Complete the questions below. The answers are listed below so you can check your answers.

Review Questions

Write the hypothesis and the conclusion for each statement.

  1. If 2 divides evenly into x, then x is an even number.
  2. If a triangle has three congruent sides, it is an equilateral triangle.
  3. All equiangular triangles are equilateral.
  4. What is the converse of the statement in exercise 1 above? Is the converse true?
  5. What is the inverse of the statement in exercise 2 above? Is the inverse true?
  6. What is the contrapositive of the statement in exercise 3? Is the contrapositive true?
  7. The converse of a statement about collinear points A, B, and C is: If AB = 5 and BC = 5, then B is the midpoint of \overline{AC}.
    • What is the statement?
    • Is it true?
  8. What is the inverse of the inverse of if p then q?
  9. What is the one-word name for the converse of the inverse of an if-then statement?
  10. What is the one-word name for the inverse of the converse of an if-then statement?

For each of the following biconditional statements:

  • Write p in words.
  • Write q in words.
  • Is p \rightarrow q true?
  • Is q \rightarrow p true?
  • Is p \leftrightarrow q true?

Note that in these questions, p and q could be reversed and the answers would be correct.

  1. A U.S. citizen can vote if and only if he or she is 18 or more years old.
  2. A whole number is prime if and only if it is an odd number.
  3. Points are collinear if and only if there is a line that contains the points.
  4. x + y = 17 if only if x = 8 and y = 9
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Review Answers

  1. Hypothesis: 2 divides evenly into x; conclusion: x is an even number.
  2. Hypothesis: A triangle has three congruent sides; conclusion: it is an equilateral triangle.
  3. Hypothesis: A triangle is equiangular; conclusion: the triangle is equilateral.
  4. If x is an even number, then 2 divides evenly into x. True.
  5. If a triangle does not have three congruent sides, then it is not an equilateral triangle. True.
  6. If a triangle is not equilateral, then it is not equiangular. True.
  7. If B is the midpoint of \overline{AC}, then AB = 5 and BC = 5. False (AB and BC could both be 6, 7, etc.).
  8. If p then q.
  9. Contrapositive
  10. Contrapositive
  11. p = he or she is 18 or more years old; q = a U. S. citizen can vote; p \rightarrow q is true; q \rightarrow p is true; p \leftrightarrow q is true.
  12. p = a whole number is an odd number; q = a whole number is prime; p \rightarrow q is false; q \rightarrow p is false; p \leftrightarrow q is false.
  13. p = a line contains the points; q = the points are collinear; is p \rightarrow q is true; q \rightarrow p is true; p \leftrightarrow q is true.
  14. p = x = 8 and y = 9; q = x + y = 17; p \rightarrow q is true; q \rightarrow p is false; p \leftrightarrow q is false.


Last modified: Sunday, September 5, 2010, 8:05 PM