Geometry Practice Test: Conditional Statements, Converse and Contrapositive
- Conditional Statements and Logic
- 10 questions
- 13 minutes
These 10 questions cover if-then statements and the rules of logical reasoning used in geometry proofs. Tip: a conditional and its contrapositive are always both true or both false.
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Question 1
Statement: If a polygon is a square, then it is a rectangle.
What is the converse of this statement?
Correct answer: A
The converse switches the hypothesis and the conclusion. This converse is false, because a 2 by 5 rectangle is not a square.
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Question 2
Statement: If a polygon is a square, then it is a rectangle.
What is the contrapositive of this statement?
Correct answer: D
The contrapositive switches and negates both parts: "if not rectangle, then not square". The option "if not square, then not rectangle" is the inverse.
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Question 3
Which statement always has the same truth value as the original conditional statement?
Correct answer: C
A conditional and its contrapositive are logically equivalent. The converse and the inverse are equivalent to each other, but not always to the original.
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Question 4
Which value of x is a counterexample to the statement "If x2 = 25, then x = 5"?
Correct answer: B
A counterexample makes the hypothesis true and the conclusion false. (−5)2 = 25, but −5 ≠ 5. The value 0 does not satisfy the hypothesis, so it proves nothing.
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Question 5
Which conditional statement can be written as a true biconditional ("if and only if") statement?
Correct answer: D
A biconditional is true only if the statement and its converse are both true. "A right angle measures 90°" is also true, but 6 is even without being divisible by 4, and congruent angles need not be vertical.
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Question 6
If it rains on Saturday, the soccer game will be canceled. It rains on Saturday.
Which conclusion follows by the Law of Detachment?
Correct answer: A
The Law of Detachment says: if "p implies q" is true and p is true, then q is true. Here p (rain) happened, so the game is canceled.
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Question 7
If a shape is a square, then it is a rhombus. If a shape is a rhombus, then its diagonals are perpendicular.
Which conclusion follows by the Law of Syllogism?
Correct answer: C
The Law of Syllogism chains "p implies q" and "q implies r" into "p implies r": square → rhombus → perpendicular diagonals. The first option is the converse of that conclusion.
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Question 8
If a car is a hybrid, then it uses less gasoline than a similar regular car. Jamal's car uses less gasoline than a similar regular car.
Which statement is correct?
Correct answer: D
Knowing that the conclusion is true does not prove the hypothesis (this error is called affirming the conclusion). Other cars, for example small or electric ones, can also use less gasoline.
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Question 9
Statement: If a triangle is equilateral, then it is isosceles.
Which is true about the inverse of this statement?
Correct answer: B
The inverse is "If a triangle is not equilateral, then it is not isosceles." A 5-5-8 triangle is not equilateral but is isosceles, so the inverse is false even though the original is true.
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Question 10
A number is divisible by 3 if the sum of its digits is divisible by 3.
What is the hypothesis of this conditional statement?
Correct answer: A
The hypothesis is the part after "if", even when it comes at the end of the sentence. Rewritten: "If the sum of its digits is divisible by 3, then the number is divisible by 3."
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