Topic 1: 1.5 – 1.8 Goals and Common Core Standards Ms. Helgeson Recognize and analyze a conditional statement. Write postulates about points, lines, and planes using conditional statements. Recognize and use definitions. Recognize and use biconditional statements.

Use symbolic notation to represent logical statements. Form conclusions by applying the laws of logic to true statements. Use properties from algebra Use properties of length and measure to justify segment and angle relationships. Justify statements about congruent segments. Write reasons for steps in a proof.

Use angle congruence properties. Prove properties about special pair of angles. CC.9-12.G.CO.9 CC.MP.3

1.5 Conditional Statements

1.5 Conditional Statements A conditional statement is a type of logical statement that has two parts, the hypothesis “if” and the conclusion “then”. Conditionals can be represented as p q, read as “If p, then q,” where p represents the hypothesis and q represents the conclusion. .

Examples If a plane exists, then it contains at hypothesis conclusion least three noncollinear points. A line contains at least two points. If a line exist, then it contains at least two points.

Finding the inverse, converse, and the contrapositive of a conditional statement. Conditional Statement: If p, then q. or p q. Inverse: If not p, then not q. or ~ p ~ q. Converse: If q, then p. or q p. Contrapositive: If not q, then not p. or ~ q ~ p.

Inverse An inverse is the statement formed when you negate the hypothesis and conclusion of a conditional statement. If a plane exist, then it contains at least three noncollinear points. Inverse: Ex: If a plane does not exist, then it is not true that it contains at least three noncollinear points.

Converse The converse of a conditional statement is formed by switching the hypothesis and the conclusion. If a plane exist, then it contains at lease three noncollinear points. Converse: Ex: If a plane contains at least three noncollinear points, then the plane exists.

Contrapositive A contrapositive is the statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement. If a plane exist, then it contains at least three noncollinear points. Contrapositive: If it is not true that a plane contains a least three noncollinear points, then the plane does not exist.

Write an [a] inverse, [b] converse, and [c] contrapositive of the following statement. If the sun is shining, then we are not watching tv.

If the sun is shining, then we are not watching TV. Inv: If the sun is not shining, then we are watching TV. Conv: If we are not watching TV, then the sun is shining. Contrapositive: If we are watching TV, then the sun is not shining.

If my allowance increases, then I can save more money. Write the [a] inverse, [b] converse, and [c] contrapositive of the conditional statement. If my allowance increases, then I can save more money. Answers on next slide

Answers!! Inverse: If my allowance does not increase, then I cannot save more money. Converse: If I can save more money, then my allowance increased. Contrapositive: If I cannot save more money, thenmy allowance does not increase.

Rewrite the conditional statements in [a] if-then form, [b] inverse, [c] converse, and [d] contrapositive. Three points are coplanar if they lie on the same plane. Water freezes at temperatures below 32ºF. An even number is divisible by 2.

A square must have four congruent sides A square must have four congruent sides. Conditional: If a polygon is a square, then it has four congruent sides. Inverse, converse, contrapositive ????

Truth value of a conditional The truth value of a statement is “true” (T) or “false” (F) according to whether the statement is true or false, respectively. A truth table lists all the possible combinations of truth values for two or more statements. p 37

To determine the truth value of a conditional, consider all the options for the hypothesis and for the conclusion. For example, assume the hypothesis is true, then determine whether the conclusion must also always be true. Ex: If a number is even, then it is divisible by 2. next page

An even number is always divisible by two, so then the hyp is true, the conclusion is always true. The conditional is true. Ex:2 If a quadrilateral has two pairs of congruent angles, then it is a parallelogram. False: trapezoid The hyp is true and conclu is false, so conditional is false.

Truth value of Converse: Cond: If you play the trumpet, then you play a brass instrument. Conv: If you play a brass instrument, then you play the trumpet. Truth value: If you play a brass instrument, then you may play a brass instrument that is not a trumpet. The converse is false.

Cond: If two whole numbers are even, then their sum is even Cond: If two whole numbers are even, then their sum is even. Truth statement for inverse and contrap. Inv: If two whole numbers are not even, then their sum is not even. 3 + 5 = 8 false Contrap: If the sum of two whole numbers is not even, then the numbers are not even. 5 + 8 = 13 false

Biconditional Statement A biconditional statement is the combination of a conditional, p q, and its converse, q p. The resulting compound statement p q is read as “p if and only if q.”

When p and q have the same truth value, the biconditional is true When p and q have the same truth value, the biconditional is true. When they have opposite truth values, it is false. p q p q T F

Example: An angle is a straight angle if and only if its measure is 180º. True

Let's rewrite our last example: Conditional statement: If a polygon has three sides, then it is a triangle. Converse statement: If a polygon is a triangle, then it has three sides. Since both are statements are true, we can go ahead and make our biconditional statements: A polygon is a triangle 'if and only if' it has three sides. A polygon has three sides 'if and only if' it is a triangle.

1.6 Deductive Reasoning p 44 Deductive Reasoning is a process of reasoning using given and previously known facts to reach a logical conclusion. Proving statements by reasoning from accepted postulates, definitions, theorems, properties, and given information.

Another definition of Deductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument. This differs from inductive reasoning, in which previous examples and patterns are used to form a conjecture.

Inductive and Deductive Reasoning Inductive: Josh knows that Brand X computers cost less than Brand Y computers. All other brands that Josh knows of cost less than Brand X. Josh reasons that Brand Y cost more than all other brands. Deductive: Josh knows that Brand X computers cost less than Brand Y computers. He also knows that Brand Y computers cost less than Brand Z. Josh reasons that Brand X cost less than Brand Z.

There are two laws of deductive reasoning There are two laws of deductive reasoning. The first is the Law of Detachment, And the second is the Law of Syllogism.

Determine whether a statement if true Determine whether a statement if true. Given that a conditional and its conclusion are true, can you use deductive reasoning to determine whether the hypothesis is true. You are given the facts that p q is true and q is true. Make a truth table for the conditional p q. When p q and q are true , p can be true or false. p q p q T F

The Law of Detachment is a low of logic that states if a conditional statement and its hypothesis are true, then its conclusion is also true. If … p q and p are true. Then… q is true.

Law of Detachment Ex: If Alicia scores 85 or greater on her test, she will earn an A as her final grade. Alicia scores 89 on her test. What can you logically conclude? The hypothesis is true because 89 > 85. Since the hypothesis is true, you can conclude that Alicia will earn an A as her final grade.

Law of Detachment If p → q is true conditional statement and p is true, then q is true. Example: If two angles are vertical, then they are congruent. < ABC and < DBE are vertical. So < ABC and < DBE are congruent. The logical argument is a valid use of the Law of Detachment. It is given that both a statement (p→q) and its hypothesis (p) are true. So, it is valid that < ABC and < DBE are congruent.

Example: Sarah knows that all sophomores take driver education in her school. Hank takes driver education. So Hank is a sophomore. This logical argument is not a valid use of the Law of Detachment. Given that a statement (p→q) and its conclusion (q) are true does not mean the hypothesis (p) is true. The argument implies that everyone that takes driver education is a sophomore.

If two angles form a linear pair, then they are supplementary; <A and <B are supplementary. So, <A and <B for a linear pair. Not valid. Given that a statement p q and its conclusion q are true does not mean the hypothesis p is true. The argument implies that all supplementary angles for a linear pair. 60 120

Law of Syllogism The Law of Syllogism is a law of logic that states that given two true conditionals with the conclusion of the first being the hypothesis of the second, there exists a third true conditional having the hypothesis of the first and the conclusion of the second. If … p q and q r are true. Then… p r is true

Law of Syllogism If p → q and q → r are true conditional statements, then p → r is true. Example: If a bird is the fastest bird on land, then it is the largest of all birds. If a bird is the largest of all birds, then it is a ostrich. Use Law of Syllogism: If a bird is the fastest bird on land, then it is an ostrich.

1.7 Writing Proofs p 51

2.4 Algebraic Properties of Equality Addition Property: if a=b, then a + c = b + c. Ex: If WX = YZ, then WX + AB = YZ + AB Ex: If m< 1 = 10˚, then 5˚ + m< 1 = 15˚ Subtraction Property: if a=b, then a – c = b - c. Ex: If PQ + ST = RS + ST, THEN PQ = RS

Reflexive Property: For any real number a, a = a. AB = AB Symmetric Property: If a = b, then b = a. If FG = EF, then EF = FG

Multiplication Property: if a=b, then ac=bc. Ex: If AB = 10, then 3(AB) = 30 Division Property: if a=b and c≠0, then a ÷ c = b ÷ c. Ex: If 10x = 20, then x = 2

Transitive Property: If a = b and b = c, then a = c. If AB = JK and JK = ST, then AB = ST Substitution Property: If a = b, then “a” can be substituted for “b” in any equation or expression. If AB + CD = 15 and CD = 5, then AB + 5 = 15.

Distributive Property of Equality: a(b + c) = ab + ac

Example Solve -2x + 1 = 56 – 3x and write a reason for each step. -2x + 1 = 56 – 3x Given x + 1 = 56 Addition prop of = x = 55 Subtr. Prop. Of =

Example 12x – 3(x + 7) = 8x 12x – 3x – 21 = 8x 9x - 21 = 8x -21 =-1x Given Distr. Prop. Subtr. prop of = Div. prop of =

Theorem is a conjecture that is proven Theorem is a conjecture that is proven. Vertical Angles Theorem: Vertical angles are congruent.

Def. of Linear Pair Def: A linear pair consists of two adjacent angles whose noncommon sides are opposite rays. Linear pairs of angles are supplementary by the Linear Pair Postulate. <1 and <2, <2 and <3, <3 and <4, and <4 and <1 are linear pairs of angles. 2 3 1 4

Example Solve for x and y. Then find the angle measures. 4x + 15

Example A) Name one pair of vertical angles and one pair of angles that form a linear pair. B) What is the measure of <GHI in the figure above? J I 5x + 30 H 2x - 4 G K

Def. of Complementary Angles Complementary angles are two angles whose measures have the sum 90°. Complement: The sum of the measures of an angle and its complement is 90°. 20 or 70

Def. of Supplementary Angles Supplementary angles are two angles whose measures have the sum 180°. Supplement: The sum of the measures of an angle and its supplement is 180°. 120 or 60

Examples 1. Given that <A is a complement of <C and m<A = 47°, find m<C. 2. Given that <P is a supplement of <R and m<R = 36°, find m<P. 3. <W and <Z are complementary. The measure of <Z is 5 times the measure of <W. Find m<W. 4. <T and <S are supplementary. The measure of <T is half the measure of <S. Find m<S.

Examples 5. When two lines intersect, the measure of one of the angles they form is 20° less than three times the measure of one of the other angles formed. What are the measures of all four angles formed by the lines?

Theorem- A theorem is a statement that is proved. Two- column proof- A two-column proof is a type of proof written as numbered statements and reasons that show the logical order of an argument.

Parts of a two column proof: Given, prove, diagram, statements, and reasons. Reasons: Given, definitions, postulates, properties of equality, and theorems.

Paragraph proof- A paragraph proof is a type of proof written in paragraph form.

Theorem: Properties of Segment Congruence Reflexive For any segment AB, AB AB Symmetric If AB CD, then CD AB Transitive If AB CD, and CD EF, then AB EF

Proving Statements about Angles Theorem 2.2 Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Reflexive For any angle A, <A=<A Symmetric If <A=<B, then <B=<A Transitive If <A=<B and <B=<C, then <A=<C.

Right Angle Congruence Theorem All right angles are congruent.

Thm. 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. If m< 1 + m< 2 = 180 and m<2 + m < 3 = 180, then < 1 = < 3. 2 1 3

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then the two angles are congruent. If m<4+m<5=90º and m<5+m<6=90º, then <4 <6 1 2 3 6 5 4

Vertical Angles Theorem Linear Pair Postulate If two angles form a linear pair, then they are supplementary. m<1+m<2=180º 1 2 Vertical Angles Theorem Vertical angles are congruent. <1 <3 and <2 <4 2 3 4

Theorem: If two angles are congruent and supplementary, they each is a right angle.

In the diagram at the right, B is the midpoint of AC and C is the midpoint of BD. Show that AB=CD.

Given: RT and PQ intersecting at S so that RS = PS and ST = SQ Given: RT and PQ intersecting at S so that RS = PS and ST = SQ. Prove: RT = PQ Given: m< AOC = m< BOD Prove: m< 1 = m< 3 P R S T Q B C D A 1 2 3 O

Theorem- A theorem is a statement that is proved. Two- column proof- A two-column proof is a type of proof written as numbered statements and reasons that show the logical order of an argument.

Parts of a two column proof: Given, prove, diagram, statements, and reasons. Reasons: Given, definitions, postulates, properties of equality, and theorems.

Paragraph proof- A paragraph proof is a type of proof written in paragraph form.

Theorem: Properties of Segment Congruence Reflexive For any segment AB, AB AB Symmetric If AB CD, then CD AB Transitive If AB CD, and CD EF, then AB EF

Ex: Given: EF = GH Prove: EG FH E F G H Statements Reasons

Given: RT WY, ST = WX Prove: RS XY R S T W X Y Statement Reasons

In the diagram at the right, <1 <5, <5 <3, m<1=103º In the diagram at the right, <1 <5, <5 <3, m<1=103º. What is the measure of <3? Explain your reasoning. 1 2 4 3 5 6 8 7

Theorem 2.3 Right Angle Congruence Theorem All right angles are congruent.

Thm. 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. If m< 1 + m< 2 = 180 and m<2 + m < 3 = 180, then < 1 = < 3. 2 1 3

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then the two angles are congruent. If m<4+m<5=90º and m<5+m<6=90º, then <4 <6 1 2 3 6 5 4

Vertical Angles Theorem Linear Pair Postulate If two angles form a linear pair, then they are supplementary. m<1+m<2=180º 1 2 Vertical Angles Theorem Vertical angles are congruent. <1 <3 and <2 <4 2 3 4

Given: < 2 < 3 Prove: < 1 < 4 Given: < 3 supp Given: < 2 < 3 Prove: < 1 < 4 Given: < 3 supp. to < 1 < 4 supp. to < 2 Prove: < 3 < 4 3 4 1 2 3 4 1 2

Given: <5 <6 Prove: <4 <7 Statements Reasons

Given: < QVW and < RWV are supplementary Prove: < QVP < RWV

In the diagram, <3 is a right angle and m<5=57º In the diagram, <3 is a right angle and m<5=57º. Find the measures of <1, <2, <3, and <4. 2 3 1 4 5

Find the measure of each angle. Find m<1 and m<2 m<1=159º, m<2=148º Find the measure of each angle. 46º , 134º 1 2 (3x-17) º (6x+8) º 32º 21º

1.8 Indirect Proof p 58 A proof that uses indirect reasoning is an indirect proof. Use indirect proof when a direct proof is impossible. Two types of indirect proof are proof by contradiction and proof by contrapositive.

Proof by Contradiction p 59 A statement is given as a conditional p q. Step 1: Assume p and ~q are true. Step 2: Show that the assumption ~q leads to a contradiction. Step 3: Conclude that q must be true.

Proof by Contrapositive: p 60 Step 1: Assume ~q is true Proof by Contrapositive: p 60 Step 1: Assume ~q is true. Step 2: Show that the assumption leads to ~p, which shows ~q ~p. Step 3: Conclude that p q must be true.