Geometry Chapter 2 Form G

Name Class Date

2-1 Practice Form G

Find a pattern for each sequence. Use the pattern to show the next two terms.

1. 5, 11, 18, 26, … 2. A, B, D, E, G, H, …

3. −3, 6, −12, 24, −48, … 4. 1, 5, 30, 210, 1680, …

5.

Use the sequence and inductive reasoning to make a conjecture.

6. How many sides does the fifth figure of Sequence A have?

7. How many sides does the tenth figure of Sequence A have?

8. How many sides does the fourteenth figure of Sequence A have?

Sequence B: −5, 4, −2, −5, 4, −2, −5, 4, −2, …

9. What is the tenth term of Sequence B?

10. What is the fifteenth term of Sequence B?

Make a conjecture for each scenario. Show your work.

11. the square of an odd number 12. the cube of a negative number

13. the product of two even 14. the product of a multiple of 5 numbers and an odd number and a multiple of 2

Find a pattern for each sequence. Use inductive reasoning to show the next two terms.

15. 3, 5, 9, 17, … 16. 1, 4, 7, 28, 31, …

17. 5, 3, 9, 7, 21, … 18. 1, –2, 2, –4, 0, …

19. 0.3, −0.09, 0.0027, … 20. .82 4, , ,..3 9 27

21. 2, 3, 5, 8, 13, 21, … 22. 4, 7, 12, 19, 28, …

Prentice Hall Gold Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

Patterns and Inductive Reasoning

Name Class Date

2-1 Practice (continued) Form G

Use inductive reasoning to make a prediction for each scenario.

23. A farmer keeps track of the water his livestock uses each month. a. Predict the amount of water used in

August.

b. Is it reasonable to use the graph to predict water consumption for October? Explain.

24. Hannah sells snow cones during soccer tournaments. She records data for snow cone sales and temperature. a. Predict the amount of snow cone sales when the

temperature is 100°F.

b. Is it reasonable to use the graph to predict sales for when the temperature is 15ºF? Explain.

Find one counterexample to show that each conjecture is false.

25. The sum of two integers is always positive.

26. The product of two mixed numbers is never a whole number.

27. All four-sided figures are rectangles.

28. Patterns Draw the next two figures in the sequence shown below.

29. Open-Ended Use letters of the alphabet and create two different sequences that

begin with the same two letters.

30. Writing Think about all of the things you did this morning. Choose one activity and explain how you used inductive reasoning to complete it.


4

Patterns and Inductive Reasoning

Name Class Date

2-2 Practice Form G

Identify the hypothesis and conclusion of each conditional.

1. If a number is divisible by 2, then the number is even.

2. If the sidewalks are wet, then it has been raining.

3. The dog will bark if a stranger walks by the house.

4. If a triangle has three congruent angles, then the triangle is equilateral.

Write each sentence as a conditional.

5. A regular pentagon has exactly five congruent sides.

6. All uranium is radioactive.

7. Two complementary angles form a right angle.

8. A catfish is a fish that has no scales.

Write a conditional statement that each Venn diagram illustrates. 9. 10.

Determine if the conditional is true or false. If it is false, find a counterexample.

11. If the figure has four congruent angles, then the figure is a square.

12. If an animal barks, then it is a seal.


13

Conditional Statements

Name Class Date

2-2 Form G

Write the converse, inverse, and contrapositive of the given conditional statement. Determine the truth value of all three statements. If a statement is false, give a counterexample.

13. If two angles are complementary, then their measures sum to 90.

14. If the temperature outside is below freezing, then ice can form on the sidewalks.

15. If a figure is a rectangle, then it has exactly four sides.

Draw a Venn diagram to illustrate each statement.

16. If a figure is a square, then it is a rectangle.

17. If the game is rugby, then the game is a team sport.

18. Open-Ended Write a conditional statement that is false and has a true converse. Then write the converse, inverse, and contrapositive. Determine the truth values for each statement.

19. Multiple Representations Use the definitions of p, q, and r to write each conditional statement below in symbolic form. p: The weather is rainy. q: The sky is cloudy. r: The ground is wet. a. If the weather is not rainy, then the sky is not cloudy. b. If the ground is wet, then the weather is rainy. c. If the sky is not cloudy, then the ground is wet.


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Practice (Continued) Form G Conditional Statements

Name Class Date

2-3

Practice Form G

Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional.

1. If a number is divisible by 2, then the number is even.

2. If two angles have the same measure, then the angles are congruent.

3. If x > 5, then |x| > 5.

4. If a closed figure is a pentagon, then it has exactly five sides.

5. If two numbers are both even, then the sum of the two numbers is even.

Write the two statements that form each biconditional.

6. Two lines are perpendicular if and only if they intersect to form four right angles.

7. A whole number is divisible by 3 if and only if the sum of the digits of the whole number is divisible by 3.

8. A whole number is an odd number if and only if it is not divisible by 2.

9. A person lives in Alaska if and only if the person lives in the northernmost state in the United States.


23

Biconditionals and Definitions

Name Class Date

2-3

Form G

Test each statement below to see if it is reversible. If so, write it as a true biconditional. If not, write not reversible.

10. If a quadrilateral is a square, then the quadrilateral has four congruent angles.

11. An isosceles triangle is a triangle with two congruent angles.

12. A circle is a figure with no sides.

13. If a quadrilateral is a trapezoid, it has exactly two sides that are parallel.

14. A person who lives in Miami is a person who lives in Florida.

Is each statement below a good definition? If not, explain.

15. Two rays intersect if and only if they lie in the same plane.

16. A redwood tree is an evergreen tree that grows very tall.

17. A rectangle is a quadrilateral with four congruent angles.

18. A hexagon is a polygon with exactly six sides.

Write each statement as a biconditional.

19. A square is a rectangle with four congruent sides.

20. An equilateral triangle is a triangle with three congruent angles.

21. A factor of a whole number is a whole number that divides evenly into the given number.

22. Open-Ended Write a definition of your choice. Then write the definition as a biconditional.


24

Biconditionals and Definitions

Practice (continued)

Name Class Date

2-4 Practice Form G

If possible, use the Law of Detachment to make a conclusion. If it is not possible to make a conclusion, tell why.

1. If a triangle is a right triangle, then the triangle has one 90º angle. ∆ABC is a right triangle.

2. If a parallelogram has four congruent sides, then the parallelogram is a rhombus. The parallelogram has four congruent sides.

3. If x > 7, then |x| > 7. x < 7

4. If cats prowl, mice will scatter. Mice are scattering.

5. If the light is flashing yellow, then you may drive with caution through

the intersection. The light is flashing yellow.

6. If a triangle has two congruent sides, then the triangle is isosceles. In

∆DEF, DE EF≅ .

If possible, use the Law of Syllogism to make a conclusion. If it is not possible to make a conclusion, tell why.

7. To take Calculus, you must first take Algebra 2.

To take Algebra 2, you must first take Algebra 1.

8. If a tree has ragged bark, then the tree is unhealthy.

If a tree has ragged bark, then the tree might be a birch tree.

9. A quadrilateral has four congruent sides if and only if it is a rhombus. A square is a rhombus.


33

Deductive Reasoning

Name Class Date

2-4

Practice (continued) Form G

If possible, use the Law of Syllogism to make a conclusion. If it is not possible to make a conclusion, tell why.

10. If you like to snow ski, then you will like Colorado. If you like to wakeboard, then you will like Florida.

11. If it is Tuesday, then the cafeteria is serving meat loaf. When the cafeteria serves meat loaf, Harlan brings a sack lunch.

12. If a polygon is a square, then it has exactly four congruent angles. If a polygon has exactly four congruent angles, then it is a rectangle.

Use the Law of Detachment and the Law of Syllogism to make conclusions from the following statements. If it is not possible to make a conclusion, tell why.

13. If you live in Fairbanks, then you live in Alaska. If you live in Alaska, then you live in the largest state in the United States. Alan lives in Fairbanks.

14. A rectangle is a quadrilateral with four congruent angles. A rectangle is a parallelogram with four congruent angles. A square is a rectangle.

15. If it is summer, the days will be warm. If people are swimming, the days are warm. If the air conditioning is turned on inside, then it is warm outside.

16. If it is raining, the temperature is greater than 32°F. If the temperature is greater than 32°F, then it is not freezing outside. It is raining.

17. During the school week, if it does not rain, the soccer team will have practice. If the soccer team has practice, the team members will warm up by jogging two miles. It does not rain on Wednesday.

18. Open-Ended Write a set of statements that uses the Law of Syllogism to make a conclusion.

19. Writing Give an example of how a police officer uses the Law of Detachment.


34

Deductive Reasoning

Name Class Date

2-5

Practice Form G

Fill in the reason that justifes each step.

1. 0.25x + 2x + 12 = 39 Given 2.25x + 12 = 39 a. 2.25x = 27 b. 225x = 2700 c.

x = 12 d.

2. Given: m∠ABC = 80

m∠ABD + m∠DBC = m∠ABC

(3x + 3) + (6x + 5) = 80

9x + 8 = 80

9x = 72

x = 8

3. Given: KL = 3(PM)

5x = 3(2x – 4)

5x = 6x – 12

–x = –12

x = 12

4. Given: XY = YZ

8m + 5 = 6m + 17

2m + 5 = 17

2m = 12

m = 6

Name the property of equality or congruence that justifies going from the first statement to the second statement.

5. XY TZ

TZ XY

≅

≅

7. 4n + 6 – 2n = 9 2n + 6 = 9

6. 3(x + 2) = 15 3x + 6 = 15

8. ∠A ≅ ∠B and ∠B ≅ ∠C ∠A ≅ ∠C


43

Reasoning in Algebra and Geometry

Angle Addition Postulate

Substitution Property

a.

b.

c.


a.

b.

c.


a.

b.

c.

Name Class Date

2-5


9. Write a two-column proof. Given: ∠QWT and ∠TWX are complementary. Prove: x = 28

Statements Reasons

1) ∠QWT and ∠TWX are complementary

1)

2) m∠QWT + m∠TWX = 90 2)

3) 2x + x + 6 = 90 3)

4) 3x + 6 = 90 4) 5) 3x = 84 5)

6) x = 28 6)

10. Developing Proof Fill in the missing statements or reasons for the two-column proof.

Given: E is the midpoint of DF . Prove: DE = 23

Statements Reasons

1) E is the midpoint of DF . 1)

2) 2) Definition of midpoint 3) 6x + 5 = 8x – 1 3)

4) 5 = 2x – 1 4)

5) 5) Addition Property of Equality 6) 6) Division Property of Equality 7) DE = 6x + 5 7) Given

8) DE = 6(3) + 5 8)

9) DE = 23 9)

11. Write a two-column proof.

Given: m∠PMN = m∠RBC

Prove: m∠ABR + m∠PMN = m∠ABC


44

Reasoning in Algebra and Geometry

Name Class Date

2-6 Practice Form G

Find the value of x.

1. 2. 3.

4. 5. 6.

Find m∠1 using the given information.

7. m∠1 = 5x, m∠4 = 2x + 90

8. m∠1 = 8x − 120, m∠4 = 4x + 16

9. m∠2 = 180 − 3x, m∠3 = 2x

Complete the proofs by filling in the blanks.

10. Given: ∠A ≅ ∠BDA Prove: x = 5

Statements Reasons

1) 1) Given

2) 2) Vertical Angles are ≅.

3) ∠A ≅ ∠CDE 3)

4) 4) Definition of Congruence

5) 11x + 20 = 12x + 15 5)

6) 6) Subtraction Property of Equality

7) 7)


53

Proving Angles Congruent

Name Class Date

2-6


11. Given: ∠5 > ∠2

Prove: ∠8 > ∠4

Statements Reasons

1) 1) Given

2) ∠2 ≅ ∠4 2)

3) 3) Transitive Property of Congruence

4) 4) Vertical Angles are ≅.

5) ∠8 ≅ ∠4 5)

12. Complete the paragraph proof below. Given: ∠1 and ∠2 are complementary

∠2 and ∠3 are complementary

BD bisects ∠ABC Prove: m∠1 = 45

We know that ______and ______are complementary and ∠2 and ∠3 are complementary because these facts are given. By the________, m∠2 + m∠3 = 90. Given that BD bisects ∠ABC, it follows that ________. Using substitution, _______, or 2(m∠3) = 90. Using the ________, m∠3 = 45. By the Congruent Complements Theorem, ________. It follows that _____, because congruent angles have the same measure and ________ by substitution.

13. Writing Look back at the proof in Exercise 11. Rewrite the proof as a paragraph proof.


54

Proving Angles Congruent

Geometry Chapter 2 Form G

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Transcript of Geometry Chapter 2 Form G