overview General Formulas Given P=(x.yi) and Q (x2,y2), distance is d • 1)istance Formula: • Midpoint Formula: Given P=(xi,yi) and Q • Slope Formula: Given P(xj,yi) and (x2,y2), slope is m 2 +(y 1 1 —x,) /(x = Q ±X 2 , midpoint is ,y 2 (x ) ‘ ) ± 2 = 1 X — Symbols Meaning Not equal Congruent Greater than or equal to Less than or equal to Not And Or ymboI E A v Meaning Triangle Angle Perpendicular Union Intersection Implies If and only if A X I U fl Symbol .. •: 3 \/ 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 (I): lbrmed 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 >): begins at an endpoint & then extends infinitely far in only 1 direction. • Ray( • • • 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 Engsh Word Algebraic Translation 90 —x Complement DitYerence = Equal +,> than Greater + Increased by Less than - — English Word Number Opposite of a number Product Sum Supplement Algebraic Translation N -N * + 180— x -, Example: Angle Measures I. Find the measures of 2 supplementary angels if the measure of I angle is 6 less than 5 times the measure of the other angle. 180—x = 5x -6; anglesl49, 31 RevB GeQrnetry A Overview Angle Relationships Vocabulary • Acute Angle: between 0° and 90° • 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 90° • Linear Pair: pair of adjacent angels whose non-common sides are opposite rays. • Obtuse Angle: between 90° and 180° • Perpendicular (I): lines that form 90° angles. Intersect to form congruent adjacent angles; rt angle symbol indicates lines are perpendicular • Right Angle: 90° • Straight Angle: 180° • Supplementary Angles: 2 angles whose measures have sum of 180° • 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 & r CEB, / AEB & / CED 3. Linear Pair: / DEA & / BEA, r DEC & / BEC 4. Given the diagram to the right, identify all angle relationships. • / I and / 5 are exterior angles • / I sup /2 / I + /2 =180, / I & r 2 are linear pair • r 4 sup /5 /4 + /5 =180. /4 & /5 are linear pair • /2±/3+Z4=180 • /1=Z3±/4,/l>/3,Z1>/4 • /5rr Z2+ /3, /5> /2, /5>3 /3 . // . < 1/2 4 5 CM 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 (A): compound statement formed by joining 2 or more statements with word and; true when both are true if-q then —p Contrapositive: if p then q Converse: statement associated with if p then q having form if q then p I)isjunction (v): 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 if —p then q Inverse: if p then q Negation ( or —n): opposite meaning as well as opposite truth value; represented by —p Statement: sentence that is either true or false; represented by p RevB 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 r, linear pairs, supplementary Z 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. ZAOC = 90° no, cannot assume right angles 2. AOOB ZFOB 3. = no, cannot assume congruency yes, given 500 Lo2ic Table Implication Converse p q Inverse Contrapositive q p p q 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 A q P Q q P T T F T F T A b. ‘-g F T F TAFr=F T ATT FAF=F L_LIT FAI= T T F F -p v—q pv’-g_— T F FvF=F F F FT FvTT TvFT T T F T TvTT F T Properties of Equality Reflexive Property yçtric Property Addition and Subtraction Properties Multiplication and Division Properties For every number a, a = a. For all numbers a & b, if a = b, then b a lfabandbcthena=c If a = b then a + c b + c and a c b c a b lfc Oanda=b,thenac=bcand—=— — C L Distributive Property Substitution Property C A(b+c) = ab + ac If a = b, then a may be replaced by b in equation Rev B Geometry A Overview Properties of Inequalities: Comparison Property a - < b, a = b or a> b If a < b and b < Transitive Property c then a < c If a> b and b> c then a> c Addition and Subtraction Properties Multiplication and Division Properties If a> b then a + c> b + c and a c > b c Ifa<bthena±c<b+canda—c<b—c b a Ifc>Oanda<b,thenac<bcand—<— — — C If c> 0 and a> b, then ac> be and C ab —> C C — ab Ifc<Oanda<b,thenac>bc and—>— C C ab Ifc<Oanda>b.thenac<bcand—<— C --_________________________________ C Properties • Reflexive: AB • • • AB (any segment/angle is congruent to itself) CD then CD AB Symmetric: if AB CD and CD EF then AB E EF Transitive: if AB if B is between A and C. then AB Postulate: Segment Addition ± BC = AC Examples: Algebraic Properties State the property that justifies each statement: Addition Property of Equality • If x v, then x + 8 y + 8 Multiplication Property of Equality • If x y, then 8x 8y Transitive Property of Equality • If a = 5 and 5 b, then a = b Theorems and Postulates Angles • • • • • • • • 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 / (or segments) then they are to each other. (Transitive Property) to same or If Z (or segments) are 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 Angle Theorem: 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 4 RevB GeQmetry A Overview Points, Lines & Planes • Through any 2 points, there is exactly 1 line. • 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 AME MB • Segment Addition Postulate: if B is between A and C, then AB + BC AC. Or if AB is between A and C. BC + = AC, then B General: • Addition Property: If segments/angles are added to segments/angles, then the sums are E. • Subtraction Property: If segment /angle is subtracted from segments/angles, the differences are their like multiples are • Multiplication Property: If segments/angles are their like divisions are • Division Property: If segments/angles 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 mt 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 mt & 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. F Which of the lines in the figure is a transversal? EF 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. B A c_. . . 3\4 C . 6 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 Rev B D .> ______________ Geometty A Overview Angles and Parallel Lines Theorems: • Alternate Interior Angles: If 2 parallel lines are cut by transversal then each pair of alternate interior angles are congruent. 4Alt mt / ) (II ifa • I b /l/2 Alternate Exterior Angles: If 2 parallel lines are cut by transversal then each pair of alternate exterior angles are congruent. ( Alt ext / ) a_____ I ifaIlb/lEZ8 / ./ /8 Corresponding Angles: If 2 parallel lines are cut by transversal then each pair of corresponding angles are congruent. + corresponding / ) 1% a ifaJ!b=’/l/5 b H/Z (II • 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. a b 5/ • a if b =r3+z5= 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 arc supplementary. I / if / 7/ • all all b Zl+/7= 180 Perpendicular Transversal: In a plane, if a line is perpendicular to I of 2 parallel lines, then it is perpendicular to the other. if a band cIa=cIb b • c Transitive Property of Parallel Lines: If 2 lines are parallel to a other. a ifa bandb c a b a b line, then they are parallel to each llc I • 31 Congruent/Supplementary: If 2 parallel lines are cut by transversal then any pair of angles x 180-x x 180-x 180-x x 180-x x Ij —>aliintZ E, alt ext Z 6 , correspondingr E, ssi = 180, sse = 180 Rev B Geometry A Overview Examples: Angles and Parallel Lines 1. 1fcd,find Z1 c 2.IfaIb,find Li a 100\ b4 Since c Id, alt intZ 2x + 10 3x + 5 This is a crook problem. Li is determined by the following 1. l00+partof Li =180(SSI) 80 2. 40 part of Li (Alt Int) 40 Therefore, Li = 80 + 40 120 , x=5 2(5)+i0=20 Since c JJd, corresponding Z So Li = 20 , Proving Lines are Parallel Two lines can be proven to be parallel by: • Alternate Interior L • Alternate Exterior L • Corresponding L • • • Same Side Interior (SSI) supplementary Same Side Exterior (SSE) supplementary Same slope Triangle Properties Vocabulary: • Exterior Angle: formed by 1 side of A & extension of another side • Interior: inside • Remote: far away • Remote Interior Angle: interior L of A not adjacent to given exteriorL. Interior L farthest from exterior L Theorems: • Angle Sum Theorem: Sum of measure of angles of a triangle is 180. • No Choice Theorem/31d Angle Theorem: if 2 angles of triangle are to 2 angles of another triangle, then the 3 angle of the triangles areE. • 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 L is exterior L of a triangle then its measure is greater than the measure of its corresponding remote interiorL. • Side Angle Theorem: longest side of a triangle is opposite the largest L in triangle. • Angle Side Theorem: largest L 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 3” 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,l4 6+8>14 2+3>4 yes no 7 Rev B ______________ Geometry A Overview Examples: Determining range d 0 r 3 f side of triangle Find the range for the measure of the 3 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. st 1 Order angles: ZV=28 ZU=70 ZW=82 List the angles in order from least to greatest measure. St 1 Scm Order sides: p ST=7RT=8RS= 13 \7cm 13 2 list sides associated with / Uw, vw, uv 2 list L associated with side ZR, ZS, rT 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 [egs: 2 sides forming right angle in right triangle; or congruent sides of isosceles tn Obtuse Triangle: triangle with I 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 . A is obtuse 2 b c Aisright + Ifa = . 2 2 <c 2+b b c 0 If a Aisacute + lfa > , 0 0 2 8 Rev B Geometry A Overview Congruent Triangles/Transformations 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/si ide 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 Add number of units to yyale • LLv• 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 A • Included Side: sides that compose an angle Methods for proving triangles are congruent SSS If there exists a correspondence between the vertices of 2 As such that 3 sides of I A arc to 3 sides of other A, then the 2 As are SAS If there exists a correspondence between the vertices of 2 As such that 2 sides and the included / of I A are to corresponding parts of the other A, then 2 s are. ASA If there exists a correspondence between the vertices of 2 As such that 2 /s and the included side of I A are to the corresponding parts of the other A, then 2 As are. AAS If there exists a correspondence between the vertices of 2 As such that 2 Zs and the non-included side of I A are to the corresponding parts of the other triangle, then 2 As are ilL If the hypotenuse and the leg of 1 right A are to the hypotenuse and corresponding leg of another right A, then As area. LL If legs of 1 right A are to corresponding legs of another right A, then As are . CPCTC What is CPCTC? • Corresponding Parts of Congruent Triangles are Congruent When is it used? • Only after 2 A have been proven or stated to be Cannot be used to prove A . Isosceles Triangles Angle-Side Theorems If sides then angles: If 2 sides of a triangle areE then the angles opposite those sides are. If angles then sides: If 2 angles of a triangle are ,then the sides opposite those angles area. 9 RevB Geometry A Overview Parts of Triangles Vocabulary • Altitude: line/segment drawn from vertex to point on opposite side making them I • Median: line/segment drawn from I vertex of triangle to midpoint of opposite side. Altitude of AKMO: LO “ / Altitude of AOLM: LN -..—. j Median of ALKO: JL a Median of AKMO: OL C 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 I 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 I. 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/definitionlpostulate or your given information. 4. State that what you assumed to start was WRONG and that the desired conclusion then must be true. 10 RevB Geometry A Overview Quadrilaterals Parallelogram • • • • • Opposite sides are parallel Opposite sides are Opposite angles are Consecutive angles are supplementary Diagonal bisect each other Rectangle • • • • All properties of parallelogram All angles are right angles Diagonals are Diagonals divide rectangle into isosceles triangles. Trapezoid Kite • • • • 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 4. • 1 pair of parallel lines I Isosceles Trapezoid Rhombus • • • • • • 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 propees of rectangle all properties of rhombus diagonals form 4 isosceles right triangles (45-45-90 triangles) 11 RevB