1. **State the problem:** Write a two-column proof for a given geometric statement or theorem. 2. **Understand the two-column proof format:** It consists of two columns: the left column lists the statements, and the right column lists the reasons or justifications for each statement. 3. **Example problem:** Prove that if two angles are vertical angles, then they are congruent. 4. **Proof:** | Statements | Reasons | |-----------------------------------|---------------------------------| | 1. \angle 1 and \angle 2 are vertical angles | 1. Given | | 2. \angle 1 and \angle 3 form a linear pair | 2. Definition of vertical angles | | 3. \angle 2 and \angle 3 form a linear pair | 3. Definition of vertical angles | | 4. \angle 1 + \angle 3 = 180^\circ | 4. Linear pair postulate | | 5. \angle 2 + \angle 3 = 180^\circ | 5. Linear pair postulate | | 6. \angle 1 + \angle 3 = \angle 2 + \angle 3 | 6. Transitive property of equality | | 7. \cancel{\angle 3} + \angle 1 = \cancel{\angle 3} + \angle 2 | 7. Subtraction property of equality (subtract \angle 3 from both sides) | | 8. \angle 1 = \angle 2 | 8. Simplification | 5. **Explanation:** - We start with the given that the angles are vertical. - Use the definition of vertical angles and linear pair postulate to set up equalities. - Use properties of equality to subtract common angles and conclude the angles are congruent. This is a typical structure of a two-column proof.