2-33. THE COLOR SQUARE GAME OBJECTIVE: FIGURE OUT THE ARRANGEMENT OF COLORED SQUARES ON A 3 × 3 GRID OR A 4 × 4 GRID USING AS FEW CLUES AS POSSIBLE. Rules: In a 3 × 3 Color Square Game, each of the nine squares are colored: three are red, three are green, and three are blue. However, all squares of the same color must be contiguous (linked along a side). The diagram below demonstrates what is meant by contiguous. In a 4 × 4 Color Square Game, there are four red squares, four green squares, four blue squares, and four yellow squares. Again, all squares of the same color must be contiguous. To get information about a Color Square, you ask, “What is in row ___?” or “What is in column ___?” You will then be given the total number of squares of each color in that row or column, but not necessarily in the order that they appear in the secret arrangement. For reference, rows are numbered from top to bottom and columns are numbered left to right.

Each member of your team should create one 3 × 3 and one 4 × 4 Color Square, then choose a partner and play.

SEPTEMBER 24, WHAT IF I ASSUME THE OPPOSITE IS TRUE?

OBJECTIVES CO: SWBAT use proof by contradiction to prove the converses of theorems they have previously studied. LO: SWBAT explain their reasoning using proof.

2-34. RIANNA IS STUCK ON THE COLOR SQUARE GAME AT RIGHT, SO SHE ASKS HER TEAMMATE WILMA FOR A HINT. WILMA SAYS, "YOU SHOULD KNOW THAT THE BOTTOM RIGHT CORNER MUST BE GREEN." RIANNA DISAGREES WITH WILMA. SHE SAYS, "I KNOW THE GREEN CAN’T GO IN THE TOP RIGHT CORNER, BUT I THINK THE MIDDLE RIGHT SQUARE COULD BE GREEN." Assume Rianna is correct and complete the Color Square Game. What happens?

2-35. PROOF BY CONTRADICTION In problem 2-34, you first assumed that Wilma’s conjecture was false (and that Rianna’s was true). However, that assumption led to a contradiction of the Color Square Game rules. That showed that your assumption was false, and therefore Wilma’s conjecture was true. This type of reasoning is called a proof by contradiction. To prove a conjecture, you start by assuming it is false. If your assumption leads to an impossibility, or a contradiction of other facts, then the conjecture must be true. You will use proof by contradiction to prove the converses of some familiar geometric relationships. a.In the diagram, what is the relationship between angles x and y? Write a conditional statement or arrow diagram to justify your answer. If lines are parallel, then same side interior angles are supplementary. b.Write the converse of the theorem you used in part (a). Is the converse true? If same side interior angles are supplementary, then the lines are parallel.

C. TO PROVE THAT THE CONVERSE IS TRUE USING PROOF BY CONTRADICTION, YOU MUST START BY ASSUMING IT IS FALSE. FOR THIS STATEMENT, THAT MEANS YOU WILL ASSUME THAT THE HYPOTHESIS (SAME-SIDE INTERIOR ANGLES ARE SUPPLEMENTARY) IS TRUE, BUT THE CONCLUSION (LINES ARE PARALLEL) IS FALSE. WITH YOUR TEAMMATES, DRAW A DIAGRAM THAT SHOWS THESE ASSUMPTIONS. THEN USE YOUR UNDERSTANDING OF GEOMETRIC RELATIONSHIPS TO IDENTIFY A CONTRADICTION. (LOOK AT PICTURE ON 2-36 IF NEEDED) If the lines are not parallel, then they will intersect. If they intersect, then it will make a triangle. The sum of the angles of a triangle equal 180. Same side interior angles add to 180. There fore the third angle of the triangle would have to be zero. That’s impossible. An angle can’t be zero. Therefore the lines must be parallel.