Geometry Workbook — Chapter 1: Logic, Conditionals, and Venn Diagrams

SOL G.RLT.1 (a, b, c, d) · Companion to Textbook Chapter 1

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 112.

PAGE 1 — Chapter opener

Chapter 1 · Logic, Conditionals, and Venn Diagrams

Standard G.RLT.1 (a, b, c, d)

In this chapter you will:

Decide what counts as a statement and write its negation
Translate between English and the five symbolic forms: $\sim p$ , $p \wedge q$ , $p \vee q$ , $p \rightarrow q$ , $p \leftrightarrow q$
Split a conditional into its hypothesis and conclusion, and know the one case that makes it false
Write the converse, inverse, and contrapositive, and disprove one with a counterexample
Recognize a biconditional as a definition, true in both directions
Shade and read union, intersection, subset, and negation on Venn diagrams

Words to know: statement · negation · conjunction · disjunction · conditional · hypothesis · conclusion · counterexample · converse · inverse · contrapositive · logically equivalent · biconditional · universe · union · intersection · subset

Convention: "or" is inclusive — $p \vee q$ is true when both parts are true. A trapezoid has exactly one pair of parallel sides.

PAGE 2 — What counts as a statement

1.1 Statements

FIGURE: fig1-statement-versus-not-a-statement.png (full width)

Fill in the blanks.

A statement is a sentence that is either ____________ or ____________ , and not both.

A sentence that is false ______ (is / is not) still a statement.

Which are statements? Give the truth value of each statement.

a) A pentagon has five sides. ____________________________

b) Construct the perpendicular bisector of $\overline{PQ}$ . ____________________________

c) $12$ is a prime number. ____________________________

d) How many diagonals does a hexagon have? ____________________________
Is "Is $\overline{AB}$ congruent to $\overline{CD}$ ?" a statement? Why or why not?

PAGE 3 — The five symbolic forms

Symbols to Know

FIGURE: fig2-symbolic-forms-reference.png (full width)

Complete the frame.

Name	Symbol	Read as
negation	____________	____________
conjunction	____________	____________
disjunction	____________	____________
conditional	____________	____________
biconditional	____________	____________

Name the symbol for each of the five forms. (Use the table above.)

PAGE 4 — Negation

Writing a Negation

$p$ : $\angle A$ is acute. Write $\sim p$ in words.
Write the negation of each.

a) $\triangle DEF$ is a right triangle. ____________________________

b) $m\angle B = 55°$ . ____________________________

c) The diagonals are not congruent. ____________________________
Write the negation of " $m\angle 1 > 90°$ ." ____________________________

Watch out. The negation of $m\angle A = 40°$ is $m\angle A \ne 40°$ , not $m\angle A = 140°$ .

PAGE 5 — Truth tables for not, and, or

When Is It True?

FIGURE: fig3-truth-tables-not-and-or.png (full width)

Fill in the blanks.

An "and" statement is true in exactly ______ row — the one where ____________________ .

An "or" statement is false in exactly ______ row — the one where ____________________ .

In which row is a conjunction true? ____________ In which row is a disjunction false? ____________
Let $p$ be true and $q$ be false.

$\sim p$ = ______ $p \wedge q$ = ______ $p \vee q$ = ______
$p$ is true, $q$ is false. $p \wedge q$ = ______ $p \vee q$ = ______

PAGE 6 — Translating both directions

English ⇄ Symbols

Let $p$ : the figure is a square. Let $q$ : the figure is a rectangle.

$p$ : the lines are parallel; $q$ : the lines are coplanar. Translate $p \wedge q$ :
Translate into English.

a) $p \wedge q$ ____________________________

b) $p \vee q$ ____________________________

c) $\sim p$ ____________________________

d) $\sim p \wedge q$ ____________________________
Translate into symbols.

a) The figure is a square and it is not a rectangle. ____________

b) The figure is neither a square nor a rectangle. ____________

PAGE 7 — Build the tables

Practice · Truth Tables

FIGURE: fig13-blank-truth-tables-and-venn-frames.png (half width, left)

Complete a truth table for $p \wedge \sim q$ . (Use Truth table A.)
Complete a truth table for $\sim p \vee q$ . (Use Truth table B.)
$p$ is false and $q$ is false. Which of these are true?

$p \wedge q$ ______ $p \vee q$ ______ $\sim p \wedge \sim q$ ______
Write a compound statement about a triangle that is true because of an "or," even though one of its parts is false.

PAGE 8 — Reasoning and application · Lesson 1.1

Apply It

Application. A gate opens when a badge is valid and the time is within business hours. On Saturday a guard uses a valid badge and the gate stays shut. Which part of the conjunction was false, and how do you know?
Application. A scholarship requires a GPA above $3.0$ or a varsity letter. Ana has a $2.8$ GPA and a varsity letter. Does she qualify? Which connective decided it?
Reasoning. Why is " $7$ is an even number" a statement while "Draw a circle" is not?
Error analysis. A student says the negation of "all rectangles are squares" is "no rectangles are squares." What is wrong, and what is the correct negation?
Reasoning. Why does $\sim(\sim p)$ have the same truth value as $p$ in every row?
In one sentence: what makes an "and" false, and what makes an "or" true?

PAGE 9 — Inside a conditional

1.2 Conditional Statements

FIGURE: fig4-anatomy-of-a-conditional.png (full width)

Label the parts.

The part after if is the ____________________ , written $p$ .

The part after then is the ____________________ , written $q$ .

On the figure, which part is the hypothesis? ____________________ The conclusion? ____________________
"If a polygon has five sides, then it is a pentagon."

Hypothesis: ____________________ Conclusion: ____________________

PAGE 10 — Rewriting into if-then form

Every Theorem Is a Conditional

Rewrite "All squares are rhombi" as a conditional.
Rewrite "A number is divisible by $4$ whenever it is divisible by $8$ " as a conditional.
Rewrite each as a conditional.

a) Every rectangle is a parallelogram. ____________________________

b) Perpendicular lines form right angles. ____________________________

c) Two points determine exactly one line. ____________________________

PAGE 11 — The conditional truth table

The One False Row

FIGURE: fig5-conditional-truth-table.png (full width)

Fill in the blanks.

A conditional is false only when the hypothesis is ______ and the conclusion is ______ .

When the hypothesis is false, the conditional is ______ , because the promise was ____________________ .

In which row is $p \rightarrow q$ false? Give that row's truth values. ____________
Why are both rows with a false hypothesis true?
Complete the truth table for $p \rightarrow q$ from memory, then check it against the figure.

$p$	$q$	$p \rightarrow q$
T	T	______
T	F	______
F	T	______
F	F	______

PAGE 12 — Practice · hypothesis, conclusion, truth

Practice · Conditionals

Identify the hypothesis and conclusion.

a) If two angles are complementary, then their measures add to $90°$ .

H: ____________________ C: ____________________

b) If $\overline{AB} \cong \overline{CD}$ , then $AB = CD$ .

H: ____________________ C: ____________________

c) A figure is a polygon if it is a triangle.

H: ____________________ C: ____________________
$p$ false, $q$ true. $p \rightarrow q$ = ______
$p$ true, $q$ false. $p \rightarrow q$ = ______ Which row is that? ______
State the one combination of truth values that makes a conditional false. ____________

PAGE 13 — Counterexamples

One Counterexample Is Enough

Decide whether each conditional is true. If false, give a counterexample.

a) If a figure is a rhombus, then it is a parallelogram. ____________________________

b) If a figure is a parallelogram, then it is a rhombus. ____________________________

c) If two angles are congruent, then they are vertical angles. ____________________________
Give a counterexample to "If a quadrilateral has four right angles, then it is a square."
Reasoning. Why do twenty supporting examples not prove a conditional, while one counterexample disproves it?

PAGE 14 — Application and error analysis · Lesson 1.2

Apply It

Application. "If you spend $50, then you get free shipping." A customer spends $60 and is charged shipping. Was the promise broken? Which row is that?
Application. The same customer spends $30 and is charged shipping. Was the promise broken? Explain from the truth table.
Error analysis. A student says "If a triangle has four sides, then it is a rectangle" must be false because triangles never have four sides. Why is the conditional actually true?
Write one true conditional about angles, and one false conditional with its counterexample.

True: _______________________________________________

False: ______________________________ Counterexample: ______________________
Write "If it is a square, then it has four right angles" in symbols. Say what $p$ and $q$ mean.

$p$ : ____________________ $q$ : ____________________ Symbols: ____________
Hypothesis and conclusion of "If a triangle is equiangular, then each angle measures $60°$ ."

H: ____________________ C: ____________________
In one sentence, explain the "promise" way of remembering the conditional truth table.

PAGE 15 — The four related conditionals

1.3 Converse, Inverse, Contrapositive

FIGURE: fig6-four-related-conditionals.png (full width)

Complete the frame.

Name	Form	Built by
Conditional	$p \rightarrow q$	the original
Converse	____________	____________________
Inverse	____________	____________________
Contrapositive	____________	____________________

Copy the symbolic form of the converse, inverse, and contrapositive. ____________
On the figure, which two are true? ____________ Which two are false? ____________
Which pairs do the dashed arrows join, and what does the label say?

PAGE 16 — Building all three

Build All Three

"If a polygon is a pentagon, then it has five sides." Converse:
Inverse: _______________________________________________
Contrapositive: _______________________________________________

Truth value, without checking separately: ____________ Why may you do that?
For "If two lines are perpendicular, then they intersect," write all three.

Converse: ____________________________

Inverse: ____________________________

Contrapositive: ____________________________
Give the truth value of all four statements in item 51, with a counterexample where one is false.

PAGE 17 — A counterexample kills a converse

The Converse Is a Different Claim

FIGURE: fig7-counterexample-kills-a-converse.png (full width)

Which statement does the rectangle disprove? Why is one figure enough?
For "If a number is divisible by $6$ , then it is divisible by $3$ ," write all four related statements and give their truth values.

Conditional: ____________________________ ______

Converse: ____________________________ ______

Inverse: ____________________________ ______

Contrapositive: ____________________________ ______

PAGE 18 — Application and reasoning · Lesson 1.3

Apply It

Application. A sign reads "If you are under $48$ inches tall, then you may not ride." A rider is $52$ inches and is turned away. Which related statement did the operator apparently use? Is it equivalent to the sign?
Application. "If the alarm sounds, then the door unlocks." The door is locked. What can you conclude, and which related statement did you use?
Error analysis. A student writes the contrapositive of "If it rains, then the game is cancelled" as "If it does not rain, then the game is not cancelled." Name the error and give the correct contrapositive.
Reasoning. Why can a conditional and its converse have different truth values? Give your own geometric example.

PAGE 19 — Practice · Lesson 1.3

Practice

Write a true conditional whose converse is also true: ____________________________

Write a true conditional whose converse is false: ____________________________
Write "All squares are parallelograms" as a conditional, then write and evaluate its converse.

Conditional: ____________________________

Converse: ____________________________ True or false? ______
Reasoning. The inverse of a statement is the contrapositive of its converse. Use that to explain why the inverse and converse always agree.
A conditional is false. What do you know about its contrapositive? ____________ About its converse? ____________
Converse of "If a figure is a rhombus, then it is a parallelogram": ____________________________ Truth value: ______
Which related statement is always logically equivalent to the original? ____________
A conditional is true and its converse is false. Inverse: ______ Contrapositive: ______
In one sentence, what must a counterexample do?

PAGE 20 — Biconditionals

1.4 Biconditionals and Definitions

FIGURE: fig8-biconditional-as-two-conditionals.png (full width)

Fill in the blanks.

A biconditional $p \leftrightarrow q$ is true exactly when the ____________________ and its ____________________ are both true.

Every ____________________ in geometry is a biconditional.

Write the two conditionals the figure contains.
Why must both halves be true for the biconditional to be true?

PAGE 21 — Splitting and testing

Split It, Then Test Both Halves

Split "A polygon is a triangle if and only if it has exactly three sides."

Forward: ____________________________

Backward: ____________________________
Is the biconditional in item 69 true? How did you check? ____________________________
Is "A figure is a square if and only if it is a rhombus" true? Which half fails?
Rewrite as a biconditional: "A perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through its midpoint."
Split each and decide whether the biconditional is true.

a) An angle is acute iff its measure is less than $90°$ and greater than $0°$ . ______

b) A quadrilateral is a parallelogram iff both pairs of opposite sides are parallel. ______

c) A triangle is right iff it has a $90°$ angle. ______

PAGE 22 — Definition or theorem?

Definition or Theorem?

Why is "A number is even if and only if it is divisible by $2$ " a definition rather than a theorem?
Which can be written as a true biconditional? For those that cannot, name the failing half.

a) A figure is a rectangle; it is a parallelogram with a right angle. ____________

b) A figure is a rhombus; it is a parallelogram. ____________

c) An angle is straight; it measures $180°$ . ____________
Write a true biconditional about circles. ____________________________
Reasoning. Why can "vertical angles are congruent" not be upgraded to a biconditional, while "right angles measure $90°$ " can?
Reasoning. Why may a definition be used in either direction inside a proof, while a theorem may only be used forwards?

PAGE 23 — Application and exit · Lesson 1.4

Apply It

Application. "A member is in good standing if and only if dues are paid and two meetings were attended." Amir paid dues and attended one meeting. Is he in good standing? Which half decided it?
Application. "The light is green if and only if the door is secured." The light is red. What do you know about the door, and why does the biconditional let you say it?
Error analysis. A student writes "A figure is a square if and only if it has four congruent sides." Give the counterexample and repair the statement.
Give a true conditional whose converse is false. What does that tell you about writing it with "if and only if"?
Write the two conditionals inside "Two segments are congruent if and only if they have equal lengths."
Is "A figure is a parallelogram if and only if it is a rectangle" true? Justify. ____________________________
Difference between a definition and a theorem, in terms of biconditionals: ____________________________
One biconditional from geometry you are confident is true: ____________________________

PAGE 24 — Union and intersection

1.5 Venn Diagrams

FIGURE: fig9-venn-union-and-intersection.png (full width)

Fill in the blanks.

$A \cup B$ is the ____________________ — it matches the English word ____________ .

$A \cap B$ is the ____________________ — it matches the English word ____________ .

Which region is shaded for $A \cup B$ ? ____________ For $A \cap B$ ? ____________
Which English word matches union? ____________ Intersection? ____________

PAGE 25 — Subset and negation

Subset and Negation

FIGURE: fig10-venn-subset-and-negation.png (full width)

Write the subset relationship shown as a conditional.
Name the two parts of the diagram shaded for $\sim A$ .
$A = \{$ triangles $\}$ , $B = \{$ right triangles $\}$ . Which is a subset of which? Draw the diagram in the box.

┌──────────────────────────────┐ │ │ │ │ │ │ └──────────────────────────────┘
Write each as a conditional.

a) Rectangles $\subseteq$ Parallelograms ____________________________

b) Squares $\subseteq$ Rectangles ____________________________

PAGE 26 — The quadrilateral family

The Quadrilateral Family

FIGURE: fig11-venn-quadrilateral-family.png (full width)

Name a set that Squares is a subset of. ____________
True or false? Cite the picture.

a) Every rhombus is a parallelogram. ______

b) Every parallelogram is a rhombus. ______

c) Every square is a rectangle and a rhombus. ______

d) Some trapezoids are parallelograms. ______
Draw a Venn diagram showing that all squares are rectangles but not all rectangles are squares. Write the conditional and its false converse beneath it.

┌──────────────────────────────┐
│                              │
│                              │
│                              │
└──────────────────────────────┘

Conditional: ____________________________

False converse: ____________________________

PAGE 27 — A Venn diagram with counts

Reading Counts

FIGURE: fig12-venn-survey-in-context.png (full width)

Fill in the blanks — the order matters.

Step 1: put the ____________________ count in the overlap.

Step 2: ____________________ to get each "only" region.

Step 3: whatever is left goes ____________________ the circles.

How many students take art only? ______ How many take neither? ______
How many are in $\sim(\text{cat owners})$ for item 96? ______
Reasoning. Why is $|A \cup B| = |A| + |B| - |A \cap B|$ rather than $|A| + |B|$ ?

_______________________________________________

PAGE 28 — Practice · filling diagrams

Practice · Fill the Diagram

FIGURE: fig13-blank-truth-tables-and-venn-frames.png (right third — Venn frame)

In a group of $40$ people, $22$ own a dog, $17$ own a cat, $8$ own both. Fill all four regions.

Dog only ______ Both ______ Cat only ______ Neither ______
How many own at least one pet? ______ How many own neither? ______
Application. Of $50$ students, $31$ took geometry, $24$ took art, and $6$ took neither. How many took both? ______
Application. $12$ bikes have a bell, $9$ have a basket, $5$ have both, $3$ have neither. How many bikes are there? ______
Error analysis. Given $18$ art, $14$ band, $7$ both, a student writes $11$ in the art circle, $7$ in the overlap, and $14$ in the band circle. What is wrong? Correct it.

_______________________________________________

PAGE 29 — Exit tickets · Lesson 1.5

Check Yourself

Reasoning. Why do " $A \subseteq B$ " and "if $x \in A$ , then $x \in B$ " say the same thing?

_______________________________________________

Shade $A \cap B$ and name the English word it matches.

○○  ____________

Universe of $20$ , with $12$ in $A$ . How many are in $\sim A$ ? ______
Write "Rhombi $\subseteq$ Parallelograms" as a conditional. ____________________________

Converse's truth value: ______
Of $25$ students, $14$ play piano, $11$ play guitar, $4$ play both. How many play neither? ______

PAGE 30 — Chapter review · Part A

Review · Symbolic Form

$p$ : the figure is a rectangle; $q$ : the figure is a rhombus. Translate each.

a) $p \wedge q$ ____________________________

b) $\sim p \vee q$ ____________________________

c) $p \rightarrow q$ ____________________________

d) $p \leftrightarrow q$ ____________________________

Figure described by (a): ____________
Negation of " $m\angle A \ge 90°$ ": ____________

Complete the truth table for $p \wedge \sim q$ :

$p$	$q$	$\sim q$	$p \wedge \sim q$
T	T	______	______
T	F	______	______
F	T	______	______
F	F	______	______

PAGE 31 — Chapter review · Parts B and C

Review · Conditionals, Biconditionals, Venn Diagrams

"If a triangle is equilateral, then it is isosceles."

a) H: ____________________ C: ____________________

b) Converse: ____________________________

Inverse: ____________________________

Contrapositive: ____________________________

c) Truth values, with a counterexample for each false one:

d) Why did (c) require only two independent checks?
Survey of $60$ students: $34$ chorus, $28$ drama, $9$ both.

a) Fill all four regions: chorus only ______ both ______ drama only ______ neither ______

b) In chorus or drama ______ In neither ______

c) Is "a student is in chorus if and only if the student is in drama" true here? Justify with the diagram.

d) Write a true subset statement about this survey, or explain why neither set is a subset of the other.

Canva production notes

Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white and need no background treatment.
Blanks: ____________ is a fill-in rule; the boxed areas on pages 25, 26, and 28 are drawing frames — leave them empty and ruled, not shaded.
Truth tables on pages 11, 30, and 31 are typed tables, not figures. The blank truth-table and Venn frames on pages 7 and 28 come from fig13-blank-truth-tables-and-venn-frames.png.
Symbols: $\sim$ , $\wedge$ , $\vee$ , $\rightarrow$ , $\leftrightarrow$ , $\cup$ , $\cap$ , $\subseteq$ must render in the body font; check the Canva font supports them before the first export.
Item numbers are continuous from 1 to 112 and must not be renumbered when pages are reordered.