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Transcript
Geometry Final Review Chapter 1: Essentials of Geometry 1 - 1 Undefined Terms Geometry - branch of math that defines and relates basic properties and measurements of line segments and angles Undefined Terms - meaning is accepted without definition Set - well defined collection of items {numbers, objects, etc.} Point - exact location in space, has no dimension Line - Infinite set of points in opposite directions forming straight path, 1 dimension - length Plane - set of points forming flat surface that extends without end in all directions 1 - 2 The Real Numbers and Their Properties Real Numbers - any number on the number line Counting Numbers - natural numbers, 1, 2, 3, 4… Whole Numbers - 0, 1, 2, 3… Integers - whole numbers and opposites, …-1, 0, 1… Rational Numbers - real number written as a fraction, can repeat and terminate 9/2, 7.12, 11.23, 3/4 Irrational Numbers - Number cannot be written as exact ratio, , 2 Rational Integers Whole Counting Irrational Number Line - a line on which each point represents real numbers Coordinate - ordered pair (x, y) identifies point on coordinate plane Graph - graphic representation used to show numeric relationships Properties: Closure - set of numbers is closed under operation if any two elements of the set yields another element of set when operation (sum/product of two real numbers is a real number) Closure property of addition - sum of two real numbers is a real number 13 + 3 = 16 1 + 79 = 80 Closure property of multiplication - product of two real numbers is a real number 3 x 8 = 24 ***Subtraction 3 - 6 = -3 NOT CLOSED, thus no closure property of subtraction Commutative: Commutative property of addition - a + b = b + a Commutative property of multiplication - a x b = b x a Associative: Associative property of addition - a + (b + c) = (a + b) + c Associative property of multiplication - a x (b x c) = (a x b) x c Identity Property: Additive identity: the number 0 a+0=a Multiplicative identity: the number 1 ax1=a Inverse property: Additive inverse: the opposite; gets sum to 0 a + (-a) = 0 Multiplicative inverse: 1/a; gets product to 1 a x 1/a = 1 Distributive property: a(b + c) = ab + ac Multiplication property of zero: ab = 0 if and only if a = 0 or b = 0 1 - 3 Definitions, Lines, and Line Segments Collinear set of points - set of points all of which lie on the same straight line Non-collinear set of points - set of points not on the same straight line Distance between two points on the real number line - absolute value of difference of coordinates of 2 points Ex) point -3 to 5: -3 - 5 = -8 = 8 Betweenness - B is between A and C is A, B, and C are distinct collinear points, AB + BC = AC Line segment - set of points consisting of two points on a line, called endpoints, and all of the points in the line between the endpoints Length of a line segment - distance between its endpoints Congruent segments - segments that have the same measure 1 - 4 Midpoints and Bisectors Midpoint of a line segment - point of that line segment that divides the segment into two congruent segments Bisector of a line segment - any line, or subset of a line, that intersects the segment at its midpoint Sum of two line segments - line segment RS is the sum of 2 segments, RP and PS, if P is between R and S 1 - 5 Rays and Angles On one side of a point - A and B are on one side of the point P is A, B, and P are collinear and P is not between A and B Half-line - consists of the set of all points on one side of the point of division Ray - consists of a point on a line and all points on one side of the point (one endpoint and goes indefinitely in one direction) Opposite Rays - 2 rays of the same line with a common endpoint and no other point in common Angles - set of points that is the union of two rays having the same endpoint Vertex - common endpoint of two rays that forms an angle Side - a line segment forming adjacent vertices of a polygon Straight angle - angle that is the union of opposite rays, 180 Acute angle - angle whose degree measure is greater than 0 and less that 90 Right Angle - angle whose degree measure is 90 Obtuse angle - angle whose degree measure is greater than 90 and less than 180 1 - 6 More Angle Definitions Congruent Angles - angles that have the same measure Bisector of an angle - ray whose endpoint is the vertex of the angle, and that divides that angle into two congruent segments Perpendicular lines - two lines that intersect to form right angles 1 - 7 Triangles Polygon - closed figure in a plane that is union of line segments such that segments intersect only at endpoints, no segments share common endpoint are collinear Triangle - polygon with exactly 3 sides Scalene Triangle - No congruent sides Isosceles Triangle - 2 congruent sides Legs - two congruent sides Base - third non-congruent sides Vertex Angle - formed by the two congruent sides Base Angles - angles whose vertices are endpoints of base of the triangle Equilateral Triangle - 3 congruent sides Acute triangle - 3 acute angles Right triangle - has a right angle Legs - two sides of a triangle that form the right angle Hypotenuse - side opposite the right angle Obtuse angle - has an obtuse angle Equiangular triangle - has 3 congruent angles Chapter 2: Logic 2 - 1 Sentences, Statements, and Truth Values Logic - study of reasoning Mathematical Sentences - sentence that contains a complete thought and can be judged true or false Ex) Tomorrow is Tuesday Albany is the capital of New York State 7>5 Non-Mathematical Sentences - sentence that does not state a fact, such as a command, question, or exclamation Ex) Do you like math? Find your book. Phrase - an expression that is only part of a sentence Ex) All sides 3+5 Open Sentence - sentence that contains a variable Variable - a symbol used to represent a number in Domain or Replacement Set - set from which replacements for the variable are taken Solution or Truth Set - set of all the replacements that will change the open sentence into a true sentence If no replacements from the domain make a true statement, then the solution set is { } Closed sentence or statement - sentence that can be judged true or false Negation: usually formed by placing the word “not” in the original statement Symbol ~ (A statement and its negation have the opposite truth values) P: Albany is the capital of New York State ~P: Albany is not the capital of NYS Truth Table: P ~P T F T F F T F T 2 - 2 Conjunctions Conjunction - A compound statement formed by combining two simple statements using the word “and” Symbol ^ P: It is raining Q: Kate is wet P^Q: It is raining and Kate is wet P^~Q: It is raining and Kate is not wet ~(P^~Q): It is not the case that it is raining and Kate is not wet Truth Table: P T T F F Q T F T F P^Q T F F F Example: P: x is odd Q: x is prime When x = 3 P Q P^Q T T F When x = 9 P Q P^Q T F F P F When x = 2 Q P^Q T F When x = 6 P Q P^Q F F F 2 - 3 Disjunctions Disjunction: a compound statement formed by combining 2 simple statements using the word “or” Symbol P: Pat eats pizza Q: Carol drinks coke PQ: Pat eats pizza or Carol drinks coke Truth Table: P T T F F Q T F T F PQ T T T F 2 - 4 Conditionals Conditional - compound statement formed by using if then to combine two simpler statements Symbol Parts of a conditional: Hypothesis - an assertion that begins an argument, follows if Conclusion - ending the argument, follows then Truth Table: P T T F F Q T F T F PQ T F T T Hidden Conditional: If, then aren’t in the statement, but suggests conditional Ex) When I finish my homework, I’ll go to the movies. 2 - 5 Inverses, Converses, and Contrapositives Inverse - Formed by negating the hypothesis and conclusion Conditional: If a number is not a whole number, then it is not an integer Inverse: If a number is a whole number, then is it an integer Truth Table: P T T F F Q T F T F ~P F F T T ~Q F T F T PQ T F T T ~P~Q T T F T Converse - formed by interchanging the hypothesis and the conclusion Conditional: If a number is an integer, then it is a whole number Converse: If a number is a whole number, then it is an integer Truth Table: P T T F F Q T F T F PQ T F T T QP T T F T Contrapositive - formed by negating and interchanging both the hypothesis and the conclusion Conditional: If a number is not an integer, then it is not a whole number Contrapositive: If a number is a whole number, then a number is an integer Truth Table: P T T F F Q T F T F ~P F F T T ~Q F T F T PQ T F T T ~Q~P T F T T Logically Equivalents - having the same truth value P T T F F Q T F T F ~P F F T T ~Q F T F T PQ T F T T ~P~Q T T F T QP T T F T ~Q~P T F T T Conditional (PQ) is logically equivalent to Contrapositive (~Q~P) Inverse (~P~Q) is logically equivalent to Converse (QP) 2 - 6 Biconditionals Biconditional - conjunction of the conditional and its converse Symbol: PQ^QP, PQ Example: P: A polygon is a triangle Q: A polygon has exactly 3 sides PQ: A polygon is a triangle is and only if it has 3 sides Truth Table: P Q T T T F F T F F PQ T F T T QP PQ^QP T T T F F F T T PQ T F F T PQ is true when: P and Q are both true or both false 2 - 7 Laws of Logic The Law of Detachment - If a conditional is true and the hypothesis is true then the conclusion is true Valid Argument - uses a series of statements called premises that have known truth values to arrive at the conclusion Ex) “If today is Friday, then I get my allowance” T “Today is Friday” T Thus: I get my allowance is True The Law of Disjunctive Inference - If the disjunction (pq) is true and the disjunct (p) is false, then the other disjunct (q) is true 2 - 8 Drawing Conclusions Example: 1. If Rachel joins the choir, then Rachel likes to sing T 2. Rachel will join the choir or Rachel will play basketball T 3. Rachel does not like to sing T Conclusions: Rachel will play basketball (She won’t join choir and doesn’t like to sing) Example: Ted, Bill, and Mary each take a different course in one of three areas for their senior year: mathematics, art, and thermodynamics. 1. Ted tutors his sibling taking the mathematics course 2. The art students and Ted have an argument over last night’s basketball game 3. Mary loves the drawing made by her sibling taking the art course T B M M X X A X X T X X Tautology - logical statement that is always true for all truth values of the simple statements that compose it Chapter 3: Proving Statements in Geometry 3 - 1 Inductive Reasoning Generalization - Going from a few specific cases to a general case Inductive Reasoning - method of reasoning in which a series of particular examples leads to a conclusion Counterexample - Example that disproves a generalization Conjectures - statements that are likely to true but have not yet been proven true by a deductive proof 3 - 2 Definitions as Biconditionals Deductive Reasoning - uses the laws of logic to combine definitions and general statements that we know to be true to reach a valid conclusion - A scalene triangle is a triangle that has no congruent sides - If a triangle is scalene, then the triangle has no congruent sides - A triangle is scalene if and only if the triangle has no congruent sides 3 - 3 Deductive Reasoning Proofs - In geometry, it is a valid argument that establishes the truth of a statement Perpendicular Lines - 2 lines that intersect to form right angles Parallel Lines - Coplanar lines that have no points or all points in common Line Segment - A set of points consisting of endpoints and all the points in between Angle Bisector - A line extending from the vertex of an angle, and it divides the segment into two congruent segments Ray - A line extending from one endpoint indefinitely Congruent angles - Two angles with the same measure Congruent segments - segments with the same measure Proof Format in Euclidean Geometry: Given: Prove: Construction of a two-column proof Examples: 1) Given: In ABC, AB is perpendicular to BC Prove: ABC is a right angle Statements Reasons 1. AB is perpendicular to BC 1. Given 2. ABC is a right angle 2. If 2 lines are perpendicular, they form right angles 2) Given: BD is the bisector of ABC Prove: mABD = mDBC Statements 1. BD is the bisector of ABC 2. mABD = mDBC Reasons 1. Given 2. If an angle is bisected then it is divided into two congruent segments 3 - 4 Direct and Indirect Proofs Direct Proofs - proof that starts with a given statement and uses laws of logic to arrive at the statement to be proved Example: Given: ABC is an acute triangle Prove: mA < 90 Statements 1. ABC is an acute triangle 2. A, B, C are acute angles 3. A < 90 Reasons 1. Given 2. An acute triangle has 3 acute angles 3. If an angle is acute, its measure is greater than 0 and less than 90 Indirect Proofs - Proof that starts with the negation of a statement to be proved false Example: Given: AB and CD such that AB CD Prove: AB and CD are not congruent segments Statements Reasons 1. AB and CD are congruent 1. Assumption segments 2. AB = CD 2. Congruent segments have the same measure 3. Given 4. Contradiction in 2 and 3 Therefore, the assumption is false and the negation is true 3. AB CD 4. AB and CD are not congruent segments 3 - 5 Postulates, Theorems, and Proof Postulate (axiom) - statement whose truth is accepted without proof Theorems - statement that is proved by deductive reasoning Reflexive Postulate - a quantity is equal to itself Ex) 7 = 7, x = x Symmetric Postulate - equality may be expressed in either order Ex) 7 = x, x = 7 Transitive Postulate - quantities equal to the same quantity are equal to each other Ex) x = y, y = z, x = z Example: Given: AB = CD A BC = AD AB = BC Prove: AD = CD D Statements 1. AB = CD, BC = AD, AB = BC 2. CD = BC 3. CD = AD B C Reasons 1. Given 2. Quantities equal to the same quantity are equal to each other 3. Transitive; same as 2 3 - 6 The Substitution Postulate Substitution Postulate - a quantity may be substituted for its equal in any statement of equality Example: Given: ABD + DBC = 90 ABD = CBE Prove: CBE + DBC = 90 Statements 1. ABD + DBC = 90 ABD = CBE 2. CBE + DBC = 90 A B D C E Reasons 1. Given 2. A quantity may be substituted for its equal in any statement of equality 3 - 7 The Addition and Subtraction Postulates Partition Postulate - a whole is equal to the sum of its parts Addition Postulate - If equal quantities are added to equal quantities, then the sums are equal A B C Example: Given: ABC and DEF with AB = DE and BC = EF D E F Prove: AC = DF Statements 1. AB = DE and BC = EF 2. AB + BC = DE + EF 3. AC = DF Reasons 1. Given 2. If equal quantities are added to equal quantities, then the sums are equal 3. A quantity may be substituted for its equal in any statement of equality Subtraction Postulate - If equal quantities are subtracted from equal quantities, the differences are equal Example: Given: x + 6 = 14 Prove: x = 8 Statements 1. x + 6 = 14 2. x + 6 - 6 = 14 - 6 3. x = 8 Reasons 1. Given 2. If equal quantities are subtracted from equal quantities, then the differences are equal 3. A quantity may be substituted for its equal in any statement of equality 3 - 8 The Multiplication and Division Postulates Multiplication Postulate - If equal quantities are multiplied by equal quantities, then the products are equal Doubles of equal quantities are equal Division Postulate - If equal quantities are divided by equal quantities, then the quotients are equal Halves of equal quantities are equal Powers Postulate - The squares of equal quantities are equal Roots Postulate - If a = b and a > 0, then a = b Positive square roots of positive equal quantities are equal Examples: 1. Given: AB = CD, RS = 3AB, LM = 3CD Prove: RS = LM Statements 1. AB = CD, RS = 3AB, LM = 3CD 2. 3AB = 3CD 3. RS = LM 2. Given: 5x + 3 = 38 Prove: x = 7 Statements 1. 5x + 3 = 38 2. 5x + 3 - 3 = 38 - 3 3. 5x = 35 Reasons 1. Given 2. If equal quantities are multiplied by equal quantities, then the products are equal 3. A quantity may be substituted for its equal in any statement of equality Reasons 1. Given 2. If equal quantities are subtracted from equal quantities, then the differences are equal 3. A quantity may be substituted for its equal in any statement of equality 4. 5x 5 = 35 5 4. If equal quantities are divided by equal quantities, then the quotients are equal 5. A quantity may be substituted for i its equal in any statement of equality 5. x = 7 Chapter 4: Congruence of Line Segments, Angles, and Triangles 4 - 1 Postulates of Lines, Line Segments, and Angles A line segment can be extended to any length in either direction Through two given points, one and only one line can be drawn Two lines cannot intersect in more than one point One and only one circle can be drawn with any given point as center and the length of any given line segment as a radius At a given point on a given line, one and only one perpendicular can be drawn to the line From a given point not on a given line, one and only one perpendicular can be drawn to the line For any two distinct points, there is only one positive real number that is the length of the line segment joining the two points The shortest distance between two points is the length of the line segment joining the two points A line segment has one and only one midpoint An angle has one and only one bisector Conditional Statements: The information that is known to be true is often stated as given and what is to be proved as prove. When the information needed for a proof is presented in a conditional statement, we use the information in the hypothesis to form a given statement, and the information in the conclusion to form a prove statement. Examples: 1. If a ray bisects a straight angle, it is perpendicular to the line determined by the straight angle Given: BD bisects ABC ABC is a straight angle Prove: BD is perpendicular to AC 2. If AB = AD and DC = AD, then AB = DC Given: AB = AD, DC = AD Prove: AB = DC Then write a formal proof… 4 - 2 Using Postulates and Definitions in Proofs Given: AB = CD Prove: AC = BD Statements 1. AB = CD 2. BC = BC 3. AB + BC = BC + CD 4. AC = BD A B C D Reasons 1. Given 2. A quantity is equal to itself 3. If equal quantities are added to equal quantities, then the sums are equal 4. A quantity may be substituted for its equal in any statement of equality A E Given: AE = AF, EB = FC Prove: ABC is isosceles Statements 1. AE = AF, EB = FC 2. AE + EB = AF + FC B 3. AB = AC 4. ABC is isosceles F C Reasons 1. Given 2. If equal quantities are added to equal quantities, then the sums are equal 3. A quantity many be substituted for its equal in any statement of equality 4. An isosceles triangle has 2 congruent sides 4 - 3 Proving Theorems About Angles If two angles are right angles, then they are congruent If two angles are straight angles, then they are congruent Adjacent Angles - two angles in the same plane that have a common vertex and a common side but do not have any interior points in common Complementary Angles - two angles, the sum of whose degree measures is 90 Supplementary Angles - two angles, the sum of whose degree measures is 180 Example: Given: 1 = 2 Prove: 3 = 4 Statements 1. 1 = 2, ABCD 2. ABC and DCB are straight angles 3. ABC = DCB 4. ABC - 1 = DCB - 2 5. 3 = 4 A 3 1 2 4 B C D Reasons 1. Given 2. If ABCD, then ABC and DCB are straight angles 3. All straight angles are congruent 4. If equal quantities are subtracted from equal quantities, then the differences are equal 5. A quantity may be substituted for its equal in any statement of equality If two angles are complements of the same angle, then they are congruent If two angles are congruent, then their complements are congruent If two angles are supplements of the same angle, then they are congruent If two angles are congruent, then their supplements are congruent Linear Pair of Angles - two adjacent angles whose sum is a straight angle If two angles form a linear pair, then they are supplementary If two lines intersect to form congruent adjacent angles, then they are perpendicular Vertical Angles - the non-adjacent angles formed by the intersection of two lines If two lines intersect, then the vertical angles are congruent A D Example: Given: ABC and DBE intersect at B B BC bisects EBF F Prove: CBF = ABD E C Statements Reasons 1. ABC and DBE intersect at B 1. given BC bisects EBF 2. EBC = CBF 2. If an angle is bisected, two congruent angles are formed 3. EBC and ABD are vertical angles 3. Vertical angles are the two nonadjacent angles formed by the intersection of two lines 4. EBC = ABD 4. If two lines intersect, then the vertical angles are congruent 5. CBF = ABD 5. Quantities equal to the same quantities are equal Example: WX and YZ intersect at V, WVY = 3x + 25 and XVZ = 10x + 4. Find WVZ W Y Let WVY = 3x + 25 Let XVZ = 10 x + 4 Let WVZ = 180 - 10x + 4 Z V X 3x + 25 = 10x + 4 21 = 7x 3=x The measure of angle WVZ is 146 10 (3) + 4 = 34 180 - 34 = 146 4 - 4 Congruent Polygons and Corresponding Parts Congruent Polygons - polygons that have the same size and shape One-to-one correspondence - each part or a polygon corresponds to exactly one part of the congruent polygon Any geometric figure is congruent to itself A congruence may be expressed in either order Two geometric figures congruent to the same geometric figure are congruent to each other 4 - 5 Proving Triangles Congruent Using Side, Angle, Side SAS - Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other A D Given: BD bisects ABC, AB = BC Prove: ADB = CDB Statements 1. BD bisects ABC, AB = BC 2. ABD = CBD 3. BD = BD 4. ADB = CDB C Reasons B 1. Given 2. If an angle is bisected, it is divided into two congruent angles 3. A quantity is equal to itself 4. SAS = SAS 4 - 6 Proving Triangles Congruent Using Angle, Side, Angle ASA - Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other Given: AB is perpendicular to BC ED is perpendicular to DC C is the midpoint of BD Prove: ABC = EDC Statements 1. AB is perpendicular to BC ED is perpendicular to DC C is the midpoint of BD 2. B and D are right angles 3. B = D 4. BC = CD 5. ACB and ECD are vertical s 6. ACB = ECD 7. ABC = EDC A C D B E Reasons 1. Given 2. Perpendicular lines form right angles 3. All right angles are congruent 4. A midpoint divides a segment into 2 congruent segments 5. Vertical angles are two non-adjacent angles formed by the intersection of 2 lines 6. All vertical angles are congruent 7. ASA = ASA 4 - 7 Proving Triangles Congruent Using Side, Side, Side SSS - Two triangles are congruent is the three sides of one triangle are congruent, respectively, to the three sides of the other K Given: Isosceles triangle JKL JK = KL M is the midpoint of JL J L Prove: JKM = LKM M Statements Reasons 1. JK = KL 1. Given M is the midpoint of JL 2. JM = ML 2. A midpoint divides a segment into two congruent segments 3. KM = KM 3. A quantity is equal to itself 4. JKM = LKM 4. SSS = SSS Chapter 5: Congruence Based on Triangles 5 - 1 Line Segments Associated with Triangles Altitude of a Triangle - line segment drawn from any vertex of a triangle perpendicular to and ending in the line that contains the opposite side Median of a Triangle - line segment that joins any vertex of a triangle to the midpoint of the opposite side Angle bisector of a Triangle - ray whose endpoint is vertex of the angle and divides the angle into two congruent segments Examples: B Given: BD is the median of AE InBDC, BE is the median to DC A = C AB = BC A C Prove: ABD = CBE D E Statements Reasons 1. BD is the median of AE 1. Given BE is the median to DC A = C AB = BC 2. D is the midpoint of AE 2. A midpoint is a line segment that joins E is the midpoint of DC vertex of a triangle to the midpoint of the Opposite side 3. AD = DE, DE = EC 3. A midpoint divides a segment into two congruent segments 4. AD = EC 4. Quantities equal to the same quantity are equal to each other 5. ABD = CBE 5. SAS = SAS B Given: BD is the altitude of ABC D is the midpoint of AC Prove: ABD = CBD Statements 1. BD is the altitude of ABC D is the midpoint of AC 2. AD = DC 3. BD is perpendicular to AC 4. ADB and CDB are right angles 5. ADB = CDB 6. BD = BD 7. ABD = CBD Reasons 1. Given A D C 2. A midpoint divides a segment into two congruent segments 3. An altitude of a triangle extends from the vertex of the triangle perpendicular to and ending in the opposite side 4. Perpendicular lines form right angles 5. All right angles are congruent 6. A quantity is equal to itself 7. SAS = SAS 5 - 2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent If two triangles are congruent, then their corresponding parts are congruent C Given: ABC, CA = CB, AD = BD Prove: ACD = BCD Statements 1. CA = CB, AD = BD 2. CD = CD 3. CAD = CBD 4. ACD = BCD 5 - 3 Isosceles and Equilateral Triangles A B Reasons D 1. Given 2. A quantity is equal to itself 3. SSS = SSS 4. Corresponding parts of congruent are congruent If two sides of a triangle are congruent, the angles opposite these sides are congruent The median from the vertex angle of an isosceles triangle bisects the vertex angle The median from the vertex angle of an isosceles triangle is perpendicular to the base Every equilateral triangle is equiangular Given: ABC is isosceles with vertex angle B B AD = CD Prove: BAD = BCD A C Statements Reasons D 1. ABC is isosceles with vertex angle B 1. Given AD = CD 2. BA = BC 2. An isosceles triangle has 2 congruent legs surrounding the vertex angle 3. BAC = BCA, ACD = CAD 3. Angles opposite congruent sides are congruent 4. BAC + CAD = BCA + ACD 4. If equal quantities are added to equal quantities, then the sums are equal 5. BAD = BCD 5. A quantity may be substituted for its equal in any statement of equality 5 - 4 Using Two Pairs of Congruent Triangles C Given: AEB, AC = AD CB = DB E Prove: CE = DE Statements 1. AC = AD, CB = DB 2. AB = AB 3. ACB = ADB 4. CBE = DBE 5. EB = EB 6. CEB = DEB 7. CE = DE A B D Reasons 1. Given 2. A quantity is congruent to itself 3. SSS = SSS 4. CPCTC 5. A quantity is equal to itself 6. SAS = SAS 7. CPCTC 5 - 5 Proving Overlapping Triangles Congruent G Given: GA = GC, AR = CE Prove: GAE = GCR R A Statements 1. GA = GC, AR = CE 2. GA - AR = GC - CE 3. RG = EG 4. G = G 5. GAE = GCR E C Reasons 1. Given 2. If equal quantities are subtracted from equal quantities, then the differences are equal 3. Substitution postulate 4. A quantity is equal to itself 5. SAS = SAS 5 - 6 Perpendicular Bisector of a Line Segment Perpendicular Bisector of a Line Segment - any line that is perpendicular to the line segment at its midpoint Concurrent - lines that intersect in one point Intersection of the perpendicular bisector of the three sides Perpendicular Bisector Concurrence Theorem - perpendicular bisectors of a triangle are concurrent Circumcenter - point where the 3 perpendicular bisectors of the sides of the triangle intersect P Given: PQ is the perpendicular bisector of RS Prove - RSP is isosceles R Statements 1. PQ is the perpendicular bisector of RS 2. PQR and PQS are right angles Q S Reasons 1. Given 2. Perpendicular lines form right angles 3. All right angles are congruent 4. A quantity is equal to itself 5. A perpendicular bisector extends to the midpoint of the opposite side 6. A midpoint divides a segment into two congruent segments 7. SAS = SAS 8. CPCTC 9. An isosceles triangle has 2 congruent sides 3. PQR = PQS 4. PQ = PQ 5. Q is the midpoint of RS 6. RQ = QS 7. PQR = PQS 8. PR = PS 9. RPS is isosceles 5 - 7 Basic Constructions Construction 1: Construct a Line Segment Congruent to a Given Line Segment Steps: 1. With a straight edge draw a ray 2. Open the compass so that the point is on A and the point of the pencil is on B 3. Using the say compass radius, draw an arc that intersects the ray A B Construction 2: Construct an Angle Congruent to a Given Angle Steps: 1. Draw a ray with a straight edge with the endpoint D 2. With A as the center, draw an arc that intersects each ray of angle A, using the same radius, draw an arc with D as the center. Where the arc and the ray intersect, call that E 3. With E as the center, draw an arc, draw an arc with the radius equal to BC that intersects that arc in step 3, Label it F 4. Draw DF C A B Construction 3: Construct the Perpendicular Bisector of a Given Line Segment and the Midpoint of A given Line Segment Steps: 1. Open the compass to a radius greater than one-half of AB 2. Place the point of the compass at A and draw an arc above AB and an arc below AB 3. Using the dame radius, place the point of the compass at B and draw an arc above AB and an arc below AB 4. Using a straight edge, draw CD intersecting AB A B Construction 4: Bisect a Given Angle Steps: 1. With B as center, draw an arc that intersects ray BA at D and ray BC at E 2. With D as the center, draw an arc in the interior or angle ABC 3. Using the same radius, and with E as the center, draw an arc that intersects the arc drawn in step two. Label it F. 4. Draw ray BF A B C Construction 5: Construct a Line Perpendicular to a Given Line Through a Given Point on a Line Steps: 1. With P as the center draw arcs that intersect PA at C and PB at D 2. With C and D as centers and a radius greater than that used in step 1, draw arcs intersecting at E 3. Draw EP A P B Construction 6: Construct a Line Perpendicular to a Given Line Through a Point Not on a Given Line Steps: 1. With P as the center, draw an arc that intersects AB in two points, C and D 2. Open the compass to a radius greater than one-half of CD. With C and D as centers, draw intersecting arcs. Label the point of intersection E. 3. Draw line PE intersecting line AB A B Chapter 6: Transformations and the Coordinate Plane 6 -1 The Coordinates of a Point in a Plane x - axis - horizontal line y - axis - vertical line Origin - point where x and y meet Coordinate - every point on the plane can be described by 2 numbers Ordered Pair - (x,y) Abscissa - x coordinate Ordinate - y coordinate Two points are on the same horizontal line if and only if they have the same y coordinates The length of the horizontal line segment is the absolute value of the difference of the x coordinates Two points are on the same vertical line if and only if they have the same x coordinates The length of a vertical line segment is the absolute value of the difference of the y coordinates Each vertical line is perpendicular to each horizontal line To find area of graphed objects: find internal shapes 6 - 2 Line Reflections Line of Reflection - line in a picture that when folded makes the object coinside Life Reflection - one to one correspondence between an object and the image points Fixed Points - point on the line of reflection Transformation - one to one correspondence between the set of points in S and S’ such that every point in S corresponds to one point one point in S’ Image - S’ is called the image of the points in S Preimage - S in called the preimage of S’ ***Under a line reflection, distance, angle measure, collinearity, and midpoint are preserved Line Symmetry - figure is its own image under a line reflection Examples: WOW KICK unlimited both ways 6 - 3 Line reflections in the Coordinate Plane Rules: Line reflection in the y-axis: (x, y) (-x, y) Line reflection in the x-axis: (x, y) (x, -y) Line reflection in y = x: (x, y) (y, x) 6 - 4 Point Reflections in the Coordinate Pane ***Under point reflection, distance, angle measure, collinearity, and midpoint is preserved Point Symmetry - a figure has point symmetry if the figure is its own image under a reflection in a point Reflection of the origin: Ro (a, b) (-a, -b) 6 - 5 Translations in the Coordinate Plane Translation - transformation of the plane that moves every point in the plane the same distance in the same direction ***Under a translation, distance, angle measure, collinearity, and midpoint is preserved Translational Symmetry - if the image of every point of a figure is a point in the figure Rule: Ta,b (x, y) (x + a, y + b) 6 - 6 Rotations in the Coordinate Plane Rotations - transformation of the plane about a fixed point P through an angle of d degrees such that a counter clockwise rotation is positive and a clockwise rotation is negative Rotation Symmetry - If a figure is its own image under a rotation and the center of the rotation is the only fixed point Rules: R90 (x, y) (-y, x) R180 (x, y) (-x, -y) R270 (x, y) (y, -x) Thus…R360 (x, y) (x, y) 6 - 7 Glide Reflections Glide Reflection - composition of transformations of the plane that consist of a line reflection and a translation in the direction of the line of reflection performed in either order Isometry - transformation that preserves distance 6 - 8 Dilations in the Coordinate Plane Dilations - a transformation in the plane that preserves angle measure but not distance Rule: Dk (x, y) (k x x, k x y) 6 - 9 Transformations as Functions Function - set of ordered pairs in which no 2 pairs have the same first element Domain - in an ordered pair, the set of first elements is the domain of the function Range - In an ordered pair, the set of second elements is the range of the function Composition of Transformations - combination of 2 transformations in which the first transformation produces an image and the second performs on that image Orientation - order of points around an object Direct Isometry - transformation that preserves distance and orientation Opposite Isometry - transformation that preserves distance but changes the order of orientation from counterclockwise to clockwise or visa versa ex) line reflection Chapter 7: Geometric Inequalities 7 - 1 Basic Inequality postulates Whole Quantity and Its Parts - The whole is greater than any of its parts Transitive Property of Inequalities - a, b, and c are real numbers such that a>b and b>c then a>c Substitution Postulate of Inequalities - A quantity my be substituted for its equal in any statement of inequality Trichotomy Postulate - a and b are real numbers then one of the following must be true: a<b, a = b, or a>b 7 - 2 Inequality Postulates Involving Addition and Subtraction Addition Postulate of Inequalities: A: I equal quantities are added to unequal quantities, the sums are unequal in the same order B: If unequal quantities are added to unequal quantities of the same order, the sums are unequal in the same order Subtraction postulate of Inequalities - If equal quantities are subtracted from unequal quantities, the differences are unequal in the same order 7 - 3 Inequality Postulates Involving Multiplication and Division Multiplication Postulates of Inequalities A: If unequal quantities are multiplied by positive equal quantities, then the products are unequal in the same order B: If unequal quantities are multiplied by negative equal quantities, then the products are unequal in the opposite order Division Postulate of Inequalities A: If unequal quantities are divided by positive equal quantities, then the quotients are unequal in the same order B: If unequal quantities are divided by negative equal quantities, then the quotients is unequal in the opposite order 7 - 4 An inequality Involving the Lengths of the Sides of a triangle Triangle Inequality Theorem - The length of one side of the triangle is less than the sum of the lengths of the other 2 sides 7 - 5 An Inequality Involving an Exterior Angle of a Triangle Exterior Angle of a Polygon - angle that forms a linear pair with one of the interior angles of the polygon Adjacent Interior angle - for each interior angle there is an adjacent interior angle Remote (non-adjacent interior angles) - other two angles in the triangle The measure of an exterior angle of a triangle is greater than the measure of either non-adjacent interior angles 7 - 6 Inequalities Involving Sides and Angles of a Triangle If the lengths of two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal and the larger angle lies opposite the longer side If the measure of two angles of a triangle are unequal, then the lengths of the sides opposite these angles are unequal and the longer side lies opposite the larger angle Chapter 8: Slopes and Equations of Lines 8 - 1 The Slope of A Line Finding the Slope of a Line: M = y = rise = y2 - y1 x run x2 - x1 To find, select two points on a line. Find the vertical change, then the horizontal. Write the ratio of change. Positive Slope Negative Slope Zero Slope No/Undefined Slope 8 - 2 The Equation of a Line Y-intercept - y-coordinate of the point at which the line intersects the x-axis X-intercept - x-coordinate of the point at which the line intersects the y-axis A, B, and C lie on the same line if and only if the slope of AB is equal to the slope of BC To write the equation of a line: First, find the slope of the line. Then, plug this slope (m) and one coordinate (x, y) into the y= m x + b formula. Solve for b. Your equation will be something like y = 2x + 1 Point-Slope Formula: Y-b=m X-a 8 - 3 Midpoint of a Line Segment Midpoint - point of a line segment that divides the segment into 2 congruent segments M= X 1 + X 2, Y 1 + Y 2 2 2 8 - 4 The Slope of Perpendicular Lines If two non-vertical lines are perpendicular, then the slope of one is the negative reciprocal of the other 8 - 5 Coordinate Proof Example 1: Prove that AB and CD bisect each other and are perpendicular to each other if the coordinates of the endpoints of these segments are A (-3,5), B (5,1), C (-2,-3), D (4,9). The Midpoint of AB and CD is (1,3). The slope of AB is -1/2 and of CD Is 2. Since these are negative reciprocals, And the two lines intersect at their midpoint, AB and are Perpendicular and bisect each other. QuickTi me™ and a TIFF ( Uncompressed) decompr essor are needed to see thi s p icture. since BC 8 - 6 Concurrence of the Altitudes of a Triangle Concurrent - 3 or more lines are concurrent is they intersect in one point The altitudes of a triangle are concurrent Orthocenter - point at which altitudes of a triangle intersect To find, find the negative reciprocal slope for each side of the triangle. Use this slope and plug it into the point that is opposite the side for which it applies. Plug into y = m x + b to find equation of line. You can plug these into a system algebraically (if two things equal y, set them equal to each other) or simply graph. This will give you the point of orthocenter Example 1) The Coordinates of the Vertices of PQR are P (0,0), Q (-2,6), And R (4,0). Find the orthocenter. mPQ = -3 m(alt) = 1/3 0 = 1/3 (4) + b 0 = 4/3 + b - 4/3 = b y = 1/3x + -4/3 mQR = -1 m (alt) = 1 0 = 1 (0) + b b=0 y=x mPR = 0 m (alt) = undefined -2 = x y=x y = -2 x = -2 QuickTi me™ and a TIFF ( Uncompressed) decompr essor are needed to see thi s p icture. Orthocenter = (-2,-2) Chapter 9: Parallel Lines 9 - 1 Proving Lines Parallel Coplanar - all points/lines are in the same plane Parallel Lines - lines in the same plane and that have no points in common or have all points in common (coincide) AB and CD Transversal - intersects two other coplanar lines, EF Interior angles - in between two given lines, 3, 4, 5, 6 QuickTi me™ and a TIFF ( Uncompressed) decompressor are needed to see thi s pi ctur e. Exterior angles - on outside of given lines, 1, 2, 7, 8 Alternate interior angles - on opposite sides of transversal and common vertex, 3 & 6, 4 & 5 don’t have Alternate exterior angles - same but on exterior, 1 & 8, 2 & 7 Interior angles on the same side of the transversal - equal 180, 3 & 5, 4 & 6 Corresponding angles - same side of transversal and don’t have common vertex, one exterior and one interior, 1 & 5, 3 & 7, 2 & 6, 4 & 8 9 - 2 Properties of Parallel Lines If two coplanar lines are cut by a transversal so that the alternate interior angles formed are congruent, then the two lines are parallel If two coplanar lines are cut by a transversal so that the corresponding angles are congruent, then the two lines are parallel If two coplanar lines are cut by a transversal so that the interior angles on the same side of the transversal are supplementary, then the lines are parallel If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other If two of three lines in the same plane are parallel to the third line, then they are parallel to each other 9 - 3 Parallel Lines in the Coordinate Plane If two non-vertical lines in the same plane are parallel, then they have the same slope 9 - 4 The Sum of the Measures of the Angles of the Triangle The sum of the measure of the angles of a triangle is 180 If two angles of one triangle are congruent to 2 angles of another triangle, the third angles are congruent The acute angles of a right triangle are complementary Each angle of an equilateral triangle measures 60 Each acute angle of an isosceles right triangle measures 45 The sum of the measure of the angles of a quadrilateral is 360 The measure of an exterior angle of a triangle is equal to the sum of the measures of the nonadjacent interior angles 9 - 5 Proving Triangles Congruent by Angle, Angle, Side If two angles and the sides opposite one of them in one triangle are congruent to the corresponding angles and sides in another triangle, then the triangles are congruent ***Two triangles cannot be proven congruent using AAA or SSA 9 - 6 The Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite these angles are congruent 9 - 7 Proving Right Triangles Congruent by Hypotenuse Leg If the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other, then the two right angles are congruent The angle bisectors of a triangle are concurrent Incenter - the point where the angle bisectors meet 9 - 8 Interior and Exterior Angles of Polygons Polygon - closed figure Triangle - 3 sides Quadrilateral - 4 sides Pentagon - 5 sides Hexagon - 6 sides Octagon - 8 sides Decagon - 10 sides n-gon - variable Convex Polygon - polygon in which at least one angle measures under 180 Concave Polygon - polygon in which at least one angle measures over 180 Diagonal - a lone segment whose endpoints are 2 non-adjacent vertices The sum of the degree measures of the interior angles of any polygon of n sides is 180(n-2) The sum of the measure of the exterior angles of a polygon is 360 Regular Polygon - polygon that’s equilateral and equilngular Chapter 10: Quadrilaterals 10 - 2 The General Quadrilateral / The Parallelogram Quadrilateral - polygon with 4 sides Q R P S Consecutive Vertices - (Q,P) (Q,R) (R,S) (S,P) Consecutive Sides - PS, SR, PQ, QR Parallelogram - quadrilateral in which two pairs of opposite sides are parallel and congruent 10 - 3 Proving a Quadrilateral is a Parallelogram - Both pairs of opposite sides are parallel Both pairs of opposite sides are congruent Two opposite sides are congruent and parallel The diagonals bisect each other Both pairs of opposite angles are congruent 10 - 4 The Rectangle Rectangle - parallelogram with one right angle A quadrilateral is a rectangle if any of the following can be proven true: - It is a parallelogram with one right angle It is equiangular It is a parallelogram whose diagonals are congruent 10 - 5 Rhombus Rhombus - parallelogram with 2 congruent consecutive sides A quadrilateral is a rhombus if any of the following can be proven true: - It is a parallelogram with two congruent consecutive sides It is equilateral It is a parallelogram whose diagonals are perpendicular to each other It is a parallelogram whose diagonals bisect opposite angles 10 - 6 Square A quadrilateral is a square if any of the following can be proven true: - It is a rectangle with two consecutive sides congruent It is a rhombus with one right angle 10 - 7 Trapezoid Trapezoid - quadrilateral in which two and only two sides are parallel Isosceles Trapezoid - quadrilateral in which the non-parallel sides are congruent Median of a Trapezoid - line segment whose endpoints are the midpoints of the non-parallel sides of a trapezoid A trapezoid is an isosceles trapezoid is any of the following can be proven true: - Base angles of a trapezoid are congruent Diagonals of a trapezoid are congruent Consecutive non-base angles are supplementary The median is parallel to the bases 10 - 8 Area of Polygons Area of a Polygon - unique real number assigned to any polygon that indicates the number of the nonoverlapping square units contained in polygon interior Formulas: Rectangle: a = L X W Triangle: a = ½ b X h Square: A = S2 Trapezoid: a = 1/3 (b1 + b2) h Parallelogram: a = b X h Rhombus: a = b X h; a = ½ X d1 X d2 Chapter 11: The Geometry of Three Dimensions 11 - 1 Points, Lines, and Planes, Perpendicular Lines and Planes, Parallel Lines and Planes Parallel Lines in Space - lines in the same plane that have no points in common If two lines intersect, then there is exactly one plane containing them Skew Lines - lines in space that are neither parallel nor intersecting If two planes intersect then they intersect in exactly one line If a line is perpendicular to each of two intersecting lines at their point of intersection then the line is perpendicular to the plane determined by these lines Dihedral Angle - union of two half planes with a common edge Measure of a Dihedral Angle - measure of the plane angle formed by two rays each in different half planes of the angle and each perpendicular to the common edge at the same point of the edge Perpendicular Planes - two planes that intersect to form a right dihedral angle A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the intersection of the line and the plane A plane is perpendicular to a plane if the line is perpendicular to the plan Trough a given point of a line, there can be only one plane perpendicular to the given line Two planes are perpendicular to the same line if and only if the planes are parallel Distance between two planes - length of the line segment perpendicular to both planes with an endpoint on each plane 11 - 2 Surface Area of a Prism Polyhedron - 3D figure formed by the union of surfaces enclosed by plane figures Prism - polyhedron in which 2 faces called bases of the prism , are congruent polygons in parallel planes Faces - portion of the planes enclosed by a plane figure Edges - intersection of the faces Vertices - intersection of edges Lateral Sides - surfaces between corresponding sides of the bases Lateral Edges - common edges of the lateral sides Altitude - line segment perpendicular to each edge of the bases with an endpoint on each base Height - length of the altitude Right Prism - prism in which lateral sides are all perpendicular to the base and all lateral sides are rectangles Parallelpiped - prism that has parallelograms as bases Rectangular Parallelpiped - parallelpiped that has rectangular bases and lateral edges Lateral Area - sum of the areas of the lateral sides Total Surface Area - sum of the lateral area and area of the bases Prism Lateral Area Surface Area Volume Bh Pyramid 1/3Bh Cone rhs L + r2 1/3r2hc Cylinder 2rh 2rh + 2r2 r2h Sphere 4r2 4/3r3 Chapter 12: Ratio, Proportion, Similarity 12 - 1 Distance Distance Formula: (x2- x1)2+ (y2 - y1)2 12 - 2 Ratio and Proportions/Proportions Involving Line Segments Ratio - a/b, a:b Proportion - equation that states two ratios are equal In a proportion, the product of the means is equal to the product of the extremes The Mean Proportional - means of proportional are equal, then each is called mean proportional between 1st and 4th term 2=8 8 32 8 = mean proportional 2, 32 = extremes A line segment joining the midpoint of two sides of a triangle is parallel to the third side and its length is ½ the length of the third side When the ratio of the lengths of the parts of one segment is equal to the ratio of the lengths of the parts of the other = divided proportionally 12 - 3 Similar Polygons/Proving Triangles Similar AA Triangle Similarity - Two triangles are similar if two angles of one triangle are congruent to the corresponding angles of the other SSS Similarity Theorem - Two triangles are similar if three ratios of corresponding sides are equal A line that is parallel to one side of a triangle and intersects the other two sides in different points cuts off a triangle similar to the given angle 12 - 4 Proportional Relations Among Segments Related to Triangles If two triangles are similar the lengths of corresponding altitudes, medians and angle bisectors have the same ration as the lengths of any two corresponding sides The perimeters of two similar triangles have the same ratio as the lengths of any pair of corresponding sides 12 - 5 Concurrence of Medians of a Triangle Centroid - If 3 medians of a triangle are drawn, they intersect at one point called the centroid Any two medians of a triangle intersect in a point that divides each median in a ration of 2:1 The medians of a triangle are concurrent 12 - 6 Proportions in a Right Triangle Projection of a Point on a Line - foot of a perpendicular drawn from the point on a line Projection of a Segment - when the segment is not perpendicular to the line, the projection of the segment whose endpoints are projection of the endpoints of the given line segment on the line The altitude to the hypotenuse of a right triangle divided the triangle into two triangles that are similar to each other and to the original triangle The length of each leg of a right triangle is the mean proportional Between the length of the projection of that leg on the hypotenuse And the length of the hypotenuse x a hyp = b a=x x b The length of the altitude to the hypotenuse of a right triangle is the mean proportional between the length of the projections of the legs on the hypotenuse a alt = x b a=x x b 12 - 7 Pythagorean Theorem Pythagorean Theorem: a2 + b2 = c2 Chapter 13: Geometry of Circles 13 - 1 Arcs and Angles All radii of the same circle are congruent Central Angle - angle whose vertex is the center of the circle Minor Arc - less than 180 Major Arc - greater than 180 In a circle or congruent circles, central angles are congruent if their arcs are congruent 13 - 2 Arcs and Chords Chord - line segment whose endpoints are points of the circle Diameter - chord that has the center of the circle as one of its points In a circle or in congruent circles, two arcs are congruent if and only if their central angles are congruent In a circle, two chords are congruent if and only if their arcs are congruent Apothem - perpendicular segment from the center of the circle to the midpoint of a chord, also length of segment Two chords are equidistant from the center of a circle if their distance away is congruent 13 - 3 Inscribed Angles and Their measures Inscribed Angle - angle whose vertex is on the circle and whose sides are chords The measure of an inscribed angle of a circle is equal to ½ the measure of its intercepted arc An angle inscribed in a semicircle is a right circle If two inscribed angles of a circle intercept the same arc, then they are congruent 13 - 4 Tangents and Secants Tangent - line in a plane of the circle that intersects the circle in one point Secant - intersects the circle in two points A line is tangent to a circle if it is perpendicular to the radius 13 - 5 Angles Formed by Tangents, Chords, and Secants The measure of an angle formed by a tangent and a chord that intersect at the point of tangency is equal to ½ the measure of the intercepted arc The measure of an angle formed by two chords intersecting within a circle is equal to ½ the sum of the measure of the arcs intercepted by the angles and its vertical angle The measure of an angle formed by a tangent and a secant, two secants, or two tangents intersection outside the circle is equal to ½ the difference of the measure of the intercepted arcs 13 - 6 Measure of Tangents Segments, Chords, and Secant Segment If two chords intersect within a circle, the product of the measure of the segments of one chord is equal to the product of the measure of the segment of the other. AE(EB) = CE(ED) If a tangent and a secant are drawn to a circle from an external point, the square of the tangent segment equals the product of the whole segment and its external segment (AB)2 = BD(BC) A D E C B A D If two secant segments are drawn to a circle from an external point, then the product of the lengths of one secant segment and its external segments is equal to the product of the lengths of the other secant segments and its external segment AC(AB) = AE(AD) B C C E B D 13 - 7 Circles in the Coordinate Plane Center-radius equation of a circle: Center: (h,k) (x - h)2 + (y - k)2 = r2 Chapter 14: Locus 14 - 1 Five Fundamental Locus Locus - set of all points, and only those points that satisfy a given set of conditions 1. The locus of points equidistant from 2 fixed points A and B is a perpendicular bisector . . 2. The locus of points equidistant from 2 intersecting lines AB and CD D A C B 3. The locus of points equidistant from 2 parallel lines AB and CD A 4. The locus of points a fixed distance d from a line AB 5. The locus of points a fixed distance d from fixed point A . 14 - 2 Parabola x = -b 2a Finds axis of symmetry Then, plug x-coordinate into the parabola formula to find the y-coordinate Make a table, using the x from the equation above, and 2 x points below and 2 above Once Solved, plot all points