To complete the proof showing that angles \( m\angle 6 \) and \( m\angle 11 \) are congruent in the parallelogram, we can fill in the blanks in the two-column proof as follows:
Two-column Proof:
| Statements | Reasons | |------------------------------------------------|----------------------------------------------| | 1. \( a \parallel b \) and \( e \parallel f \) | Given | | 2. \( m\angle 5 + m\angle 6 = 180^\circ \) and \( m\angle 9 + m\angle 11 = 180^\circ \) | Definition of linear pair | | 3. \( m\angle 5 = m\angle 9 \) | Corresponding Angles Theorem | | 4. \( m\angle 13 + m\angle 6 = 180^\circ \) and \( m\angle 13 + m\angle 11 = 180^\circ \) | Same-Side Interior Angles Theorem | | 5. \( m\angle 6 = m\angle 11 \) | Transitive Property of Equality | | 6. \( m\angle 6 \cong m\angle 11 \) | Definition of Congruence |
Explanation for the omitted statements:
- Statement 2 refers to the angles formed by lines intersecting a transversal, which are supplementary.
- Statement 3 uses the fact that corresponding angles are equal when two parallel lines are cut by a transversal.
- Statement 4 confirms that angles on the same side of the transversal and inside the parallel lines are supplementary.
- Statement 5 uses the Transitive Property, which states if \( a = b \) and \( b = c \), then \( a = c \).
- Statement 6 concludes that since the measures of angles \( m\angle 6 \) and \( m\angle 11 \) are equal, the angles themselves are congruent.
This structured proof demonstrates the congruence of angles in the given parallelogram based on properties of parallel lines and transversals.