Lesson 2.1 Inductive Reasoning Geometry Answers

Performance Task: Induction And The Next Dimension

Mathematical Practices

Explain the purpose of justifying each step in Exercises 5-14 on page 96.Answer:The purpose of justifying each step in exercises 5-14 is to learn algebraic properties of equality given below

• Addition Property of Equality

Create a diagram to model each statement in Exercises 5-10 on page 103.Answer:

Question 3.Explain why you would not be able to prove the statement in Exercise 21 on page 113 if you were provided with the given information or able to use an postulates or theorems.Answer:

Big Ideas Math Geometry Answers Chapter 2 Reasoning And Proofs

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Reasoning And Proofs Cumulative Assessment

Question 1.Use the diagram to write an example of each postulate.a. Two Point Postulate : Through any two points, there exists exactly one line.Answer:Read the different postulates and use the diagram to look for an example for each postulate.Passing through two points, A and F there exists exactly one line which is \

b. Line Intersection Postulate : If two lines intersect, then their intersection is exactly one point.Answer: \ intersects \ at exactly one point which is point C.

c. Three Point Postulate : Through any three noncollinear points, there exists exactly one plane.Answer:Points E, B, C there exists exactly one plane which is plane S.

d. Plane-Line Postulate : If two points lie in a plane, then the line containing them lies in the plane.Answer: Points A and F lies in plane T, therefore \ also lies in the plane T.

e. Plane Intersection Postulate : If two planes intersect, then their intersection is a lineAnswer: Planes S and T at \

Question 2.Enter the reasons in the correct positions to complete the two-column proof.Answer:2. AX = DX according to definition of congruent segments.4. XB = XC according to definition of congruent segments.5. AX + XC = AC according to segment addition postulate.6. DX + XB = DB according to segment addition postulate.7. AC = DX + XB according to substitution property of equality.8. AC = BD according to substitution property of equality.9. \ = \ according to substitution property of equality and definition of congruent segments.

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Presentation On Theme: 21 Inductive Reasoning Review Geometry R/h Inductive Reasoning When You Make A Prediction Based On Several Examples You Are Applying Inductive Reasoning Presentation Transcript:

1 2.1 Inductive Reasoning Review Geometry R/H

3 Inductive Reasoning When you make a prediction based on several examples, you are applying inductive reasoning. Formal Definition: Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true.

4 Making a Conjecture A statement you believe to be true based on inductive reasoning is called a conjecture. Example 2: Complete the conjecture. The sum of two positive numbers is ? List some examples and look for a pattern. 1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917 The sum of two positive numbers is positive.

5 Counterexamples To show that a conjecture is true you must prove it, but to show a conjecture is false, you need only to find one example for which the conjecture is false. This is called a counterexample. In the following conjecture, show that it is not true by finding a counterexample. For any real number x, x 2 x A counterexample could be x = ½ x 2 = 2 = ¼ ¼ ½ therefore, the conjecture is not true.

7 Steps in Inductive Reasoning Step 1: Look for a Pattern Step 2: Make a Conjecture Step 3: Is it True? Verify the Conjecture Find a Counter- example yesno

9 Go over Sequences Packet Arithmetic, Quadratic, Triangular, Geometric Sequences **

10 2.2 Logic and Venn Diagrams Geometry R/H

12 Conjunction and Disjunction Let p: I am tired. Let q: I will fall asleep

13 Now you try! Let p: Let q:

Extra: Practice: Key Ch2

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Inductive reasoning patterns geometry figure triangle squares triangles would many align there end each. Patterns reasoning inductive practice 1a worksheet geometry ws pdf reached ve end. Lesson reasoning inductive

Reasoning And Proofs Mathematical Practices

Monitoring Progress

Question 1.Figure ABC is a triangle.Therefore, figure ABC is a polygon.Answer:Yes, all the triangles are examples of polygons the name itself tells how many sides the shape has.Thus all the triangles are polygons.

Question 2.Some rectangles are not squares.Therefore, some squares are not trapezoids.Answer:No, all the squares are not trapezoids.A trapezoid is a quadrilateral with at least one pair of parallel sides.In square there are always two pairs of parallel sides.

Question 3.If polygon ABCD is a square. then ills a rectangle.Polygon ABCD is a rectangle.Therefore, polygon ABCD is a square.Answer:Yes, Polygon ABCD is a square.

Question 4.If polygon ABCD is a square, then it is a rectangle.Polygon ABCD is not a square.Therefore, polygon ABCD is not a rectangle.Answer:No, Polygon ABCD is not a rectangle.Polygons are plane figures made up of line segment.

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Exercise 25 Proving Statements About Segments And Angles

Vocabulary and Core Concept Check

Question 1.How is a theorem different from a postulate?Answer:A postulate is a rule that is accepted to be true without proof and a theorem is a statement that can be proven by using definitions, postulates, and previously proven theorems.

Question 2.COMPLETE THE SENTENCEIn a two-column proof, each __________ is on the left and each __________ is on the right.Answer: In a two-column proof, each statement is on the left and each reason is on the right.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4. copy and complete the proof.

Question 3.

Answer: Reflexive Property of Angle Congruence

Question 7.If G H. then H G.Answer:

Question 8.\Answer: The statement is talking about reflexity involving line segments, we can say that this observes the Reflexive Property of Segment Congruence

Question 9.If \, then \.Answer:

Question 10.If L M and M N, then L N.Answer: Since the if-then statement is talking about transitivity involving angles, we can say that this observes the Transitive Property of Angle Congruence

PROOFIn Exercises 11 and 12, write a two-column proof for the property.

Question 11.Reflexive Property of Segment Congruence Answer:

Transitive Property of Angle Congruence Answer:

From AB = BC and AB = FG substitution property of equalityBC = FGFrom BC = FG and DF = FG substitution property of equality\ \

Question 19.Explain why you do not use inductive reasoning when writing a proof.Answer:

Answer Key Lesson 21 Inductive Reasoning Answers

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Exercise 22 Inductive And Deductive Reasoning

Vocabulary and Core Concept Check

Question 1.How does the prefix counter help you understand the term counterexample?Answer:Because the prefix counter means opposing, a counter example opposes the truth of the statement.

Question 2.Explain the difference between inductive reasoning and deductive reasoning.Answer:Inductive reasoning is finding a pattern in specific case and then writing a conjecture for the general case.Deductive reasoning uses facts, definitions, accepted properties and the laws of logic to form a logical argument.Inductive reasoning would be like generalizing and deductive reasoning would be like concluding.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 8, describe the pattern. Then write or draw the next two numbers, letters, or figures.

Question 3.1, 2, 3, 4, 5, ..Answer:

0, 2, 6, 12, 20, ..Answer:This is sequence of numbers where first number is 0, and every next number is obtained by adding 2, 4, 6, to previous number.Next two numbers are: 30, 42

Question 5.Z, Y, X, W, V, ..Answer:

J, F, M, A, M, ..Answer:J = January, F = February, M = March, A = April, M = MayThe next two letters would be J and JJ = June, J = July

The quotient of two negative integers is positive rational number.

In Exercises 13 16, find a counter example to show that the conjecture is false.

Question 13.The product of two positive numbers is always greater than either number,Answer:

Question 29.the sum of two odd integersAnswer:

Question 47.

Lesson 21 Conditional Statements

Monitoring Progress

Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form.

Question 1.All 30° angles are acute angles.Answer: If an angle measures 30°, then it is acute angle.

Question 2.2x + 7 = 1. because x = 3.Answer: If x = -3, then 2x + 7 = 1

In Exercises 3 and 4, write the negation of the statement.

Question 3.Answer: The shirt is not green

Question 4.The Shoes are not red.Answer: The shoes are red

Question 5.Repeat Example 3. Let p be the stars are visible and let q be it is night.Answer:p q If the stars are visbile, then it is night Trueq p If it is night, then the stars are visible Falsep q If the stars are not visbile, then it is not night Falseq p If it is not night, then the stars are not visible True

Use the diagram. Decide whether the statement is true. Explain your answer using the definitions you have learned.

Question 6.JMF and FMG are supplementary.Answer: True. They are a linear pair

Question 7.Point M is the midpoint of \.Answer: False. There is mo marking to show \ \

Question 8.JMF and HMG arc vertical angles.Answer: True. They share a vertex and their sides for, opposite rays.

Question 9.Answer: False. You cannot assume their intersection is a right angle without markings.

Question 14.Make a truth table for the conditional statement p ~ q.Answer:

Vocabulary and Core Concept Check

Question 3.If a polygon is a pentagon, then it has five sides.Answer:

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< strong> Lesson< /strong> < strong> 2.1< /strong> < strong> < /strong> < strong> Inductive< /strong> < strong> Reasoning< /strong> Name Period Date For Exercises 17, use inductive reasoning to find the next two terms in each sequence. 1. 4, 8, 12, 16, _____, _____ 2. 400, 200, 100, 50, 25, _____, _____ 3. 1 , 2 , 1 , 4 , 8 7 2 5 _____, _____ 4. 5, 3, 2, 1, 1, 0, _____, _____ 5. 360, 180, 120, 90, _____, _____ 6. 1, 3, 9, 27, 81, _____, _____ 7. 1, 5, 14, 30, 55, _____, _____ For Exercises 810, use inductive reasoning to draw the next two shapes in each picture pattern. 10. 8. 9. y x For Exercises 1113, use inductive reasoning to test each conjecture. Decide if the conjecture seems true or false. If it seems false, give a counterexample. 11. The square of a number is larger than the number. y 12. Every multiple of 11 is a palindrome, that is, a number that reads the same forward and backward. 13. The difference of two consecutive square numbers is an odd number. x 10 CHAPTER 2 Discovering Geometry Practice Your Skills y x ©2008 Kendall Hunt Publishing

Lesson 21 Inductive Reasoning

Transcription

1 Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10, 90,, 6. 1,, 9, 7, 81,, 7. 1,, 14, 0,,, For Eerises 8 10, use inutive resoning to rw the net two shpes in eh piture pttern For Eerises 11 1, use inutive resoning to test eh onjeture. Deie if the onjeture seems true or flse. If it seems flse, give ounteremple. 11. he squre of numer is lrger thn the numer. 1. Ever multiple of 11 is plinrome, tht is, numer tht res the sme forwr n kwr. 1. he ifferene of two onseutive squre numers is n o numer. 10 CAPER Disovering Geometr Prtie Your Skills 008 Kenll unt Pulishing

2 Lesson. Fining the nth erm Nme Perio Dte For Eerises 1 4, tell whether the rule is liner funtion. 1.. f h g j For Eerises n 6, omplete eh tle.. 6. f 7n 1 g 8n For Eerises 7 9, fin the funtion rule for eh sequene. hen fin the 0th term in the sequene n… 0 f n… 0 g n… 0 h Use the figures to omplete the tle…. n… 0 Numer of tringles Use the figures ove to omplete the tle. Assume tht the re of the first figure is 1 squre unit…. n… 0 Are of figure Disovering Geometr Prtie Your Skills CAPER Kenll unt Pulishing

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Presentation On Theme: Lesson 21 Inductive Reasoning In Geometry Presentation Transcript:

1 Lesson 2.1 Inductive Reasoning in GeometryHOMEWORK: Lesson 2.1/1-15 odds,22-24, 31-40, 42EC: Due WednesdayPage 104 Improve Reasoning Skills #1-8

2 Vocabulary Inductive reasoning:make conclusions based on patterns you observeConjecture:conclusion reached by inductive reasoning based on evidenceGeometric Pattern:arrangement of geometric figures that repeat

3 We will be doing this ALOT this year!!Objectives:Use inductive reasoning to find the next term in a number or picture patternTo use inductive reasoning to make conjectures.Mathematicians use Inductive Reasoning to find patterns which will then allow them to conjecture.We will be doing this ALOT this year!!

4 A generalization made with inductive reasoning ConjecturesA generalization made with inductive reasoning EXAMPLES:Bell rings M, T, W, TH at 7:40 amConjecture about Friday?Chemist puts NaCl on flame stick and puts into flame and sees an orange-yellow flame. Repeats for 5 other substances that also contain NaCl also producing the same color flame.Conjecture?

5 reasoning that is based on patterns you observe.Inductive Reasoning reasoning that is based on patterns you observe.Ex. 1: Find the next term in the sequence:A) 3, 6, 12, 24, ___, ___B) 1, 2, 4, 7, 11, 16, 22, ___, ___C) ,___, ___

6 Solutions Ex. 1: Find the next term in the sequence:A) 3, 6, 12, 24, ___, ___B) 1, 2, 4, 7, 11, 16, 22, ___, ___C)4896Rule: x22937Rule: +1, +2, +3, +4, Rule: divide each section by half

30 Solutions

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Lesson 21 Inductive Reasoning Geometry Answer Key

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Lesson 23 Postulates And Diagrams

Monitoring progress

Question 1.Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane P and plane Q is a line?Answer:If two planes intersect, then their intersection is a line is satisfy the given intersection of plane P and plane Q is a line.

Question 2.Use the diagram in Example 2 to write an example of the postulate.a. Two Point PostulateAnswer:The two-point postulate states that when two points exist there exists a line that passes through the points. In plane P there lies points A and B through which a n line this is known as the two point postulate.

b. Line-Point PostulateAnswer: The line-point postulate states that a line must have at least two points in it. Applying the postulate to the given figure it can be seen that the line has two points A and B.

c. Line Intersection PostulateAnswer: The line intersection postulate states that a line will intersect another line only at one point. Here it can be seen that line n intersects the line m at point A only which illustrates the line intersection postulate.

Refer back to Example 3.

Question 3.If the given information states that \ and \ arc congruent. how can you indicate that in the diagram?Answer:

Question 4.Name a pair of supplementary angles in the diagram. Explain.Answer: There are actually four sets of supplementary angles in this diagram. You could choose any two angles that are next to each other as they would equal 180 degrees.

Use the diagram in Example 4.

Question 35.

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