Transcript 2.1.4 What if I assume the opposite is true? HW

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.
2.1.4 WHAT IF I ASSUME THE
OPPOSITE IS TRUE?
S E P T E M B E R 24 , 2 015
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)
Not parallel -> cross -> triangle -> triangle = 180 -> SSI angles = 180 -> 3rd angle = 0
Angle ≠ 0
∴ Parallel
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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.