Geometry A Common Core Curriculum

Ron Larson, Laurie Boswell

Chapter 2

Reasoning and Proofs

Educators

MH

Problem 1

VOCABULARYWhat type of statements are either both true or both false?

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Problem 2

WHICH ONE DOESN'T BELONg Which statement does not belong with the other three? Explain your reasoning.

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Problem 3

In Exercises $3-6,$ copy the conditional statement. Underline the hypothesis and circle the conclusion.
If a polygon is a pentagon, then it has ve sides.

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Problem 4

In Exercises $3-6,$ copy the conditional statement. Underline the hypothesis and circle the conclusion.
If two lines form vertical angles, then they intersect.

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Problem 5

In Exercises $3-6,$ copy the conditional statement. Underline the hypothesis and circle the conclusion.
If you run, then you are fast.

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Problem 6

In Exercises $3-6,$ copy the conditional statement. Underline the hypothesis and circle the conclusion.
If you like math, then you like science.

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Problem 7

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
$9 x | 5=23,$ because $x=2$

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Problem 8

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
Today is Friday, and tomorrow is the weekend.

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Problem 9

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
You are in a band, and you play the drums.

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Problem 10

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
Two right angles are supplementary angles.

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Problem 11

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
Only people who are registered are allowed to vote.

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Problem 12

In Exercises $7-12$ , rewrite the conditional statement in if-then form. (See Example 1.)
The measures of complementary angles sum to $90^{\circ} .$

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Problem 13

In Exercises $13-16,$ write the negation of the statement.
The sky is blue.

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Problem 14

In Exercises $13-16,$ write the negation of the statement.
The lake is cold.

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Problem 15

In Exercises $13-16,$ write the negation of the statement.
The ball is not pink.

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Problem 16

In Exercises $13-16,$ write the negation of the statement.
The dog is not a Lab.

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Problem 17

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.
Let p be "two angles are supplementary" and let q be "the measures of the angles sum to $180^{\circ}$ .

Luca A.
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Problem 18

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let $\mathrm{p}$ be "you are in math class" and let $\mathrm{q}$ be "you are in Geometry."

Kashif Q.
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Problem 19

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be "you do your math homework" and let q be "you will do well on the test."

Luca A.
Numerade Educator

Problem 20

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be "you are not an only child" and let $q$ be "you have a sibling.

Kashif Q.
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Problem 21

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be "it does not snow" and let $\mathrm{q}$ be "I will run outside."

Luca A.
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Problem 22

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be "the Sun is out" and let $\mathrm{qe}$ "it is daytime."

Kashif Q.
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Problem 23

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be $" 3 x-7=20 "$ and let $q$ be $^{\prime} x=9 "$

Luca A.
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Problem 24

In Exercises $17-24,$ write the conditional statement $p \rightarrow q,$ the converse $q \rightarrow p,$ the inverse $\sim p \rightarrow \sim q$ , and the contrapositive $\quad \mathrm{q} \rightarrow \sim \mathrm{p}$ in words. Then decide whether each statement is true or false.

Let p be "it is Valentine's Day" and let q be "it is February.

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Problem 25

In Exercises $25-28,$ decide whether the statement about the diagram is true. Explain your answer using the de nitions you have learned. (See Example $4 . )$

$\mathrm{m} \angle \mathrm{ABC}=90^{\circ}$

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Problem 26

In Exercises $25-28,$ decide whether the statement about the diagram is true. Explain your answer using the de nitions you have learned. (See Example $4 . )$

$\mathrm{PQ}_{\perp} \mathrm{ST}$

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Problem 27

In Exercises $25-28,$ decide whether the statement about the diagram is true. Explain your answer using the de nitions you have learned. (See Example $4 . )$

$\mathrm{m} \angle 2 \mathrm{Im} \angle 3=180^{\circ}$

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Problem 28

In Exercises $25-28,$ decide whether the statement about the diagram is true. Explain your answer using the de nitions you have learned. (See Example $4 . )$

$\mathrm{Mis}$ the midpoint of $\overline{\mathrm{AB}}$

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Problem 29

In Exercises $29-32,$ rewrite the de nition of the term as a biconditional statement. (See Example $5 . )$
The midpoint of a segment is the point that divides the segment into two congruent segments.

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Problem 30

In Exercises $29-32,$ rewrite the de nition of the term as a biconditional statement. (See Example $5 . )$

Two angles are vertical angles when their sides form two pairs of opposite rays.

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Problem 31

In Exercises $29-32,$ rewrite the de nition of the term as a biconditional statement. (See Example $5 . )$

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

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Problem 32

In Exercises $29-32,$ rewrite the de nition of the term as a biconditional statement. (See Example $5 . )$

Two angles are supplementary angles when the sum of their measures is $180^{\circ} .$

Kashif Q.
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Problem 33

In Exercises $33-36,$ rewrite the statements as a single bi conditional statement. (See Example $5 . )$
If a polygon has three sides, then it is a triangle.
If a polygon is a triangle, then it has three sides.

Luca A.
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Problem 34

In Exercises $33-36,$ rewrite the statements as a single bi conditional statement. (See Example $5 . )$
If a polygon has four sides, then it is a quadrilateral.
If a polygon is a quadrilateral, then it has four sides.

Kashif Q.
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Problem 35

In Exercises $33-36,$ rewrite the statements as a single bi conditional statement. (See Example $5 . )$
If an angle is a right angle, then it measures $90^{\circ} .$
If an angle measures $90^{\circ},$ then it is a right angle.

Luca A.
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Problem 36

In Exercises $33-36,$ rewrite the statements as a single bi conditional statement. (See Example $5 . )$
If an angle is obtuse, then it has a measure between $90^{\circ}$ and $180^{\circ}$ .
If an angle has a measure between $90^{\circ}$ and $180^{\circ},$ then it is obtuse.

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Problem 37

ERROR ANALYSISDescribe and correct the error in rewriting the conditional statement in if-then form.

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Problem 38

ERROR ANALYSISDescribe and correct the error in writing the converse of the conditional statement.

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Problem 39

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
\sim p \rightarrow q
$$

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Problem 40

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
\sim q \rightarrow p
$$

Kashif Q.
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Problem 41

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
\sim(\sim p \rightarrow \sim q)
$$

Luca A.
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Problem 42

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
\sim(p \rightarrow \sim q)
$$

Kashif Q.
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Problem 43

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
q \rightarrow \sim p
$$

Luca A.
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Problem 44

In Exercises $39-44$ , create a truth table for the logical statement. (See Example $6 .$ )
$$
\sim(\mathrm{q} \rightarrow \mathrm{p})
$$

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Problem 45

UsING STRUCTURThe statements below describe three ways that rocks are formed.

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Problem 46

MAKING AN ARGUMEN Your friend claims the statement "If I bought a shirt, then I went to the mall"
can be written as a true biconditional statement. Your sister says you cannot write it as a biconditional. Who is correct? Explain your reasoning.

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Problem 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.

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Problem 48

PROBLEM SOLVINGJe 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.

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Problem 49

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

If you tell the truth,
you don'thave to
remember anything.
Mark Twain

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Problem 50

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

You have to expect things of yourself before you can do them.Michael Jordan

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Problem 51

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

If one is lucky, a solitary fantasy can totally transform one million realities. Maya Angelou

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Problem 52

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

Whoever is happy will make others happy too. Anne Frank

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Problem 53

MATHEMATICAL CONNECTION on the statement "If $x^{2}-10=x | 2,$ then $x=4 "$ be combined with
its converse to form a true biconditional statement?

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Problem 54

CRITICAL THINKING The largest natural arch in the United States is Landscape Arch, located in
Thompson, Utah. It spans 290 feet.
a. Use the information to write at least two true conditional statements.
b. Which type of related conditional statement must also be true? Write the related conditional statements.
.c.What are the other two types of related conditional statements. Write the related conditional
statements. Then determine their truth values. Explain your reasoning.

Kashif Q.
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Problem 55

REASONING Which statement has the same meaninig as the given statement?
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.

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Problem 56

THOUGHT PROVOKINGWrite three conditional statements, where one is always true, one is always
false, and one depends on the person interpreting the statement.

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Problem 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.

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Problem 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.

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Problem 59

MULTIPLE REPRESENTATIONSEate 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.

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Problem 60

DRAWING CONCLUSIONS Of measure the heights of your classmates to get a data set.
a. Tell whether this statement is true: If $x$ and $y$ are the least and greatest values in your data set, then the mean of the data is between $x$ and $y$ .
b. Write the converse of the statement in part (a). Is the converse true? Explain your reasoning.
c. Copy and complete the statement below using mean, median, or mode to make a conditional
statement that is true for any data set. Explain your reasoning.

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Problem 61

WRITING Write a conditional statement that is true, but its converse is false.

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Problem 62

CRITICAL THINKING ON Write a series of if then statements that allow you to I nd the measure of each angle, given that $\mathrm{m} \angle 1=90^{\circ} .$ Use the de nition of linear pairs.

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Problem 63

WRITINGAdvertising 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.

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Michael H.
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Problem 64

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

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Problem 65

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

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Problem 66

Find the pattern. Then write the next two numbers.
$$
1,3,5,7, \dots
$$

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Problem 67

Find the pattern. Then write the next two numbers.
$$
12,23,34,45, \dots
$$

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Problem 68

Find the pattern. Then write the next two numbers.
$$
2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \dots
$$

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Problem 69

Find the pattern. Then write the next two numbers.
$$
1,4,9,16, \dots
$$

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