Two-column Proofs Logic Test Answer Key with Scoring – Geometry

Question 1 Converse and contrapositive (2 points)

1. If a whole number is divisible by 3, then it is divisible by 6.
2. If a whole number is not divisible by 3, then it is not divisible by 6.

(a) converse above, (b) contrapositive above

Scoring: 1 point for each correctly formed statement.

Question 2 Types of reasoning (2 points)

1. Inductive: the rider generalizes from observed cases.
2. Deductive: a general fact is applied to a specific shape.

(a) inductive, (b) deductive

Scoring: 1 point per part (label with a valid reason).

Question 3 Properties of equality (2 points)

1. Division Property of Equality (divide both sides by 3).
2. Transitive Property.

(a) division, (b) transitive

Scoring: 1 point per property.

Question 4 Solve and justify (3 points)

Check: 3(14) − 7 = 35 and 2(10) + 15 = 35.

x = 10

Scoring: 1 point for the correct value, 2 points for the reasons (1 point if two or more are missing).

Question 5 Quarter of a segment (4 points)

Scoring: 1 point for each correct pair of statements 1–2, 3–4, 5 and 6 with valid reasons.

Question 6 Vertical angles (3 points)

1. Vertical angles are equal: 9x − 14 = 6x + 16, so 3x = 30 and x = 10.
2. 9(10) − 14 = 76°, and 6(10) + 16 = 76°.
3. A neighbor forms a linear pair: 180° − 76° = 104°.

x = 10; 76°; 104°

Scoring: 1 point per part.

Question 7 Using a true statement (2 points)

1. By the contrapositive, ABCD is not a square.
2. No. A rectangle with unequal sides is not a square, so the converse is not guaranteed.

(a) not a square, (b) no

Scoring: 1 point per part.

Question 8 Counterexamples (2 points)

1. x = −4: (−4)² = 16 > 9, but −4 is not greater than 3.
2. Two 40° angles in different triangles are congruent but not vertical angles.

(a) x = −4, (b) two separate 40° angles

Scoring: 1 point for each valid counterexample.