EOC Review: Conditional Statements and Counter Examples

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?

• A. 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.

• B. 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.

• C. 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.

• D. 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.

Rewrite the definition of congruent segments as a single biconditional statement.

• 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.

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.

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.

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.

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.

The Isosceles Triangle Theorem states: “A triangle is isosceles if and only if it has at least two congruent sides.” Which of the following statements correctly rewrite this biconditional statement as two conditional statements? Select all that apply.

• If a triangle is not isosceles, then it does not have at least two congruent sides.

• If a triangle does not have at least two congruent sides, then it is not isosceles.

• If a triangle is isosceles, then it has at least two congruent sides.

• If a triangle has at least two congruent sides, then it is isosceles.

• If a triangle has exactly two congruent sides, then it is always isosceles.

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

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 not congruent, then the two triangles are similar.

• Inverse: If two triangles are similar, then 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.

The Triangle Midsegment Theorem states: "If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side and half its length." Which of the following represents the converse of this theorem?

• If a segment is parallel to the third side of a triangle and is half its length, then it joins the midpoints of two sides.

• If a segment joins two midpoints of a triangle, then it is not parallel to the third side.

• If a segment is parallel to one side of a triangle, then it divides the other two sides into equal parts.

• If a segment is half the length of one side of a triangle, then it must be a midsegment.