Chapter 2: Logical Arguments and Line Relationships

Determine if the following conjecture is true or false? Explain.

Given: x² = 4

Conjecture: x = 4 or x = -4

• False; the square root of 4 is 2 or -2.

• True; squaring a positive number results in a positive number and squaring a negative number results in a positive number.

• True; two answers are always more accurate than one answer.

• False; the square root of a positive number is always a positive number.

Refer to the figure. If G is the midpoint of , which conjecture is true?

• line EG intersects line DG at exactly two points.

• angle DGE is congruent to angle EFG

• line DF is exactly 5 feet long

• None of the statements are true

In the figure, ABCD is a square. Which of the following is a valid conjecture about points A, B, C, and D?

• AB + CD = 12

• AB = CD

• AB = BD

• None of the statements are valid conjectures.

Use the following statements to write a compound statement for the conjunction p and ∼q. Then find its truth value.

p: The triangle has at least two congruent sides.

q: The triangle has an obtuse angle.

• p and ∼q: The triangle has at least two congruent sides and does not have an obtuse angle.

  p and ∼q: False

• p and ∼q: The triangle has at least two congruent sides and does not have an obtuse angle.

  p and ∼q: True

• p and ∼q: The triangle does not have at least two congruent sides and does have an obtuse angle.

  p and ∼q: False

• p and ∼q: The triangle does not have at least two congruent sides and does have an obtuse angle.

  p and ∼q: True

Determine the contrapositive of the following if-then statement.

If three points are noncollinear, then they form a triangle.

• If three points form a triangle, then they are noncollinear.

• If three points do not form a triangle, then they are noncollinear.

• If three points do not form a triangle, then they are not noncollinear.

• If three points are not noncollinear, then they do not form a triangle.

Rewrite the statement below as a biconditional statement. Then determine whether the biconditional is true or false.

A quadrilateral is a polygon with four sides.

• A polygon is a quadrilateral if and only if it has four sides. The biconditional is false.

• A polygon is a quadrilateral if and only if it has four sides. The biconditional is true.

• A quadrilateral is a polygon if and only if it has four sides. The biconditional is false.

• A quadrilateral is a polygon if and only if it has four sides. The biconditional is true.

Determine the logical conclusion given the information. Then determine whether it is based on inductive or deductive reasoning.

If Kaya makes 50 sales this month at his company, then she will receive a bonus of $1000 on her paycheck. This month Kaya completed 62 sales.

• Through inductive reasoning, Kaya will receive a $1000 bonus on her paycheck this month.

• Through deductive reasoning, Kaya will not receive a $1000 bonus on her paycheck this month.

• Through deductive reasoning, Kaya will receive a $1000 bonus on her paycheck this month.

• Through inductive reasoning, Kaya will not receive a $1000 bonus on her paycheck this month

''If two numbers are even, then their sum is even'' is a true conditional, and 8 and 24 are even numbers. Use the Law of Detachment to reach a logical conclusion.

• The sum of 8 and 24 must be even.

• If the numbers 8 and 24 are even, then their sum is 32.

• If the numbers 8 and 24 are odd, then their sum is 32.

• The sum of 8 and 24 must be odd.

Determine whether the statement is always, sometimes, or never true.

Lines AB, CD, and EF determine plane P. If line GH intersects line AB, then line GB lies on plane P.

• Always

• Never

• not enough information

• Sometimes

Which of the following statements is not a postulate?

• Through any two points, there is exactly one line.

• If two lines intersect, then their intersection is exactly one point.

• Points that lie of the same plane are said to be coplanar.

• A plane contains at least three points not on the same line.