EOC Review: Logic and Reasoning

Given the biconditional: Two lines are perpendicular if and only if they intersect at right angles.

Write the conditional statement that could be written from the biconditional. What is the converse of that conditional statement?

• Conditional statement: If two lines intersect at right angles, then they are perpendicular.

  Converse: If two lines are not perpendicular, then they do not intersect at right angles.

• Conditional statement: If two lines are perpendicular, then they intersect at right angles.

  Converse: If two lines do not intersect at right angles, then they are not perpendicular.

• Conditional statement: If two lines intersect at right angles, then they are perpendicular.

  Converse: If two lines are perpendicular, then they intersect at right angles.

• Conditional statement: If two lines are perpendicular, then they intersect at right angles.

  Converse: If two lines intersect at right angles, then they are perpendicular.

Answer explanation

Rewrite the definition of congruent segments as a single biconditional statement. Definition: If two line segments have the same length, then they are congruent segments.

• If two line segments are congruent, then they have the same length.

• If two line segments are not congruent, then they don’t have the same length.

• Two line segments have the same length if and only if they are congruent segments.

• Two line segments do not have the same length if and only if they are not congruent segments.

Answer explanation

biconditional statements are written with "if and only if" only when the conditional and converse are both true

Rewrite the following definition as a biconditional statement. Definition: The midpoint of a segment is the point that divides the segment into two congruent segments.

• If a point is not the midpoint of a segment, then the point doesn’t divide the segment into two congruent segments.

• A point is the midpoint of a segment if and only if the point divides the segment into two congruent segments.

• If a point divides the segment into two congruent segments, then the point is the midpoint.

• A point divides the segment into two congruent segments if and only if the point is the midpoint.

Answer explanation

biconditional statements are written with "if and only if" only when the conditional and converse are both true

What is the inverse of the statement, “If a parallelogram has a right angle, then the parallelogram is a rectangle”?

• If a parallelogram is a rectangle, then the parallelogram has a right angle.

• If a parallelogram is not a rectangle, then the parallelogram does not have right angle.

• If a parallelogram does not have a right angle, then the parallelogram is not a rectangle.

• If a parallelogram has a right angle, then the parallelogram is not a rectangle.

Answer explanation

For inverse...think, "inverse operations"...the opposite of addition is subtraction, so change the positive phrasing to negative phrasing

What is the converse of the statement, “If two angles are congruent, then they have the same measure”?

• If two angles are not congruent, then they have the same measure.

• If two angles are not congruent, then they don’t have the same measure.

• If two angles have the same measure, then they are congruent.

• If two angles don’t have the same measure, then they are not congruent.

Answer explanation

The converse is simply the hypothesis and conclusion's order switched. The inverse is when it stays in the same order but both are negated.

What is the contrapositive of the statement, “If a quadrilateral is a rectangle, then it has two pairs of parallel sides”?

• If a quadrilateral is not a rectangle, then it has two pairs of parallel sides.

• If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides.

• If a quadrilateral has two pairs of parallel sides, then it is a rectangle.

• If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle.

Given the statement “If two triangles are not similar, their corresponding angles are not congruent”, write the inverse, converse, and contrapositive statements.

• Inverse: If two triangles are similar, their corresponding angles are congruent.

  Converse: If the corresponding angles are congruent, then the two triangles are similar.

  Contrapositive: If the corresponding angles are not congruent, then the two triangles are not similar.

• Inverse: If the corresponding angles are congruent, then the two triangles are similar.

  Converse: If two triangles are similar, their corresponding angles are congruent.

  Contrapositive: If the corresponding angles are not congruent, then the two triangles are not similar.

• Inverse: If the corresponding angles are not congruent, then the two triangles are not similar.

  Converse: If two triangles are similar, their corresponding angles are congruent.

  Contrapositive: If the corresponding angles are congruent, then the two triangles are similar.

• Inverse: If two triangles are similar, their corresponding angles are congruent.

  Converse: If the corresponding angles are not congruent, then the two triangles are not similar.

  Contrapositive: If the corresponding angles are congruent, then the two triangles are similar.

Answer explanation

Which figure can serve as a counterexample to the conjecture below? “If one pair of opposite sides of a quadrilateral is parallel, then the quadrilateral is a parallelogram.”

• Rectangle

• Rhombus

• Square

• Trapezoid

Which of the options best describes a counterexample to the assertion below?

“Two lines in a plane always intersect in exactly one point.”

• coplanar lines

• parallel lines

• perpendicular lines

• intersecting lines

Answer explanation

A counterexample proves the statement false with just a single example!

Parallel lines never intersect!

Which situation would provide a counterexample to this statement?

“Alternate interior angles are never supplementary.”

• A line that is parallel to two parallel lines.

• A transversal that forms 45° angle with two parallel lines.

• A transversal that is perpendicular to two parallel lines.

• A line that has a slope that is the reciprocal of the slopes of two parallel lines.

Answer explanation

A counterexample proves the statement false with just a single example!

Here, alternate interior angles would be supplementary because 90+90=180.