See Problem 2.

Identify the two statements that contradict each other.

10. 1. Δ P Q R cap delta p q r  is equilateral.
    2. Δ P Q R cap delta p q r  is a right triangle.
    3. Δ P Q R cap delta p q r  is isosceles.
11. 1. l ║ m l box drawings double vertical m
    2. l and m do not intersect.
    3. l and m are skew.
12. 1. Each of the two items that Val bought costs more than $10.
    2. Val spent $34 for the two items.
    3. Neither of the two items that Val bought costs more than $15.
13. 1. In right Δ A B C , m ∠ A = 60 . cap delta eh b c comma m angle , eh equals 60 , .
    2. In right Δ A B C , ∠ A ≅ ∠ C . cap delta eh b c comma angle eh approximately equal to angle c .
    3. In right Δ A B C , m ∠ B = 90 . cap delta eh b c comma m angle , b equals 90 , .

See Problem 3.

14. Developing Proof Fill in the blanks to prove the following statement.

    If the Yoga Club and Go Green Club together have fewer than 20 members and the Go Green Club has 10 members, then the Yoga Club has fewer than 10 members.

    Given: The total membership of the Yoga Club and the Go Green Club is fewer than 20. The Go Green Club has 10 members.

    Prove: The Yoga Club has fewer than 10 members.

    Proof: Assume temporarily that the Yoga Club has 10 or more members.
    This means that together the two clubs have a. ? ¯ modified question mark with under bar below  members.
    This contradicts the given information that b. ? ¯ . modified question mark with under bar below , .
    The temporary assumption is false. Therefore, it is true that c. ? ¯ . modified question mark with under bar below , .

15. Developing Proof Fill in the blanks to prove the following statement.

    In a given triangle, Δ L M N , cap delta l m n comma  there is at most one right angle.

    Given: Δ L M N cap delta l m n

    Prove: Δ L M N cap delta l m n  has at most one right angle.

    Proof: Assume temporarily that Δ L M N cap delta l m n  has more than one a. ? ¯ . modified question mark with under bar below , .  That is, assume that both ∠ M angle m  and ∠ N angle n  are b. ? ¯ . modified question mark with under bar below , .  If ∠ M angle m  and ∠ N angle n  are both right angles, then m ∠ M = m ∠ N = m angle , m equals , m angle , n equals  c. ? ¯ . modified question mark with under bar below , .  By the Triangle Angle-Sum Theorem, m ∠ L + m ∠ M + m ∠ N = m angle , l plus , m angle , m plus , m angle , n equals  d. ? ¯ . modified question mark with under bar below , .  Use substitution to write the equation m ∠ L + m angle , l plus  e. ? ¯ + modified question mark with under bar below , plus  f. ? ¯ = 180 . modified question mark with under bar below . equals 180 , .  When you solve for m ∠ L , m angle l comma  you find that m ∠ L = m angle , l equals  g. ? ¯ . modified question mark with under bar below , .  This means that there is no Δ L M N , cap delta l m n comma  which contradicts the given statement. So the temporary assumption that Δ L M N cap delta l m n  has h. ? ¯ modified question mark with under bar below  must be false. Therefore, Δ L M N cap delta l m n  has i. ? ¯ . modified question mark with under bar below , .

B Apply

16. History Use indirect reasoning to eliminate all but one of the following answers. In what year was George Washington born?
    1. 1492
    2. 1732
    3. 1902
    4. 2002

17. Proof Think About a Plan Write an indirect proof.

    Given: ∠ 1 ≇ ∠ 2 angle 1 neither approximately nor actually equal to angle 2

    Prove: l ∦ p l not parallel to p

    

    • What assumption should be the first step of your proof?
    • In the figure, what type of angle pair do ∠ 1 angle 1  and ∠ 2 angle 2  form?

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