Reasoning and Proofs Chapter 2

Conditional Statements I can write conditional and bi-conditional statements.

Conditional Statements Vocabulary (page 35 in Student Journal) conditional: an if-then statement hypothesis: the part of the statement following if conclusion: the part of the statement following then negation: the opposite of a statement converse: exchange the hypothesis and the conclusion of a conditional statement

Conditional Statements inverse: negate both the hypothesis and the conclusion of a conditional statement contrapositive: negate both the hypothesis and the conclusion of the converse statement equivalent statements: statements with the same truth value biconditional: a single statement that combines a true conditional statement with its true converse, which uses the phrase if and only if truth value: determining if the conditional statement is true or false

Conditional Statements Example (page 38 in Student Journal) #3) Let p be “Quadrilateral ABCD is a rectangle” and let q be “the sum of the angle measures is 360 degrees.” Write conditional, converse, inverse, and contrapositive statements, then decide if each statement is true or false.

Conditional Statements Solution Conditional: If quadrilateral ABCD is a rectangle, then the sum of its angle measures is 360 degrees. (true) Converse: If the sum of the angle measures is 360 degrees, then quadrilateral ABCD is a rectangle. (false) Inverse: If quadrilateral ABCD is not a rectangle, then the sum of its angle measures is not 360 degrees. (false) Contrapositive: If the sum of the angle measures is not 360 degrees, then quadrilateral ABCD is not a rectangle. (true)

Conditional Statements Additional Example (space on pages 36 and 37 in Student Journal) Write the following statement as a conditional statement. a) dolphins are mammals

Conditional Statements Solution a) If an animal is a dolphin, then it is a mammal.

Conditional Statements Additional Examples (space on pages 36 and 37 in Student Journal) Determine the truth value of the conditional statement. Note: If we can find one example (counterexample) for which the hypothesis is true but the conclusion is false then the truth value of the conditional statement is false. b) If you live in a country that borders the United States, then you live in Canada. c) If 2 angles form a linear pair, then they are supplementary.

Conditional Statements Solutions b) false, Mexico borders the United States as well c) true, for angles to form a linear pair they must be supplementary

Conditional Statements Additional Example (space on pages 36 and 37 in Student Journal) Determine if the converse of the conditional below is true and write a biconditional statement if possible. Note: We can write a biconditional statement by joining the 2 parts of the conditional statement (hypothesis and conclusion) with the phrase if and only if. d) If 2 angles have equal measures, then the angles are congruent.

Conditional Statements Solution d) converse: If 2 angles are congruent, then the angles have equal measures. (true) biconditional: Two angles have equal measure if and only if the angles are congruent.

Conditional Statements Additional Example (space on pages 36 and 37 in Student Journal) Write the biconditional statement below as 2 conditional statements that are converses. e) Two numbers are reciprocals if and only if their product is 1.

Conditional Statements Solution e) If 2 numbers are reciprocals, then their product is 1. If the product of 2 numbers is 1, then the numbers are reciprocals.

Inductive and Deductive Reasoning I can use reasoning to solve problems.

Inductive and Deductive Reasoning Vocabulary (page 41 in Student Journal) conjecture: an unproven statement based on observation inductive reasoning: using a pattern for a specific situation to then make a conjecture for a general situation counterexample: a specific case that shows a conjecture is false deductive reasoning: the process of reasoning logically from given statements or facts to a conclusion

Inductive and Deductive Reasoning Core Concepts (page 42 in Student Journal) Law of Detachment If the hypothesis of a true conditional statement is true, then the conclusion is true. Law of Syllogism Allows us to state a conclusion from 2 true conditional statements when the conclusion of 1 statement is the hypothesis of another statement.

Inductive and Deductive Reasoning Examples (page 42 and 43 in Student Journal) Describe the pattern. The find the next 2 terms in the sequence. #1) 20, 19, 17, 14, 10, … Make a conjecture about the given quantity. #5) the sum of 2 negative integers

Inductive and Deductive Reasoning Solutions #1) down 1, down 2, down 3, down 4, …, 5, -1 #5) is always a negative integer

Inductive and Deductive Reasoning Example (page 43 in Student Journal) Find a counterexample to show the conjecture is false. #8) Line k intersects plane P at point Q on the plane. Plane P is perpendicular to line k.

Inductive and Deductive Reasoning Solution #8) line k can intersect plane P at any angle

Inductive and Deductive Reasoning Examples (page 43 in Student Journal) Use the Law of Detachment, if possible, to determine what you can conclude. #9) If a triangle has equal side lengths, then each interior angle measure is 60 degrees. Triangle ABC has equal side lengths. Use the Law of Syllogism to write a new conditional statement, if possible. #12) If x > 1, the 3x > 3. If 3x > 3, then (3x)2 > 9.

Inductive and Deductive Reasoning Solutions #9) Triangle ABC has each interior angle measure of 60 degrees. #12) If x > 1, the (3x)2 > 9.

Inductive and Deductive Reasoning Additional Example (space on pages 41 and 42 in Student Journal) What can you conclude from the given information? Note: In some cases we can use both the Law of Detachment and the Law of Syllogism to form a conclusion. a) If a river is more than 4000 miles long, then it is longer than the Amazon. If a river is longer than the Amazon, then it is the longest river in the world. The Nile River is 4132 miles long.

Inductive and Deductive Reasoning Solution a) the Nile River is the longest river in the world

Postulates and Diagrams I can identify postulates using diagrams and sketch and interpret diagrams.

Postulates and Diagrams Core Concepts (pages 46 and 47 in Student Journal) Two Point Postulate (Postulate 2.1) Through any 2 points, there exists exactly 1 line. Line-Point Postulate (Postulate 2.2) A line contains at least 2 points. Line Intersection Postulate (Postulate 2.3) If 2 lines intersect, then their intersection is exactly 1 point

Postulates and Diagrams Three Point Postulate (Postulate 2.4) Through any 3 noncollinear points, there exists exactly 1 plane. Plane-Point Postulate (Postulate 2.5) A plane contains at least 3 noncollinear points. Plane-Line Postulate (Postulate 2.6) If 2 points lie in a plane, then the line containing them lies in the plane. Plane Intersection Postulate (Postulate 2.7) If 2 planes intersect, then their intersection is a line.

Postulates and Diagrams Examples (pages 47 and 48 in Student Journal) State the postulate illustrated in the diagram. #2) Use the diagram to give an example of the postulate. #6) Plane Intersection Postulate

Postulates and Diagrams Solutions #2) Plane-Line Postulate #6) Plane P and Plane Q intersect in line k

Postulates and Diagrams Examples (page 48 in Student Journal) Sketch a diagram of the description. #7) ray RS bisects segment KL at point R Use the diagram to determine if you can assume the statement. #10) Points C and D are collinear.

Postulates and Diagrams Solutions #7) sample answer: #10) yes

Algebraic Reasoning I can use algebraic properties to to help solve equations.

Algebraic Reasoning Core Concepts (pages 51 and 52 in Student Journal) Properties of Equality (for real numbers a, b and c) Addition: if a = b, then a + c = b + c Subtraction: if a = b, then a - c = b - c Multiplication: if a = b, then ac = bc Division: if a = b, then a/c = b/c Substitution: if a = b, then b can replace a Distributive: a(b + c) = ab + ac Reflexive: a = a Symmetric: if a = b, then b = a Transitive: if a = b and b = c, then a = c

Algebraic Reasoning Example (page 53 in Student Journal) Solve the equation. Justify each step. Note: We can use deductive reasoning when solving an equation by justifying each step with a postulate, property or definition. #3) -5(2u + 10) = 2(u – 7)

Algebraic Reasoning Solution #3) -5(2u + 10) = 2(u – 7) given -10u – 50 = 2u – 14 Distributive Property -50 = 12u – 14 Addition Property -36 = 12u Addition Property -3 = u Division Property u = -3 Symmetric Property

Proving Statements about Segments and Angles I can prove mathematical statements through two-column proofs.

Proving Statements about Segments and Angles Vocabulary (page 56 in Student Journal) proof: an argument that uses deductive reasoning to logically show why a conjecture is true two-column proof: lists each statement in the left column and the justification on the right theorem: a conjecture or statement that can be proven true

Proving Statements about Segments and Angles Core Concepts (page 56 in Student Journal) Properties of Segment Congruence (Theorem 2.1) Reflexive: segment AB is congruent to segment AB Symmetric: if segment AB is congruent to segment CD, then segment CD is congruent to segment AB Transitive: if segment AB is congruent to segment CD and segment CD is congruent to segment EF, then segment AB is congruent to segment EF These theorems also apply to angles (Theorem 2.2).

Proving Statements about Segments and Angles Example (page 58 in Student Journal) Complete the proof. #2)

Proving Statements about Segments and Angles Solutions #2) 2. m<AEB + m<BEC = 90 degrees 3. Angle Addition Postulate 4. Transitive Property of Equality 6. m<AED + 90 degrees = 180 degrees 7. Subtraction Property of Equality

Proving Statements about Segments and Angles Example (page 58 in Student Journal) Write a 2-column proof. #5) Given: M is the midpoint of segment RT Prove: MT = RS + SM

Proving Statements about Segments and Angles Solution #5) Statements Reasons 1. M is the midpoint of 1. given segment RT 2. segment RM is congruent to segment MT 2. def. of midpoint 3. RM = MT 3. def. of congruence 4. RM = RS + SM 4. segment add. post. 5. MT = RS + SM 5. substitution prop.

Proving Statements about Segments and Angles Additional Example (space on pages 56 and 57 in Student Journal) a) Prove that vertical angles are congruent using a two-column proof (Angle 1 and angle 3 are vertical angles).

Proving Statements about Segments and Angles Solution a) Statements Reasons 1. angle 1 and angle 3 are vertical angles 1. given 2. angle 1 and angle 2 are supplementary, angle 2 and angle 3 are supplementary 2. def. linear pair 3. measure of angle 1 + measure of angle 2 = 180, measure angle 2 + measure angle 3 = 180 3. def. suppl.

Proving Statements about Segments and Angles Solution (continued) Statements (cont.) Reasons (cont.) 4. measure of angle 1 + measure of angle 2 = measure of angle 2 + measure of angle 3 4. trans. equal. 5. measure angle 1 = measure angle 3 5.subtrac. equal. 6. angle 1 is congruent to angle 3 6. def. congruent

Proving Geometric Relationships I can use flowcharts and paragraphs to prove geometric relationships.

Proving Geometric Relationships Vocabulary (page 61 in Student Journal) flowchart proof: uses boxes and arrows to show a flow of a logical argument paragraph proof: a proof written in sentences in a paragraph

Proving Geometric Relationships Core Concepts (pages 61 and 62 in Student Journal) Right Angles Congruence Theorem (Theorem 2.3) All right angles are congruent. Congruent Supplements Theorem (Theorem 2.4) If 2 angles are supplementary to the same angle (or congruent angles), then they are congruent. Congruent Complements Theorem (Theorem 2.5) If 2 angles are complementary to the same angle (or congruent angles), then they are congruent.

Proving Geometric Relationships Linear Pair Postulate (Postulate 2.8) If 2 angles form a linear pair, then they are supplementary. Vertical Angles Congruence Theorem (Theorem 2.6) Vertical angles are congruent.

Proving Geometric Relationships Example (page 63 in Student Journal) Complete the flowchart proof. #1) Given: <ACB and <ACD are supplementary, <EGF and <ACD are supplementary Prove: <ACB is congruent to <EGF

Proving Geometric Relationships Solution #1) m<ACB + m<ACD = m<EGF + m <ACD def. supplementary <EGF and <ACD are supplementary def. congruent

Proving Geometric Relationships Additional Example (space on pages 61 and 62 in Student Journal) a) Prove the Congruent Supplements Theorem using a paragraph proof (Angles 1 and 3 are supplementary and angles 2 and 3 are supplementary).

Proving Geometric Relationships Solution a) Angles 1 and 3 are supplementary because it is given. The measure of angle 1 + the measure of angle 3 = 180 by the definition of supplementary. Angles 2 and 3 are supplementary because it is given. The measure of angle 2 + the measure of angle 3 = 180 by the definition of supplementary. The measure of angle 1 + the measure of angle 3 = the measure of angle 2 + the measure of angle 3 by the transitive property of equality. The measure of angle 1 = the measure of angle 2 by the subtraction property of equality. Angle 1 is congruent to angle 2 by the definition of congruent.