Geometry Chapter 2

Conditional Statements A conditional statement is a type of logical statement in the form of if → then A statement is a sentence whose truth value can be determined If a bird is a pelican, then it eats fish. hypothesisconclusion pq→ 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. If a bird eats fish then it is a pelican. hypothesisconclusion Truth value of a converse statement is not necessarily the same as the original statement. What does true or false mean? True only if all cases true. False if one counter example exists. pq→

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. task, write the converse of this statement If it is afternoon, then the sun is in the west. combined 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. pq↔

practice, write the following as a conditional statement All apples are a fruit. If a food is an apple then it is a fruit. 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 P 5 – Through any two distinct points, there exists exactly one line. P 6 – A line contains at least two points

Basic Postulates P 7 – Through any three non collinear points, there exists exactly one plane. P 8 – A plane contains at least 3 non-collinear points

Basic Postulates P 9 – If two distinct points lie in a plane, then the line containing them lies in the plane.

Basic Postulates P 10 – 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 EqualityIf a = b, then a + c = b + c Adding the same value to equivalent expressions maintains the equality If NK = 8, then NK + 4 = 12Each side is increased by 4. or 1. NK = 8 2. NK + 4 = 12 addition property of equality statementreason

Subtraction Property of EqualityIf a = b, then a - c = b - c Subtracting the same value from equivalent expressions maintains the equality If m  A = 55, then m  A - 5 = 50 Each side is decreased by 5. or 1. m  A = m  A - 5 = 50 subtraction property of equality statementreason

Multiplication Property of EqualityIf a = b, then ac = bc Multiplying equivalent expressions by the same value maintains the equality If m  A = 36, then 2(m  A) = 72 Each side multiplied by 2 or 1. m  A = (m  A) = 72 multiplication property of equality statementreason

Division Property of EqualityIf a = b, then a ÷ c = b ÷ c Dividing equivalent expressions by the same value to maintains the equality If YD = 15, then YD/3 = 5Each side divided by 3 or 1. YD = YD/3 = 5 division property of equality statementreason

Distributive property of multiplication over addition ab + ac = a(b + c) The distributive property in reverse allows us to combine like terms 1. 3x + 6x 2. x(3 + 6) 3.x(9) 4.9x distributive property number fact symmetric propertry statementreason

Reflexive Property of Equalityfor any real number a = a Any number is equal to itself 1. AB = ABreflexive property of equality statementreason AB

Symmetric Property of EqualityIf a = b, then b = a Equivalent expressions maintains their equality regardless of order If AB = CD then CD = AB or 1. AB = CD 2. CD = AB symmetric property of equality statementreason Sally likes Vijay and Vijay likes Sally AB CD

Transitive Property of EqualityIf a = b, and b = c, then a = c Two expressions equal to the same expression are equal to each other. If m  A = m  B and m  B = m  C then m  A = m  C or 1. m  A = m  B 2. m  B = m  C 3. m  A = m  C transitive property of equality statementreason ABC  A and  C are both equivalent to  B so they are equal to each other

Substution Property of EqualityIf 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 = ACCD replaces AB or 1. AB = CD 2.AB + BC = AC 3.CD + BC = AC substitution property of equality statementreason segment addition postulate

statementreason Given: AC = BD A B C D Prove: AB = CD 1. Given statement 1. Given #. Prove statement 1.AC = BD 2.AB + BC = AC 3.BC + CD = BD 4.AB + BC = BD 5.AB + BC = BC + CD 6.AB = CD 1. Given 2. Segment Addition Postulate (from picture) 3. Segment Addition Postulate 4. Substitution Property 5. Transitive Property 6. Subtraction Property 2 Column Proof Format #. reason varies Definitions Postulates Algebraic Properties Theorems

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.

statementreason Given: AC = BD A B C D Prove: AB = CD 1.AC = BD 2.AB + BC = AC 3.BC + CD = BD 4.AB + BC = BD 5.AB + BC = BC + CD 6.AB = BD 1. Given 2. Segment Addition Postulate (from picture) 3. Segment Addition Postulate 4. Substitution Property 5. Transitive Property 6. Subtraction Property 2 Column Proof Format Theorem Argument Body 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 &  4 are also a pair of vertical angles. 5 6  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. 12 statementreason 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. 1 2 If m  1 + m  2 = 90, then the angles are complementary statementreason 1.m  1 + m  2 =  1 &  2 are complementary 2. Definition of Complementary angles. Reversible Each angle is the complement of the other

Supplementary Angles If the sum of the measures of two angles is 180, then the angles are supplementary. 1 2 If m  1 + m  2 = 180, then the angles are supplementary statementreason 1.m  1 + m  2 =  1 &  2 are supplementary 2. Definition of Supplementary angles. Reversible Each angle is the supplement of the other

Linear Pair Postulate from picture 4.  1 &  2 are a linear pair. 5. m  1 + m  2 = Definition of linear pair. 5. Linear Pair Postulate If two angles form a linear pair, then they are supplementary. (m  1 + m  2 = 180) 12 statementreason

1. Solve : 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 1.  A &  C are supplementary 2.  B &  C are supplementary 3.  A   B If two angles are supplementary to the same or to congruent angles, then they are congruent. ABC If  A &  C are supplementary and  B &  C are supplementary then  A &  B are congruent. statement reason 3. Congruent Supplements Theorem

Congruent Supplements Theorem 1.  A &  C are supplementary 2.  B &  D are supplementary 3.  C   D 4  A   B If two angles are supplementary to the same or to congruent angles, then they are congruent. ABC If  A &  C are supplementary and  B &  D are supplementary then  A &  B are congruent. statement reason 4. Congruent Supplements Theorem D and  C &  D are congruent.

Congruent Supplements Theorem If two angles are supplementary to the same or to congruent angles, then they are congruent. ABC  A &  C are supplementary  B &  C are supplementary statement reason Given: Prove:  A   B 1.  A &  C also,  B &  C are supplementary 2. m  A + m  C = 180 m  B + m  C = m  A + m  C = m  B + m  C 4. m  A = m  B 5.  A   B 1. Given 2. Def. of Supplementary angles 3. Transitive Prop. of = 4. Subtraction Prop of = 5. Def. of Congruent angles

Congruent Complements Theorem 1.  A &  C are complementary 2.  B &  D are complementary 3.  C   D 4  A   B If two angles are complementary to the same or to congruent angles, then they are congruent. ABC If  A &  C are complementary and  B &  D are complementary then  A &  B are congruent. statement reason 4. Congruent Complements Theorem D and  C &  D are congruent.

Congruent Complements Theorem 1.  A &  C are complementary 2.  B &  C are complementary 3.  A   B If two angles are complementary to the same or to congruent angles, then they are congruent. ABC If  A &  C are complementary and  B &  C are complementary then  A &  B are congruent. statement reason 3. Congruent Complements Theorem

Vertical Angles Theorem 1. lines intersect at A 2.  1 &  3 are a linear pair  3 &  2 are a linear pair 3. m  1 + m  3 = 180 m  3 + m  2 = m  1 + m  3 = m  3 + m  2 5. m  1 = m  2 6.  1   2 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 A If two angles are vertical, then they are congruent. Given: lines m  l at point A m l Prove:  1   2 statementreason

Vertical Angles and Linear Pair Postulate Applications 10x x - 50 What type of angles? What is the relationship? What is the equation? m  1 = m  2 10x + 40 = 20x = 10x x = 9 m  = 10(9) + 40 = 130

Vertical Angles and Linear Pair Postulate Applications 10x x - 50 What type of angles? What is the relationship? What is the equation? m  1 + m  2 = x x – 50 = x -10 = x = 190 x = 19 / 3 m  = 10( 19 / 3 ) + 40 = 310 / 3 = / 3

1 2 m  1 = 2x + 15 m  2 = 6x – 5  1 &  2 are complementary m  1 + m  2 = 90 (2x + 15) + (6x – 5) = 90 8x + 10 = 90 8x = 80 X = 10 m  1 = 2x + 15 = 2(10) + 15 = 35 m  2 = 6x – 5 = 6(10) – 5 = 55