12 multiple-choice questions, progressively harder.
A conditional says that every member of family A belongs to family B. Which situation would refute it?
Solution
Correct answer: D
Being in A makes the hypothesis true.
Being outside B makes the conclusion false, which is exactly the failure a counterexample must exhibit.
Suppose p implies q and q implies p have both been justified. Which statement is now supported?
Correct answer: B
The two established directions form a biconditional.
This says that p and q agree, not that they are always true for every object.
You know p implies q and q implies r, and an object does not satisfy p. What follows about whether it satisfies r?
Correct answer: C
The forward chain supplies r when p holds.
Failing the starting condition supplies no conclusion about r. Either truth value of r could fit these implications.
The original conditional and its converse are both true. Which description fits its inverse and contrapositive?
Correct answer: A
The inverse matches the converse, so it is true.
The contrapositive matches the original, so it is also true.
Three distinct points A, B, and C are given to lie on one line. Someone invokes “exactly one plane passes through three points.” What is missing from that proposed foundation?
The incidence foundation requires three noncollinear points.
The supplied points are collinear, so the stated uniqueness reason does not apply.
Let p mean “the selected angle is acute” and q mean “its measure is less than 90 degrees.” Which wording is the converse?
The converse reverses the hypothesis and conclusion without negating them.
It begins with the measure condition and ends with the angle classification.
A conjecture is based on measurements of all figures currently on a worksheet. Why does the word “all” not make this a general geometric proof?
All cases on a finite worksheet need not be all possible figures covered by the claim.
A general justification can establish the conjecture; a counterexample can refute it. A search that finds no counterexample is not itself a proof.
Every square is a rectangle. Figure J is not a rectangle. Which deduction is valid?
The original conditional has the equivalent contrapositive: every nonrectangle is a nonsquare.
Applying that conditional to J gives the conclusion.
A figure has four congruent sides. Which additional condition would make the square classification follow directly from the definition, assuming it is a quadrilateral?
The square definition needs a rectangle with four congruent sides.
Four right angles supply the rectangle condition. Neither appearance nor an unspecified diagonal condition supplies it.
The implication p→qp\to qp→q is true. Which additional result would justify q→pq\to pq→p without assuming it?
The inverse is the contrapositive of the converse, so those two forms are logically equivalent.
A proof of the inverse would therefore justify the converse. The other information does not establish the reverse direction.
You know p implies q, q implies r, and an object satisfies p. Which explanation correctly reaches a fact about this object?
First connect the two conditional links.
Then apply the resulting p-to-r conditional to the matching given p.
A statement says: “If a ray is an angle bisector, its two selected parts have equal measure.” A second says: “If the parts do not have equal measure, the ray is not that angle bisector.” How are they related?
The second statement reverses the two parts and negates each.
It is the contrapositive, so it carries the same logical promise as the original.
Reset this practice set?
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