Crime Scene Decoders Geometry: Logic and Reasoning

Unit 2: Crime Scene Decoders Geometry: Logic and Reasoning

Standard Focus: Geometry and Spatial Sense Time Range : 1-3 Days Supplies : Pencil and Paper

Topics of Focus :

- Logic and Conjectures

- Compound Statements

- Venn Diagrams

- Deductive Reasoning

This particular was mapped to the Logic and Reasoning curriculum of most geometry textbooks and can be used as an enrichment or review activity.

9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate Congruence G-CO interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles Congruence G-CO are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a Congruence G-CO parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Name:______1.

Clue

______2.

Clue

______3.

Clue

______

Clue

______5.

Clue

______6.

Clue

______Cryptic Text Message

Suspect

______

Crime Scene Decoders Geometry: Logic & Reasoning

Detectives,

What’s another month without a series of high profile robberies at the hands of the Mathemagicians? The evil genius terrorist group has pulled off another elaborate crime spree that has left the country of France in disarray. As the Mathemagicians work to build their world conquering device, our investigators are trying to piece together the thefts by an anonymous associate, Quasi Truthful.

Cocky as always, the Mathemagicians have left behind a trail of mathematical puzzles and a cryptic text message that must be solved. After solving the puzzles, you can decode the message which will lead to Quasi’s favorite number. So far there are six suspects that police have questioned. It is hoped that someone with relatively strong geometry and reasoning skills can crack the codes that have puzzled the detectives on the case so far.

Since you are being brought in as a specialist you have to have definitive proof in order for any arrest to hold up in court. You need to be prepared to state your case and demonstrate your understanding of the following skills that Quasi is known to use in the notes.

- Logic and Conjectures

- Compound Statements

- Venn Diagrams

- Deductive Reasoning

Be sure to include:

- Other examples of the concepts

- Definitions

- Any other relevant information.

Keep in mind, the slightest miscalculation or illegible footnote could result in a not guilty verdict. Oh, did I mention that use of a calculator might prematurely set off his world conquering device?

Good luck to you, gumshoe.

Chief Harris

Name: Marie Name: Aminata

Occupation: Hairdresser Occupation: Restaurant Owner Favorite Number: -21 Favorite Number: 0

Name: Coco Name: Jessica

Occupation: Trust Fund Baby Occupation: Musician

Favorite Number: 97 Favorite Number: 44

Name: Napoleon XI Name: Napoleon XII

Occupation: Pizzeria Owner Occupation: Pizza Boy

Favorite Number: 11 Favorite Number: 12

© 21st Century Math Projects Scene #1 Eiffel Tower –- Paris, France In the middle of the night, Quasi Truthful helicoptered in and stole the top deck to the Eiffel Tower. In its place, investigators found this note.

Bonjour, I’m sure some will be salty that I stole the top of the Eiffel Tower. Others will be salty about this geometric proof. Prepare yourselves for a lot more French style sodium chloride-ness (SALT!)

Plug in the correct definitions or properties. (You will use one more than once)

Given: 퐸퐺̅̅̅̅ ≅ 퐹퐻̅̅̅̅, 퐵퐸̅̅̅̅ ≅ 퐷퐹̅̅̅̅ Prove: 퐵퐺̅̅̅̅ ≅ 퐷퐻̅̅̅̅

1. ̅퐸퐺̅̅̅ ≅ 퐹퐻̅̅̅̅, ̅퐵퐸̅̅̅ ≅ 퐷퐹̅̅̅̅ 1. 2. EG = FH, BE = DF 2.

3. EG + BE = FH + DF 3.

4. BG = EG + BE, DH = FH + DF 4.

5. BG = DH 5.

6. 퐵퐺̅̅̅̅ ≅ 퐷퐻̅̅̅̅ 6. Which one is leftover? This will give you your first clue.

Addition Property f = 1 Transitive Property a = 3 Given c = 5 Substitution r = 2 Def. of Congruent Segments n = 4 Segment Addition Postulate e = 6

The leftover… ______=______

Scene #2 The Louvre –- Paris, France

Investigators believe that Quasi dug a tunnel into the Louvre and stole the legs from the Venus de Milo.

Someone already stole the arms, so I had to go for the legs. Let’s see what kind of deductive skills you have. The four tourists below went to four different landmarks around France. See if you can figure out who went where with just three clues.

CLUE 1: Emma, Felix and Clara visited landmarks where they could go inside a structure.

CLUE 2: A man took a picture of a Clara Nate Emma Felix landmark with a glass pyramid. Palace of Versailles

CLUE 3: Nate and Emma visited landmarks Dune of Pyla outside of Paris. Arc de Triumph The first letter of the person’s name who Louvre went to the Arc de Triumph is equal to -5. ______= -5

The Museum of French Military History was rocked with cannon fire and Quasi Truthful escaped with two replica guillotines.

I wanted to rob the Bastille, but I found out it was torn to the ground! Saltiness! Anywho, I stumbled into this museum. Great military leaders have to do multiple things correctly at one time. Whoever is truthful for both p and q will lead you to your next clue.

(p) If p ⋀ q Military (q) are true… Angle 1 is equal to 35 Leader My statement is My Clue for always true You Joan of Arc m∠1 = 5x + 10 m∠2 = 30x – 5 If AB = BC then B must be the

midpoint of 퐴퐶̅̅̅̅.

m∠1 = 9x + 8 Charlemange m∠2 = 10x + 115 If two congruent angles are supplementary then they are right angles.

m∠1 = -2x + 21 Charles de m∠2 = 6x + 37 Gauille Through any two points, there is exactly one line.

m∠1 = 7x + 28 Napoleon m∠2 = 63 – 8x If two planes intersect, then their intersection is a point.

The correct clue… ______=______

Quasi Truthful sailed over to the famed prison fortress and made off with 16,000 pounds of stone. Carved into a cell, investigators found this note:

I really have ~(enjoyed) spending this time with you. I’m sure you are ~(not salty) by now. I think you have a ~(great) chance of catching me. ~(Good) Luck.

Before England stole the idea with Sherlock Holmes, C. Auguste Dupin and Monsieur Lecoq were the first real fiction detectives. They could analyze cases and smell the truth. Can you?

Use the statements and find the conjunction or disjunction that is true. (You may need to check your literature book)

p The Three Musketeers are named Athos, Porthos and Thanos q Quasimodo was an underdog halfback on the Notre Dame Football Team r The Man in the Iron Mask and the Count of Monte Cristo are brothers s Ratatouille was based on a true story

p ⋁ (r ⋁ q) (q ⋁ s ) ⋀ ~r q ⋁ (~p ⋀ ~s) (~r ⋁ q) ⋀ p

Which is true?... ______= ______

Scene #5 Mirazur –- Menton, France

A discarded supply of mutated oysters were found to be picked out of the trash. While it’s unclear how these oysters may factor into the world conquering device, Quasi Truthful left this note.

While my French is a little rusty, I’m pretty sure they like cheese.

After the appetizer and entrée in a classic French meal, there is a course of cheese. 200 customers could choose any combination of three different cheeses. The results can be seen in the Venn diagram.

How many people ate Camembert or Brie de Meaux, but did not eat any Roquefort cheese? The answer will be equal to r. r =____

French police were stunned to find all of the shrubs from the famed Axe Historique line of buildings and monuments were uprooted. Written in sidewalk chalk they found this note. Later, they were sent a cryptic text message.

I might come back to visit France just for kicks! I’ve had to spend so much time stealing stuff, I didn’t get to stop and smell the roses. For the final puzzle…

Napoleon made a change to the French flag. Are the blue, white and red equal? Au revoir!

Which proof is correct?

Option Un Given: Option Deux ̅̅̅̅ By the definition of midpoint B is the midpoint of 퐴퐶, C is Since B is a midpoint of a of a segment, AB = BC and BC the midpoint of 퐵퐷̅̅̅̅. segment, by its definition, AB = CD. By the transitive Prove: = BC and BC = CD. By the property, AB = CD. Therefore 퐴퐵̅̅̅̅ ≅ 퐵퐶̅̅̅̅ ≅ 퐶퐷̅̅̅̅ converse of the segment by the definition of addition postulate this means congruence, if segments have the segments have the same the same measure then they are measure. Therefore by the congruent. Thus, 퐴퐵̅̅̅̅ ≅ 퐵퐶̅̅̅̅ ≅ definition of congruence, 퐶퐷̅̅̅̅ . 퐴퐵̅̅̅̅ ≅ 퐵퐶̅̅̅̅ ≅ 퐶퐷̅̅̅̅ .

Which is correct?... ______=______

CRYPTIC PUZZLE SOLVER TEXT MESSAGE Haha. You salty. F – (R + A) (N + C) + E Quasi ~(Truthful)

Name: Marie Name: Aminata

Occupation: Hairdresser Occupation: Restaurant Owner Favorite Number: -21 Favorite Number: 0

Name: Coco Name: Jessica

Occupation: Trust Fund Baby Occupation: Musician

Favorite Number: 97 Favorite Number: 44

Name: Napoleon XI Name: Napoleon XII

Occupation: Pizzeria Owner Occupation: Pizza Boy

Favorite Number: 11 Favorite Number: 12

© 21st Century Math Projects Scene #1 Eiffel Tower –- Paris, France In the middle of the night, Quasi Truthful helicopter in and stole the top deck to the Eiffel Tower. In its place, investigators found this note.

Bonjour, I’m sure some will be salty that I stole top of the Eiffel Tower. Others will be salty about this geometric proof. Prepare yourselves for a lot more French style sodium chloride-ness (SALT!)

Plug in the correct definitions or properties. (You will use one more than once)

Given: , Prove:

𝐸𝐸𝐸𝐸�� ≅ �𝐹𝐹𝐹𝐹�� 𝐵𝐵��𝐵𝐵� ≅ 𝐷𝐷��𝐹𝐹� 𝐵𝐵��𝐵𝐵� ≅ �𝐷𝐷��𝐷𝐷�

1. , 1. Given �� 2. Definition of Congruent Segments 2. EG𝐸𝐸𝐸𝐸 ≅= FH,𝐹𝐹𝐹𝐹 BE𝐵𝐵𝐵𝐵 =≅ DF𝐷𝐷 𝐷𝐷 3. Addition Property 3. EG + BE = FH + DF 4. Segment Addition Postulate 4. BG = EG + BE, DH = FH + DF 5. Substitution 5. BG = DH 6. Definition of Congruent Segments 6. Which one is leftover? This will give you your first clue. 𝐵𝐵��𝐵𝐵� ≅ �𝐷𝐷��𝐷𝐷�

Addition Property f = 1 Transitive Property a = 3 Given c = 5

Substitution r = 2 Def. of Congruent Segments n = 4 Segment Addition Postulate e = 6

______=______a = 3

Scene #2 The Louvre –- Paris, France

Investigators believe that Quasi dug a tunnel into the Louvre and stole the legs from the Venus de Milo.

CLUE 1: Emma, Felix and Clara visited landmarks where they could go inside a building.

Clara Nate Emma Felix CLUE 2: A man took a picture of a landmark with a glass pyramid. Palace of Versailles Right

CLUE 3: Nate and Emma visited landmarks Dune of Pyla Right outside of Paris. Arc de Triumph Right The first letter of the person’s name who Louvre Right went to the Arc de Triumph is equal to -5. ______= -5 c = -5 © 21 st Century Math Projects

Scene #3 Les Invalides -- Paris, France

In the museum of French military history, the building was rocked with cannon fire and Quasi Truthful escaped with two replica guillotines.

Military (p) If p q are Angle 1 is equal to 35 (q) true… My statement is My Clue for Leader always true ⋀You m Joan of Arc m – ∠1 = 5x + 10 If AB = BC then B must be the ∠2 = 30x 5 midpoint of . a = 12 X = 5, so ∠1 = 35 TRUE Sometimes �𝐴𝐴𝐴𝐴��

m Charlemange m x ∠= 1 3,= 9x + 8 If two congruent ∠2 = 10x + 115 angles are ∠1 = 35 supplementary then f = -12 TRUE they are right angles. Always

m - Charles de m ∠1 = 2x + 21 Gauille - Through any two ∠2 = 6x + 37 points, there is x = 2, ∠1 = 25 So exactly one line. c = 21 THIS IS FALSE Always

m Napoleon m – x ∠1= 1, = 7x + 28 If two planes ∠2 = 63 8x intersect, then their ∠1 = 35 intersection is a n = -21 TRUE point. Never

The correct clue ______=______f = -12

Quasi Truthful sailed over to the famed prison fortress and made off with 16,000 pounds of stone. Carved into a cell, investigators found this note.

I really have ~(enjoyed) spending this time with you. I’m sure you are ~(not salty) by now. I think you have a ~(great) chance of catching me. ~(Good) Luck.

Before England stole the idea with Sherlock Holmes, C. Auguste Dupin and Monsieur Lecoq were the first real fiction detectives. They could analyze cases and smell the truth. Can you?

Use the statements and find the conjunction or disjunction that is true. (You may need to check your literature book)

(r q) (q ) ~r (~p (~r p

p ⋁ ⋁ ⋁ s ⋀ q ⋁ ⋀ ~s) ⋁ q) ⋀ e = 5 f = 6 n = 14 r = 18

______= ______n = 14

Scene #5 Mirazur –- Menton, France

A discarded supply of mutated oysters were found to be picked out of the trash. While it’s unclear how these oysters may factor into the world conquering device, Quasi Truthful left this note.

While my French is a little rusty, I’m pretty sure they like cheese.

How many people ate Camembert or Brie de Meaux, but did not eat any Roquefort cheese? The answer will be equal to r.r =____ 87

I might come back to visit France just for kicks! I’ve had to spend so much time stealing stuff, I didn’t get to stop and smell the roses. For the final puzzle…

Napoleon made a change to the French flag. Are the blue, white and red equal? Au revoir.

Which proof is correct?

Option Un Given: Option Deux By the definition of midpoint B is the midpoint of , C is Since B is a midpoint of a of a segment, AB = BC and BC the midpoint of . segment, by its definition, AB = CD. By the transitive Prove: 𝐴𝐴𝐴𝐴�� = BC and BC = CD. By the property, AB = CD. Therefore 𝐵𝐵�� 𝐵𝐵� converse of the segment by the definition of addition postulate this means congruence, if segments have 𝐴𝐴��𝐴𝐴� ≅ 𝐵𝐵��𝐵𝐵� ≅ 𝐶𝐶𝐶𝐶�� the segments have the same the same measure then they are measure. Therefore by the congruent. Thus, definition of congruence, . . 𝐴𝐴��𝐴𝐴� ≅ 𝐵𝐵��𝐵𝐵� ≅ 𝐶𝐶𝐶𝐶�� 𝐴𝐴 ��𝐴𝐴� ≅ 𝐵𝐵��𝐵𝐵� ≅ 𝐶𝐶𝐶𝐶�� e = 1 n = 2

______=______e = 1

CRYPTIC PUZZLE SOLVER TEXT MESSAGE Haha. You salty. F – (R + A) ÷ (N + C) + E

- From Puzzle 1: a = -3 From Puzzle 2: c = 5 From Puzzle 3: rf = 12 From Puzzle 4: n = 14 From Puzzle 5: = 87 From Puzzle 6: Fe =– 1 (R + A) ÷ (N + C) + E (-12) – (87 + 3) ÷ (14 + -5) + 1 (-12) – (90) ÷ (9) + 1 (-12) – 10 + 1 -22 + 1 = -21