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Geometry Chapter 2 Conditional Statements A statement is a sentence whose truth value can be determined A conditional statement is a type of logical statement in the form of if → then If a bird is a pelican, then it eats fish. p hypothesis → q conclusion The conclusion is assured to happen only on the condition the hypothesis has been met. Converse of conditional statement The converse of a conditional statement has the hypothesis and the conclusion reversed. Truth value of a converse statement is not necessarily the same as the original statement. If a bird eats fish then it is a pelican. q hypothesis → p conclusion What does true or false mean? True only if all cases true. False if one counter example exists. Biconditional Statements A biconditional statement is one in which the original conditional statement and its converse have the same truth value. If the sun is in the west, then it is afternoon. If it is afternoon, then the sun is in the west. combined statement task, write the converse of this statement The sun is in the west if and only if it is afternoon. iff The sun is in the west iff it is afternoon. p ↔ q All apples are a fruit. practice, write the following as a conditional statement If a food is an apple then it is a fruit. Truth Value? practice, write the converse of the conditional statement If a food is a fruit then it is an apple. Truth Value? Is this a biconditional? Assignment 81/1 – 4, 9 – 20 Basic Postulates P5 – Through any two distinct points, there exists exactly one line. P6 – A line contains at least two points Basic Postulates P7 – Through any three non collinear points, there exists exactly one plane. P8 – A plane contains at least 3 non-collinear points Basic Postulates P9 – If two distinct points lie in a plane, then the line containing them lies in the plane. Basic Postulates P10 – If two distinct planes intersect, their intersection is a line. 80/communicate about geometry A – F 82/21 – 24, 27 – 33 83/mixed review 1 – 16 Reasoning with Properties of Algebra • • • • • • • • • Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Reflexive Property Symmetric Property Transitive Property Substitution Property Addition Property of Equality If a = b, then a + c = b + c Adding the same value to equivalent expressions maintains the equality If NK = 8, then NK + 4 = 12 or statement Each side is increased by 4. reason 1. NK = 8 2. NK + 4 = 12 addition property of equality Subtraction Property of Equality If a = b, then a - c = b - c Subtracting the same value from equivalent expressions maintains the equality If mA = 55, then mA - 5 = 50 or statement Each side is decreased by 5. reason 1. mA = 55 2. mA - 5 = 50 subtraction property of equality Multiplication Property of Equality If a = b, then ac = bc Multiplying equivalent expressions by the same value maintains the equality If mA = 36, then 2(mA) = 72 or statement Each side multiplied by 2 reason 1. mA = 36 2. 2(mA) = 72 multiplication property of equality If a = b, then a ÷ c = b ÷ c Division Property of Equality Dividing equivalent expressions by the same value to maintains the equality If YD = 15, then YD/3 = 5 or statement Each side divided by 3 reason 1. YD = 15 2. YD/3 = 5 division property of equality Distributive property of multiplication over addition ab + ac = a(b + c) The distributive property in reverse allows us to combine like terms statement reason 1. 3x + 6x 2. x(3 + 6) distributive property 3. x(9) number fact 4. 9x symmetric propertry Reflexive Property of Equality for any real number a = a Any number is equal to itself A B statement 1. AB = AB reason reflexive property of equality Symmetric Property of Equality If a = b, then b = a Equivalent expressions maintains their equality regardless of order Sally likes Vijay and Vijay likes Sally A B C D If AB = CD then CD = AB or statement reason 1. AB = CD 2. CD = AB symmetric property of equality Transitive Property of Equality If a = b, and b = c, then a = c Two expressions equal to the same expression are equal to each other. A B C If mA = mB and mB = mC then mA = mC A and C are both equivalent to B so they are equal to each other or statement reason 1. mA = mB 2. mB = mC 3. mA = mC transitive property of equality Substution Property of Equality If a = b, and a ± c = d, then b ± c = d If two expressions are equivalent, one can replace the other in any equation If AB = CD, and AB + BC = AC, then CD + BC = AC or statement CD replaces AB reason 1. AB = CD 2. AB + BC = AC segment addition postulate 3. CD + BC = AC substitution property of equality 2 Column Proof Format Given: AC = BD Prove: AB = CD A B statement C D reason 1.1. AC Given = BD statement • 1. Given 1. Given 2. AB + BC = AC • 2. Segment Addition Postulate (from picture) Definitions 3. BC + CD = BD • 3. Segment Addition Postulate Postulates 4. AB + BC = BD • 4. Substitution Property Algebraic Properties 5. AB + BC = BC + CD • 5. Transitive Property Theorems 6. AB = CD #. Prove statement • 6. Property #. Subtraction reason varies Structure of a Logical Argument 1. Theorem – Hypothesis, Conclusion 2. Argument body – Series of logical statements, beginning with the Hypothesis and ending with the Conclusion. 3. Restatement of the Theorem. (I told you so) If you are careless with fire, Then a fish will die. • • • • • • • • If you are careless with fire, Then there will be a forest fire. If there is a forest fire, Then there will be nothing to trap the rain. If there is nothing to trap the rain, Then the mud will run into the river. If the mud runs into the river, Then the gills of the fish will get clogged with silt If the gills of the fish get clogged with silt, Then the fish can’t breathe. If a fish can’t breath, Then a fish will die If you are careless with fire, Then a fish will die. 2 Column Proof Format Given: AC = BD Theorem Prove: AB = CD B A statement C D Argument Body reason 1. AC = BD • 1. Given 2. AB + BC = AC • 2. Segment Addition Postulate (from picture) 3. BC + CD = BD • 3. Segment Addition Postulate 4. AB + BC = BD • 4. Substitution Property 5. AB + BC = BC + CD • 5. Transitive Property 6. AB = BD 6. Subtraction Property • Restating Theorem left out Angle Relationships • • • • • • • • Vertical Angles Linear Pair (of angles) Complementary Angles Supplementary Angles Linear Pair Postulate Congruent Supplements Theorem Congruent Complements Theorem Vertical Angles Theorem Vertical Angles Vertical Angles are the non adjacent angles formed by two intersecting lines. 1 & 2 are a pair of vertical angles. 3 2 1 4 3 & 4 are also a pair of vertical angles. 6 5 5 & 6 are not a pair of vertical angles. Linear Pair If the noncommon sides of adjacent angles are opposite rays then the angles are a linear pair. 1 statement 2 reason from picture 4. 1 & 2 are a linear pair. 4. Definition of linear pair. Complementary Angles If the sum of the measures of two angles is 90, then the angles are complementary. Each angle is the complement of the other 1 2 If m1 + m2 = 90, then the angles are complementary statement reason 1. m1 + m2 = 90 Reversible 2. 1 & 2 are complementary 2. Definition of Complementary angles. Supplementary Angles If the sum of the measures of two angles is 180, then the angles are supplementary. 1 2 Each angle is the supplement of the other If m1 + m2 = 180, then the angles are supplementary statement Reversible reason 1. m1 + m2 = 180 2. 1 & 2 are supplementary 2. Definition of Supplementary angles. Linear Pair Postulate If two angles form a linear pair, then they are supplementary. (m1 + m2 = 180) 1 statement • • • from picture 4. 1 & 2 are a linear pair. 5. m1 + m2 = 180 2 reason • 4. Definition of linear pair. • 5. Linear Pair Postulate 1. Solve: x 4 3x 2 12 28 2. If the product of the slopes of two lines is -1, then the lines are perpendicular. a) write the converse b) write the statement represented by p↔q 3. Write an example of the transitive property. Congruent Supplements Theorem If two angles are supplementary to the same or to congruent angles, then they are congruent. A B C If A & C are supplementary and B & C are supplementary then A & B are congruent. statement • • • 1. A & C are supplementary 2. B & C are supplementary 3. A B reason 3. Congruent Supplements Theorem Congruent Supplements Theorem If two angles are supplementary to the same or to congruent angles, then they are congruent. A B C D If A & C are supplementary and B & D are supplementary and C & D are congruent. statement • • • • 1. A & C are supplementary 2. B & D are supplementary 3. C D 4 A B then A & B are congruent. reason 4. Congruent Supplements Theorem Congruent Supplements Theorem If two angles are supplementary to the same or to congruent angles, then they are congruent. A B C Given: A & C are supplementary B & C are supplementary Prove: A B statement • • • • • 1. A & C also, B & C are supplementary 2. mA + mC = 180 mB + mC = 180 3. mA + mC = mB + mC 4. mA = mB 5. A B reason • 1. Given • 2. Def. of Supplementary angles 3. Transitive Prop. of = 4. Subtraction Prop of = 5. Def. of Congruent angles • • • Congruent Complements Theorem If two angles are complementary to the same or to congruent angles, then they are congruent. A B C D If A & C are complementary and B & D are complementary and C & D are congruent. statement • • • • 1. A & C are complementary 2. B & D are complementary 3. C D 4 A B then A & B are congruent. reason 4. Congruent Complements Theorem Congruent Complements Theorem If two angles are complementary to the same or to congruent angles, then they are congruent. A B C If A & C are complementary and B & C are complementary then A & B are congruent. statement • • • 1. A & C are complementary 2. B & C are complementary 3. A B reason 3. Congruent Complements Theorem Vertical Angles Theorem If two angles are vertical, then they are congruent. m 3 Given: lines m l at point A Prove: 1 2 statement • • • • • • 1. lines intersect at A 2. 1 & 3 are a linear pair 3 & 2 are a linear pair 3. m1 + m3 = 180 m3 + m2 = 180 4. m1 + m3 = m3 + m2 5. m1 = m2 6. 1 2 2 1 A l reason • • 1. Given 2. Def. of Linear Pair (picture) • 3. Linear Pair Postulate • • • 4. Transitive Property of = 5. Subtraction Prop. of = 6. Def. of congruent angles Vertical Angles and Linear Pair Postulate Applications What type of angles? 10x + 40 What is the relationship? 20x - 50 What is the equation? m1 = m2 10x + 40 = 20x - 50 90 = 10x x=9 m = 10(9) + 40 = 130 Vertical Angles and Linear Pair Postulate Applications What type of angles? 10x + 40 20x - 50 What is the relationship? What is the equation? m1 + m2 = 180 10x + 40 + 20x – 50 = 180 30x -10 = 180 30x = 190 x = 19/3 m = 10(19/3) + 40 = 310/3 = 103 1/3 m1 = 2x + 15 m2 = 6x – 5 1 m1 = 2x + 15 = 2(10) + 15 = 35 2 m2 = 6x – 5 = 6(10) – 5 = 55 1 & 2 are complementary m 1 + m 2 = 90 (2x + 15) + (6x – 5) = 90 8x + 10 = 90 8x = 80 X = 10