Problem 1 Writing the First Step of an Indirect Proof

Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?

1. An integer n is divisible by 5.

   The opposite of “is divisible by” is “is not divisible by.”

   Assume temporarily that n is not divisible by 5.

2. You do not have soccer practice today.

   Think

   How do you find the opposite of a statement?

   Write the negation of the statement. This often involves adding or removing the word not.

   The opposite of “do not have” is “do have.”

   Assume temporarily that you do have soccer practice today.

✓ Got It?

1. Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?
   1. Δ B O X cap delta b o x  is not acute.
   2. At least one pair of shoes you bought cost more than $25.

Problem 2 Identifying Contradictions

Which two statements contradict each other?

1. F G ¯ ║ K L ¯ f g bar , box drawings double vertical , k l bar
2. F G ¯ ≅ K L ¯ f g bar , approximately equal to , k l bar
3. F G ¯ ⊥ K L ¯ f g bar , up tack , k l bar

Segments can be parallel and congruent. Statements I and II do not contradict each other.

Segments can be congruent and perpendicular. Statements II and III do not contradict each other.

Parallel segments do not intersect, so they cannot be perpendicular. Statements I and III contradict each other.

✓ Got It?

2. 1. Which two statements contradict each other?
      1. Δ X Y Z cap delta x y z  is acute.
      2. Δ X Y Z cap delta x y z  is scalene.
      3. Δ X Y Z cap delta x y z  is equiangular.
   2. Reasoning Statements I and II below contradict each other. Statement III is the negation of Statement I. Are Statements II and III equivalent? Explain your reasoning.
      1. Δ A B C cap delta eh b c  is scalene.
      2. Δ A B C cap delta eh b c  is equilateral.
      3. Δ A B C cap delta eh b c  is not scalene.