Two-column Proofs Logic Worksheet with Answers – Geometry (Free PDF)

Problem 1 Hypothesis and conclusion ★★★

Separate the following statement into its hypothesis and its conclusion: “If a whole number ends in 0, then it is divisible by 10.”

Problem 2 Writing a converse ★★★

Form the converse of “If a quadrilateral is a square, then it has four right angles.” Then judge whether that converse holds.

Problem 3 Inverse and contrapositive ★★★

Form the inverse and the contrapositive of “If an animal is a robin, then it is a bird.” State the truth value of each.

Problem 4 Find a counterexample ★★★

Disprove “If a number is prime, then it is odd” with a single counterexample.

Problem 5 Spot the pattern ★★★

Look at the sequence below.

1. Find the differences between consecutive terms.
2. Make a conjecture for the next two terms.

Problem 6 Inductive or deductive ★★★

Label each argument inductive or deductive.

1. Priya has met five golden retrievers, and all were friendly, so she concludes every golden retriever is friendly.
2. All squares are rectangles. Figure ABCD is a square. So ABCD is a rectangle.
3. A student measures the base angles of three isosceles triangles, finds them equal each time, and concludes base angles are always equal.
4. The angles of a triangle add to 180°. Triangle PQR has m∠P = 70° and m∠Q = 60°. So m∠R = 50°.

Problem 7 Name the property ★★★

Match each statement with the property of equality behind it.

1. If AB = 9, then 9 = AB.
2. m∠P = m∠P.
3. If m∠1 = m∠2 and m∠2 = 47°, then m∠1 = 47°.
4. 3(2x − 5) = 6x − 15.
5. If 5x = 40, then x = 8.

Problem 8 Reasons for solving steps ★★★

Supply a reason for every line when solving 4(x − 3) + 5 = 2x + 9.

Problem 9 Biconditional statements ★★★

For each conditional, write the converse. If both are true, write a biconditional; if not, say why none exists.

1. If 3x + 1 = 22, then x = 7.
2. If a whole number is divisible by 4, then it is even.

Problem 10 Given and Prove ★★★

Rewrite the theorem “Angles supplementary to the same angle have equal measures” with a Given and a Prove statement. Then check it with numbers when the shared angle ∠3 measures 64°.

Problem 11 Complete a segment proof ★★★

Fill in the missing reasons. Given: point M bisects segment AB, AM = 5x − 3 and MB = 3x + 9. Prove that x = 6.

Problem 12 Complements of one angle ★★★

Given: ∠AOB and ∠BOC are complementary, and ∠BOC and ∠COD are complementary. Prove: m∠AOB = m∠COD. Write a two-column proof.

Problem 13 Use the vertical angles theorem ★★★

Two lines cross. A pair of vertical angles measures (8x − 17)° and (5x + 19)°. Find x, the measure of each vertical angle, and the measure of a neighboring angle.

Problem 14 Paragraph proof of perpendicular lines ★★★

Lines m and n cross, and two neighboring angles ∠POR and ∠ROQ share a side while their outer sides form a line. Given: ∠POR ≅ ∠ROQ. Prove: m ⊥ n. Present the argument as a paragraph.

Problem 15 Camp rules and logic ★★★

A summer camp posts the rule: “If it is raining, then the archery range is closed.” Assume the rule is true.

1. It is raining on Tuesday. What follows?
2. The range is open on Wednesday. What follows?
3. The range is closed on Thursday. Can anyone conclude it is raining?
4. Write the converse of the rule and say whether the rule guarantees it.

Problem 16 Disproving three statements ★★★

Give a counterexample for each statement.

1. Whenever two angles add to 180°, they are neighbors on a line.
2. If x² = 36, then x = 6.
3. If a whole number is divisible by 5, then it ends in 5.

Problem 17 Odd numbers and a false conjecture ★★★

Add the first n odd numbers for n = 1 through 5 and state a conjecture. Use it to predict the sum of the first 12 odd numbers. Then test the conjecture “n² − n + 41 is prime for every whole number n” with n = 41.

Problem 18 Linear pair and bisector steps ★★★

∠AOB and ∠BOC form a linear pair, with m∠AOB = (7x + 11)° and m∠BOC = (3x + 19)°. Ray OD bisects ∠BOC. Find x, both angles, and m∠BOD.

Problem 19 Equal segments with a shared piece ★★★

Points A, B, C and D lie on a line in that order. Given: AB = CD. Prove: AC = BD. Write a two-column proof.

Problem 20 Bisectors of a linear pair ★★★

∠AOB and ∠BOC form a linear pair. Ray OM bisects ∠AOB and ray ON bisects ∠BOC. Prove that ∠MON is a right angle. Then verify with m∠AOB = 76°.