Logic and Two-Column Proofs: Geometry Math Worksheet – PDF Download

Geometry Reasoning Proof Worksheet: problems 1–5

Problem 1: Hypothesis and conclusion

Identify the hypothesis and the conclusion: “If it rains on Saturday, then the Little League game is canceled.”

Problem 2: Writing if-then form

Rewrite as an if-then statement and underline the hypothesis in your answer: “A number is divisible by 6 if it is divisible by both 2 and 3.”

Problem 3: Rewriting a statement

Rewrite in if-then form: “All squares have four congruent sides.”

Problem 4: Writing a converse

Write the converse of “If a figure is a square, then it is a rectangle.” Decide whether the converse is true or false. If false, give a counterexample.

Problem 5: Inverse and contrapositive

Write the inverse and the contrapositive of “If a polygon is a square, then it has four sides.”

Geometry Reasoning Proof Worksheet: problems 6–10

Problem 6: Truth values of related statements

Consider “If a number is a multiple of 4, then it is even.” Write the converse, inverse and contrapositive, and decide whether each of the four statements is true or false. Give a counterexample for each false one.

Problem 7: Counterexample with primes

Find a counterexample to: “If a number is prime, then it is odd.”

Problem 8: Counterexample with squares

Find a counterexample to: “If \(x^{2}=49\), then \(x=7\).”

Problem 9: Counterexample with angles

Maya claims: “If two angles are supplementary, then they form a linear pair.” Use the figure to show that her claim is false.

Problem 10: Inductive pattern of dots

The figure shows the first four dot patterns. Use inductive reasoning to find the number of dots in Pattern 5 and Pattern 6, and write a rule for Pattern n.

Geometry Reasoning Proof Worksheet: problems 11–15

Problem 11: Inductive or deductive?

Decide whether each argument uses inductive or deductive reasoning.

1. The first five Saturdays of the season were sunny, so Maya concludes the next Saturday will be sunny.
2. Every even number is divisible by 2. The number 148 is even, so 148 is divisible by 2.
3. Diego measures the angles of three triangles, finds each sum is \(180^{\circ}\), and concludes every triangle has an angle sum of \(180^{\circ}\).
4. Every triangle has an angle sum of \(180^{\circ}\). Polygon XYZ is a triangle, so its angle sum is \(180^{\circ}\).

Problem 12: Making a conjecture

Look at these sums: \(3+5=8\), \(7+9=16\), \(11+5=16\), \(1+13=14\). Write a conjecture about the sum of two odd numbers.

Problem 13: Disproving a conjecture

Ethan conjectures: “The sum of two numbers is always greater than either number.” Find a counterexample.

Problem 14: Naming properties of equality

Name the property that justifies each statement.

1. If \(AB=CD\), then \(CD=AB\).
2. If \(m\angle 1=m\angle 2\) and \(m\angle 2=m\angle 3\), then \(m\angle 1=m\angle 3\).
3. \(PQ=PQ\).
4. If \(3x=15\), then \(x=5\).

Problem 15: Supplying reasons

Copy and complete the reasons.

Given: \(3x+7=22\). Prove: \(x=5\).

Statements	Reasons
1. \(3x+7=22\)	1. ____
2. \(3x=15\)	2. ____
3. \(x=5\)	3. ____

Geometry Reasoning Proof Worksheet: problems 16–20

Problem 16: Supplying statements

Angles 1 and 2 together form a right angle, so they are complementary. Copy and complete the statements.

Given: \(\angle 1\) and \(\angle 2\) are complementary; \(m\angle 1=35^{\circ}\). Prove: \(m\angle 2=55^{\circ}\).

Statements	Reasons
1. ____	1. Given
2. ____	2. Definition of complementary angles
3. ____	3. Substitution Property of Equality
4. ____	4. Subtraction Property of Equality

Problem 17: Midpoint proof

Copy and complete the reasons.

Given: M is the midpoint of \(\overline{ST}\); \(ST=24\). Prove: \(SM=12\).

Statements	Reasons
1. M is the midpoint of \(\overline{ST}\); \(ST=24\)	1. Given
2. \(SM=MT\)	2. ____
3. \(SM+MT=ST\)	3. ____
4. \(SM+SM=24\)	4. ____
5. \(2(SM)=24\)	5. ____
6. \(SM=12\)	6. ____

Problem 18: Vertical angles proof

Lines ℓ and m intersect to form angles 1, 2, 3 and 4. Copy and complete the reasons to prove that \(\angle 2\) ≅ \(\angle 4\).

Statements	Reasons
1. Lines ℓ and m intersect	1. Given
2. \(\angle 2\) and \(\angle 3\) are a linear pair; \(\angle 3\) and \(\angle 4\) are a linear pair	2. ____
3. \(\angle 2\) and \(\angle 3\) are supplementary; \(\angle 3\) and \(\angle 4\) are supplementary	3. ____
4. \(m\angle 2+m\angle 3=180^{\circ}\); \(m\angle 3+m\angle 4=180^{\circ}\)	4. Definition of supplementary angles
5. \(m\angle 2+m\angle 3=m\angle 3+m\angle 4\)	5. ____
6. \(m\angle 2=m\angle 4\)	6. ____
7. \(\angle 2\) ≅ \(\angle 4\)	7. ____

Problem 19: Paragraph proof

Write a paragraph proof. Given: \(\angle A\) and \(\angle B\) are complementary; \(\angle C\) and \(\angle B\) are complementary. Prove: \(\angle A\) ≅ \(\angle C\).

Problem 20: Contrapositive in context

A school rule says: “If a student is on the honor roll, then the student has a GPA of at least 3.5.” Priya’s GPA is 3.2. What can you conclude about Priya, and which related statement did you use?

Geometry Reasoning Proof Worksheet: problems 21–23

Problem 21: Chaining conditionals

Two rules are true: “If it snows more than 6 inches, then school is closed” and “If school is closed, then the bake sale moves to Sunday.” Write one conditional that links them. Aisha hears that 8 inches of snow fell. What follows?

Problem 22: Solve and justify

Lines ℓ and m intersect. \(m\angle 1\)\(=(3x+14)^{\circ}\) and \(m\angle 3\)\(=(5x-6)^{\circ}\). Find x and \(m\angle 2\), and give a reason for each step.

Problem 23: Writing a biconditional

Consider “If a polygon is a triangle, then it has exactly three sides.” Write the converse, decide whether it is true, and then write a biconditional statement.

How to use this geometry reasoning proof worksheet

Print this geometry reasoning proof worksheet as a PDF, solve every problem on paper, then compare with the answer key.

See every chapter on the Geometry math page.