Big ideas math answers geometry chapter 2

Reasoning and Proofs Chapter Answers provided are aligned as per the Big Ideas Math Geometry Textbooks. Have an overview of the concepts you need to learn in BIM Geometry Ch 2 Reasoning and Proofs and test your understanding. Master the topics of BIM Geometry Chapter 2 Reasoning and Proofs by practicing from the quick links available below. Identify the knowledge gap and concentrate on the areas you are lagging and improvise on them accordingly. You will find the Big Ideas Math Geometry Answers and attempt the exam with utmost confidence.

Big Ideas Math Book Geometry Answer Key Chapter 2 Reasoning and Proofs

Become proficient in the concepts of BIM Geometry Chapter 2 Reasoning and Proofs by referring to the quick links available. Solve the Questions available in BIM Book Geometry Chapter 2 Reasoning and Proofs Answer Key on a frequent basis and get a good hold of the concepts. Geometry Big Ideas Math Chapter 2 Reasoning and Proofs Answers here include questions from Lessons, Review Tests, Cumulative Practice, Assessment Tests, Practice Tests, etc. aligned as per the Textbooks. Download the Big Ideas Math Book Geometry Ch 2 Reasoning and Proofs Solution Key for free of cost and ace your preparation.

• • Reasoning and Proofs Maintaining Mathematical Proficiency – Page 63
  • Reasoning and Proofs Mathematical Practices – Page 64
  • 2.1 Conditional Statements – Page(65-74)
  • Lesson 2.1 Conditional Statements – Page(66-70)
  • Exercise 2.1 Conditional Statements – Page(71-74)
  • 2.2 Inductive and Deductive Reasoning – Page(75-82)
  • Lesson 2.2 Inductive and Deductive Reasoning – Page(76-79)
  • Exercise 2.2 Inductive and Deductive Reasoning – Page(80-82)
  • 2.3 Postulates and Diagrams – Page(83-88)
  • Lesson 2.3 Postulates and Diagrams – Page(84-86)
  • Exercise 2.3 Postulates and Diagrams – Page(87-88)
  • 2.1 – 2.3 Study Skills: Using the Features of Your Textbook to Prepare for Quizzes and Tests – Page 89
  • 2.1 – 2.3 Quiz – Page 90
  • 2.4 Algebraic Reasoning – Page(91-98)
  • Lesson 2.4 Algebraic Reasoning – Page(92-95)
  • Exercise 2.4 Algebraic Reasoning – Page(96-98)
  • 2.5 Proving Statements about Segments and Angles – Page(99-104)
  • Lesson 2.5 Proving Statements about Segments and Angles – Page(100-102)
  • Exercise 2.5 Proving Statements about Segments and Angles – Page(103-104)
  • 2.6 Proving Geometric Relationships – Page 105
  • Lesson 2.6 Proving Geometric Relationships – Page(106-114)
  • Exercise 2.6 Proving Geometric Relationships – Page(111-114)
  • 2.4 – 2.6 Performance Task: Induction and the Next Dimension – Page 115
  • Reasoning and Proofs Chapter Review – Page(116-118)
  • Reasoning and Proofs Test – Page 119
  • Reasoning and Proofs Cumulative Assessment – Page(120-121)

Reasoning and Proofs Maintaining Mathematical Proficiency

Write an equation for the nth term of the arithmetic sequence. Then find a50.

Question 1.
3, 9, 15, 21, ……..
Answer:
an = a1 + (n – 1)d
a1 = 1
d = 6
d =  the difference between the two numbers
a1 = first number in the series
a50 = 3 + (50 – 1)6
= 3 + (49)6
= 3 + 296 = 299

Question 2.
– 29, – 12, 5, 22, ……..
Answer:
an = a1 + (n – 1)d
a1 = -29
d = 17
d =  the difference between the two numbers
a1 = first number in the series
a50 = -29 + (50 – 1)17
= -29 + 833
= 804

Question 3.
2.8, 3.4, 4.0, 4.6, ………
Answer:
an = a1 + (n – 1)d
a1 = 2.8
d = 0.6
d =  the difference between the two numbers
a1 = first number in the series
a50 = 2.8 + (50 – 1)0.6
= 2.8 + 29.6
= 32.4

Question 4.
\(\frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{5}{6}\), ………
Answer:
an = a1 + (n – 1)d
a1 = \(\frac{1}{3}\)
d = 0.16
d =  the difference between the two numbers
a1 = first number in the series
a50 = \(\frac{1}{3}\) + (50 – 1)0.16
= \(\frac{1}{3}\) + (49)0.16
= 8.17

Question 5.
26, 22, 18, 14, ………
Answer:
an = a1 + (n – 1)d
a1 = 26
d = -4
d =  the difference between the two numbers
a1 = first number in the series
a50 = 26 + (50 – 1)-4
= 26 + (-196)
= -170

Question 6.
8, 2, – 4, – 10, ………
Answer:
an = a1 + (n – 1)d
a1 = 8
d = -6
d =  the difference between the two numbers
a1 = first number in the series
a50 = 8 + (50 – 1)(-6)
= 8 + (-294)
= -286

Solve the literal equation for x.

Question 7.
2y – 2x = 10
Answer:
Given equation
2y – 2x = 10
2(y – x) = 10
y – x = 10/2
y – x = 5
x – y = -5
x = -5 + y

Question 8.
20y + 5x = 15
Answer:
Given equation
20y + 5x = 15
5(4y + x) = 15
4y + x = 15/5
4y + x = 3
x = 3 – 4y

Question 9.
4y – 5 = 4x + 7
Answer:
Given equation
4y – 5 = 4x + 7
4y – 4x = 7 + 5
y – x = 3
x – y = -3

Question 10.
y = 8x – x
Answer:
Given equation
y = 8x – x
7x = y
x = y/7

Question 11.
y = 4x + zx + 6
Answer:
Given equation
y = 4x + zx + 6
y – 6 = 4x + zx
y – 6 = x(4 + z)
x = (y – 6)/(4 + z)

Question 12.
z = 2x + 6xy
Answer:
Given equation
z = 2x + 6xy
z = x(2 + 6y)
z/(2 + 6y) = x

Question 13.
ABSTRACT REASONING
Can you use the equation for an arithmetic sequence to write an equation for the sequence 3, 9, 27, 81. . . . ? Explain our reasoning.
Answer:
3, 9, 27, 81
3, 3², 3³, 34
The equation is 3n
Its not an arithmetic sequence it is a geometric sequence.

Reasoning and Proofs Mathematical Practices

Monitoring Progress

Decide whether the syllogism represents correct or flawed reasoning, If flawed, explain why the conclusion Is not valid.

Question 1.
All triangles are polygons.
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.
No trapezoids are rectangles.
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 segments.

2.1 Conditional Statements

Exploration 1

Determining Whether a Statement is True or False

Work with a partner: A hypothesis can either be true or false. The same is true of a conclusion. For a conditional statement to be true, the hypothesis and conclusion do not necessarily both have to be true. Determine whether each conditional statement is true or false. Justify your answer.

a. If yesterday was Wednesday, then today is Thursday.
Answer: Yes the statement is true.

b. If an angle is acute. then it has a measure of 30°.
Answer: False

c. If a month has 30 days. then it is June.
Answer: 30 days has September, April, June and November. The statement is false.

d. If an even number is not divisible by 2. then 9 is a perfect cube.
Answer: False, All even numbers are divisible by 2,9 is not a perfect cube.

Exploration 2

Determining Whether a Statement is True or False

Work with a partner: Use the points in the coordinate plane to determine whether each statement is true or false. Justify your answer.

a. ∆ABC is a right triangle.
Answer: True

b. ∆BDC is an equilateral triangle.
Answer: True

c. ∆BDC is an isosceles triangle.
Answer: False

d. Quadrilateral ABCD is a trapezoid.
Answer: True

e. Quadrilateral ABCD is a parallelogram.
Answer: False

Exploration 3

Determining Whether a Statement is True or False

Work with a partner: Determine whether each conditional statement is true or false. Justify your answer.

CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to distinguish correct logic or reasoning from that which is flawed.

a. If ∆ ADC is a right triangle, then the Pythagorean Theorem is valid for ∆ADC.
Answer: Yes it is a correct logic.

b. If ∠A and ∠B are complementary, then the sum of their measures is 180°.
Answer: No

c. If figure ABCD is a quadrilateral, then the sum of its angle measures is 180°.
Answer:

d. If points A, B, and C are collinear, then the lie on the same line.
Answer:

e. It

and intersect at a point, then they form two pairs of vertical angles.
Answer:

Communicate Your Answer

Question 4.
When is a conditional statement true or false?
Answer:
The logical connector in a conditional statement is denoted by the symbol. The conditional is defined to be true unless a true hypothesis leads to a false conclusion

Question 5.
Write one true conditional statement and one false conditional statement that are different from those given in Exploration 3. Justify your answer.
Answer:
True conditional statement:
If ∠A and ∠B are vertical angles, then they are equal.
False Conditional statement
If all the sides of a quadrilateral are equal, then the sum of its angles measures is 180 degrees.

Lesson 2.1 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.
The shirt is green.
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 visible, then it is night True
q → p If it is night, then the stars are visible False
∼p → ∼q If the stars are not visible, then it is not night False
∼q → ∼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 \(\overline{F H}\).
Answer: False. There is no marking to show \(\overline{F H}\) ≅ \(\overline{M H}\)

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 10.
Rewrite the definition of a right angle as a single biconditional statement.
Definition: If an angle is a right angle. then its measure is 90°.
Answer: An angle is a right angle if and only if its measure is 90°.

Question 11.
Rewrite the definition of congruent segments as a single biconditional statement.
Definition: If two line segments have the same length. then they are congruent segments.
Answer: Two line segments are congruent if and only if they have the same length.

Question 12.
Rewrite the statements as a single biconditional statement.
If Mary is in theater class, then she will be in the fall play. If Mary is in the fall play. then she must be taking theater class.
Answer: Mary will be in the fall play if and only if she is in theater class.

Question 13.
Rewrite the statements as a single biconditional statement.
If you can run for President. then you are at least 35 years old. If you are at least 35 years old. then you can run for President.
Answer: You can run for president if and only if you are at least 35 years old.

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

Answer:

Conditionalpqp → qTTTTFFFTTFFT

 

Conversepqq→pTTTTFTFTFFFT

Question 15.
Make a truth table for the conditional statement ~(p → q).
Answer:

Conditionalpqp → qTTTTFFFTTFFTpq∼p∼q∼p→∼qTTFFTTFFTTFTTFFFFTTT

Exercise 2.1 Conditional Statements

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What type of statements are either both true or both false?
Answer:

Question 2.
WHICH ONE DOESN’T BELONG?
Which statement does not belong with the other three? Explain your reasoning.
If today is Tuesday, then tomorrow is Wednesday
If it is Independence Day, then it is July.
If an angle is acute. then its measure is less than 90°.
If you are an athlete, then you play soccer.
Answer:
Statement: “If you are an athlete, then you play soccer.” is difficult from others because it is not true statement. If you are an athlete, maybe you play basketball or volleyball.

In Exercises 3 – 6. copy the conditional statement. Underline the hypothesis and circle the conclusion.

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

Question 4.
If two lines form vertical angles, then they intersect.
Answer: Hypothesis is underlined and the conclusion is colored in red.
If two lines from vertical angles, then they intersect

Question 5.
If you run, then you are fast.
Answer:

Question 6.
If you like math. then you like science.
Answer:
Hypothesis is underlined and the conclusion is colored in red.
If you like math, then you like science.

In Exercises 7 – 12. rewrite the conditional statement in if-then form.

Question 7.
9x + 5 = 23, because x = 2.
Answer:

Question 8.
Today is Friday, and tomorrow is the weekend.
Answer:
Hypotesis: Today is Friday
Conclusion: Tomorrow is the weekend.
Sentence:
If today is Friday, then tomorrow is the weekend.

Question 9.
You are in a hand. and you play the drums.
Answer:

Question 10.
Two right angles are supplementary angles.
Answer: If two angles are right angles, then they are supplementary.

Question 11.
Only people who are registered are allowed to vote.
Answer:

Question 12.
The measures complementary angles sum to 90°
Answer: If two angles are complementary, then their measures sum to 90°

In Exercises 13 – 16. write the negation of the statement.

Question 13.
The sky is blue.
Answer:

Question 14.
The lake is cold.
Answer: The lake is not cold.

Question 15.
The ball is not pink.
Answer:

Question 16.
The dog is not a Lab.
Answer: The dog is a Lab.

In Exercises 17 – 24. write the conditional statement p → q. the converse q → p, the inverse ~ p → ~ q, and the contrapositive ~ q → ~ p in words. Then decide whether each statement is true or false.

Question 17.
Let p be “two angles are supplementary” and let q be “the measures of the angles sum to 180°
Answer:

Question 18.
Let p be “you are in math class” and let q be “you are in Geometry:”
Answer:
p → q
If you are in math class, then you in Geometry.
Statement is not true, because you might be in Algebra.
q → p
If you are in Geometry, then you are in math class.
Statement is true
∼p → ∼q
If you are not in math class, then you are not in Geometry.
Statement is true.
∼q → ∼p
If you are not in Geometry, then you are not in math class.
Statement is not true, because you might be in Algebra.

Question 19.
Let p be “you do your math homework” and let q be “you will do well on the test.”
Answer:

Question 20.
Let p be “you are not an only child” and let q be “you have a sibling.
Answer:
p → q
If you are not an only child, then you have a sibling.
Statement is true.
q → p
If you have a sibling, then you are not an only child.
Statement is true.
∼p → ∼q
If you are an only child, then you do not have a sibling.
Statement is true.
∼q → ∼p
If do not have a sibling, then you are an only child.
Statement is true.

Question 21.
Let p be “it does not snow” and let q be I will run outside.”
Answer:

Question 22.
Let p be “the Sun is out” and let q be “it is day time”
Answer:
p → q
If the sun is out, then it is day time.
Statement is true.
q → p
If it is day time, then the sun is out.
Statement is true.
∼p → ∼q
If the sun is not out, then it is not day time.
Statement is true.
∼q → ∼p
If it is not day time, then the sun is not out.
Statement is true.

Question 23.
Let p be “3x – 7 = 20” and let q be “x = 9.”
Answer:

Question 24.
Let p be “it is Valentine’s Day” and let q be “it is February.
Answer:
p → q
If it is Valentine’s Day, then it is February.
q → p
If it is February, then it is Valentine’s Day.
∼p → ∼q
If it is not Valentine’s Day, then it is not February.
∼q → ∼p
If it is not February, then it is not Valentine’s Day.

In Exercises 25 – 28, decide whether the statement about the diagram is true. Explain your answer using the definitions you have learned.

Question 25.
m∠ABC = 90°

Answer:

Question 26.

Answer: If intersecting lines from a right angle, then they are perpendicular.

Question 27.
m∠2 + m∠3 = 180°

Answer:

Question 28.
M is the midpoint of \(\overline{A B}\).

Answer: It cannot be assumed that M is the midpoint unless \(\overline{A M}\) and \(\overline{B M}\) are marked as congruent.

In Exercises 29 – 32. rewrite the definition of the term as a biconditional statement.

Question 29.
The midpoint of a segment is the point that divides the segment into two congruent segments.
Answer:

Question 30.
Two angles are vertical angles when their sides form two pairs of opposite rays.
Answer: Two angles are vertical angles if and only if their sides form two pairs of opposite rays.

Question 31.
Adjacent angles are two angles that share a common vertex and side but have no common interior points.
Answer:

Question 32.
Two angles are supplementary angles when the sum of their measures 180°.
Answer: Two angles are supplementary angles if and only if the sum of their measures 180°.

In Exercises 33 – 36. rewrite the statements as a single biconditional statement.

Question 33.
If a polygon has three sides. then it is a triangle.
If a polygon is a triangle, then it has three sides.
Answer:

Question 34.
If a polygon has four sides, then it is a quadrilateral.
If a polygon is a quadrilateral, then it has four sides.
Answer: A polygon is a quadrilateral, if and only if it has four sides.

Question 35.
If an angle is a right angle. then it measures 90°.
If an angle measures 90°. then it is a right angle.
Answer:

Question 36.
If an angle is obtuse, then ii has a measure between 90° and 180°.
If an angle has a measure between 90° and 180°. then it is obtuse.
Answer: An angle is obtuse if and only if it has measure between 90° and 180°.

Question 37.
ERROR ANALYSIS
Describe and correct the error in rewriting the conditional statement in if – then form.

Answer:

Question 38.
ERROR ANALYSIS
Describe and correct the error in writing the converse of the conditional statement.

Answer:
Converse statement should just change a premise and conclusion. It should go like:
If I bring an umbrella, then it is raining.

In Exercises 39 – 44. create a truth table for the logical statement.
Question 39.
~ p → q
Answer:

Question 40.
~ q → p
Answer:

Question 41.
~(~ p → ~ q)
Answer:

Question 42.
~ (p → ~ q)
Answer:

Question 43.
q → ~ p
Answer:

Question 44.
~ (q → p)
Answer:

Question 45.
USING STRUCTURE
The statements below describe three ways that rocks are formed.

Igneous rock is formed from the cooling of Molten rock.

Sedimentary rock is formed from pieces of other rocks.

Metamorphic rock is formed by changing, temperature, pressure, or chemistry.

a. Write each sLaternenl in if-then form.
b. Write the converse of each of the statements in part (a). Is the converse of each statement true? Explain your reasoning.
c. Write a true if-then statement about rocks that is different from the ones in parts (a) and (b). Is the converse of our statement true or false? Explain your reasoning
Answer:

Question 46.
MAKING AN ARGUMENT
Your friend claims the statement “If I bought a shirt, then I went to the mall’ can he written as a true biconditional statement. Your sister says you cannot write it as a biconditional. Who is correct? Explain your reasoning.
Answer:
Converse of statement “If I bought a shirt, then I went to the mall” is If I went to the mall, then I bought a shirt.
As you can see, converse is false, so it cannot be written as biconditional statement, because both must be true or for that.
Your sister is correct.

Question 47.
REASONING
You are told that the contrapositive of a statement is true. Will that help you determine whether the statement can be written as a true biconditional statement’? Explain your reasoning.
Answer:

Question 48.
PROBLEM SOLVING
Use the conditional statement to identify the if-then statement as the converse. inverse. or contrapositive of the conditional statement. Then use the symbols to represent both statements.
Conditional statement: It I rode my bike to school, then I did not walk to school.
If-then statement: If did not ride my bike to school, then I walked to school.
p      q     ~    →    ↔
Answer:
Both premise and conclusion are negated, so if then statement is invetse of conditional statement.
Premise: p = “I rode my bike to school”
Conclusion: q = “I walked to school”
Conditional: p → ~q
If-then statement:~p → q

USING STRUCTURE
In Exercises 49 – 52. rewrite the conditional statement in if-then form. Then underline the hypothesis and circle the conclusion.
Question 49.

Answer:

Question 50.

Answer: If you expect things from yourself, then you can do them.

Question 51.

Answer:

Question 52.

Answer: If someone is happy, then he will make others happy too.

Question 53.
MATHEMATICAL CONNECTIONS
Can the statement “If x2 – 10 = x + 2. then x = 4″ be combined with its converse to form a true biconditional statement?
Answer:

Question 54.
CRITICAL THINKING
The largest natural arch in the United States is Landscape Arch. located in Thompson, Utah. h spans 290 feet.

a. Use the information to write at least two true conditional statements.
Answer:
Two true conditional statements are:
1. If the largest natural arch is the Landscape Arch, then it spans 290 feet
2. If the largest natural arch is in the United States, then it is located in Thompson, Utah.

b. Which type of related conditional statement must also be true? Write the related conditional statements.
Answer:
One true related conditional statement is its contrapositive.
We can say the following based on the answers in a:
If a natural arch does not span 290 feet, then it is not the Landscape Arch.

C. What are the other two types of related conditional statements? Write the related conditional statements. Then determine their truth values. Explain your reasoning.
Answer:
Inverse and Converse: These statements are false because other natural archs in different countries can also span in 290 feet.

Question 55.
REASONING
Which statement has the same meaning as the given statement?
Given statement:
You can watch a movie after you do your homework.
(A) If you do your homework, then you can watch a movie afterward.
(B) If you do not do your homework, then you can watch a movie afterward.
(C) If you cannot watch a movie afterward. then do your homework.
(D) If you can watch a movie afterward, then do not do your homework.
Answer:

Question 56.
THOUGHT PROVOKING
Write three conditional statements. where one is always true, one is always false, and one depends on the person interpreting the statement.
Answer:
Always true: If the sun is up, then it is day.
Always false: If the sun is up, then it is night.
Depends: If the sun is up, then it is warm.

Question 57.
CRITICAL THINKING
One example of a conditional statement involving dates is “If today is August 31, then tomorrow is September 1 Write a conditional statement using dates from two different months so that the truth value depends on when the statement is read.
Answer:

Question 58.
HOW DO YOU SEE IT?
The Venn diagram represents all the musicians at a high school. Write three conditional statements in if-then form describing the relationships between the various groups of musicians.

Answer:
If you are in the jazz band, then you are in the band.
If you are in the chorus you are not in the band.
If you are in the band or chorus you are a musician.

Question 59.
MULTIPLE REPRESENTATIONS
Create a Venn diagram representing each conditional statement. Write the converse of each conditional statement. Then determine whether each conditional statement and its converse are true or false. Explain your reasoning.
a. If you go to the zoo to see a lion, then you will see a Cat.
b. If you play a sport. then you wear a helmet.
c. If this month has 31 days. then it is not February.
Answer:

Question 60.
DRAWING CONCLUSIONS
You measure the heights of your classmates to gel a data set.
a. Tell whether this statement is true: If s and y are the least and greatest values in your data set, then the mean of the data is between x and y.

Answer: It is true

b. Write the converse of the statement in part (a). Is the converse true? Explain your reasoning.
Answer: If the mean of the data is between x and y, then x and y are the least and greatest values in your data set. No, because the x and y values are not always the least and the greatest numbers in the data set.

c. Copy and complete the statement below using mean, median, or mode to make a conditional statement that is true for an data set. Explain your reasoning.
If a data set has a mean. median, and a mode. then the _____________ of the data set will always be a data value.
Answer: If a data set has a mean. median, and a mode. then the mode of the data set will always be a data value.

Question 61.
WRITING
Write a conditional statement that is true, but its converse is false.
Answer:

Question 62.
CRITICAL THINKING
write a series of if-then statements that allow you to find the measure of each angle, given that m∠1 = 90° Use the definition of linear pairs.

Answer:
∠1 = 90° then ∠2 = 180° – 90° = 90° as they are supplementary angles and form a linear pair.
∠2 = 90° then ∠3 = 180° – 90° = 90° as they are supplementary angles and form a linear pair.
∠1 = 90° then ∠4 = 180° – 90° = 90° as they are supplementary angles and form a linear pair.

Question 63.
WRITING
Advertising slogans such as “Buy these shoes! They will make you a better athlete!” often imply conditional statements. Find an advertisement or write your own slogan. Then write it as a conditional statement.
Answer:

Maintaining Mathematical Proficiency

Find the pattern. Then draw the next two figures in the sequence.

Question 64.

Answer:
Sequence continues with pentagon and hexagon.

Question 65.

Answer:

Find the pattern. Then write the next two numbers.

Question 66.
1, 3, 5, 7 ……..
Answer:
This is a sequence of odd numbers
Next two numbers: 9, 11

Question 67.
12, 23, 34, 45 ……..
Answer:

Question 68.
2, \(\frac{4}{3}, \frac{8}{9}, \frac{16}{27}\), ……..
Answer:
Each number gets multiplied with 2/3 to obtain next number.
Next two numbers: 32/81, 64/243

Question 69.
1, 4, 9, 16, ……..
Answer:

2.2 Inductive and Deductive Reasoning

Exploration 1

Writing a Conjecture

Work with a partner: Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern.
a.

Answer: The circle is rotating from one vertex to the next in a clockwise direction

b.

Answer: The pattern alternates between a curve in an odd quadrant and a line with a negative slope in the even quadrant.

c.

Answer: The pattern alternates between the first three arrangements, then their respective mirror images.

Exploration 2

Using a Venn Diagram

Work with a partner: Use the Venn diagram to determine whether the statement is true or false. Justify your answer. Assume that no region of the Venn diagram is empty.

CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to justify your conclusions and communicate them to others.

a. If an item has Property B. then it has Property A.

Answer: The given statement is true because property B is a part of property A.

b. If an item has Property A. then it has Property B.
Answer: The given statement is true because property B is a part of property A and not vice versa.

c. If an item has Property A, then it has Property C.
Answer: The given statement is true because property C is not completely part of property A.

d. Some items that have Property A do not have Property B.
Answer: The given statement is true because property B is a part of property A and not vice versa.

e. If an item has Property C. then it does not have Property B.
Answer: The given statement is true because property C is not a part of property B.

f. Sonic items have both Properties A and C.
Answer: The given statement is true because some part of property C is included in the part of property A.

g. Some items have both Properties B and C.
Answer: The given statement is false because property C is not a part of property B.

Exploration 3

Reasoning and Venn Diagrams

Work with a partner: Draw a Venn diagram that shows the relationship between different types of quadrilateral: squares. rectangles. parallelograms. trapezoids. rhombuses, and kites. Then write several conditional statements that are shown in your diagram. such as “If a quadrilateral is a square. then it is a rectangle.”
Answer:

Communicate Your Answer

Question 4.
How can you use reasoning to solve problems?
Answer:

Question 5.
Give an example of how you used reasoning to solve a real-life problem.
Answer:

Lesson 2.2 Inductive and Deductive Reasoning

Monitoring Progress

Question 1.
Sketch the fifth figure in the pattern in Example 1.
Answer:

Question 2.

Answer:
The fifth figure in the pattern is shown below.

Question 3.

Answer:

Question 4.
Make and test a conjecture about the sign o1 the product of any three negative integers.
Answer: If you multiply two negative numbers, it will be equal to a positive number. If you multiply a positive number and a negative number, it will be equal to a negative number. So, multiplying three negative numbers is like multiplying a positive and negative number.
Example: (-1) × (-1) × (-1) = -1

Question 5.
Make and test a conjecture about the sum of any five consecutive integers.
Answer:
Statement: ‘The sum of any 5 consecutive integers.’
Let us consider 1, 2, 3, 4, 5
1 + 2 + 3 + 4 + 5 = 15
8 + 9 + 10 + 11 + 12 = 50
Hence the sum of any five consecutive integers is five times the third number

Find a counterexample to show that the conjecture is false.

Question 6.
The value of x2 is always greater than the value of x.
Answer: Yes, the statement is true.
Example: If x = 2
x² = 2² = 4

Question 7.
The sum of two numbers is always greater than their difference.
Answer: yes, the sum of two whole numbers is always greater than either number because even if one of the numbers to be added is 0, the answer will not be greater, but at least equal to it.

Question 8.
If 90° ∠ m ∠ R ∠ 180°, then ∠R is obtuse. The measure of ∠R is 155°. Using the Law of Detachment. what statement can you make?
Answer:
Given: The measure of ∠R=155
We know that an obtuse angle has a measurement greater than 90 degrees but less than 180 degrees.
Since ∠R=155 satisfies the hypothesis of a true conditional statement
Hence the conclusion is also true.
Hence the given angle R is obtuse.

Question 9.
Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements.
If you get an A on your math test. then you can go to the movies.
If you go to the movies, then you can watch your favorite actor.
Answer:

Question 10.
Use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to sIm that the conjecture is true.