Here’s how you can organize the statements and reasons into a two-column proof to show that the opposite angles, ∠6 and ∠11, are congruent in the parallelogram:
| Statements | Reasons | |-----------------------------------------------------|----------------------------------------| | Statement 1: a || b and e || f | Reason 1: Given | | Statement 2: m∠5 + m∠7 = 180 degrees and m∠9 + m∠11 = 180 degrees. | Reason 2: Corresponding Angles Theorem | | Statement 3: m∠5 + m∠6 = 180 degrees and m∠9 + m∠11 = 180 degrees. | Reason 3: Definition of Linear Pair | | Statement 4: m∠5 = m∠13 and m∠9 = m∠13 | Reason 4: CPCIC Theorem | | Statement 5: m∠13 + m∠6 = m∠13 + m∠11 | Reason 5: Substitution Property of Equality | | Statement 6: m∠6 = m∠11 | Reason 6: Subtraction Property of Equality |
Let’s clarify the logic used in the proof:
- Statements 1 and 2 utilize the properties of parallel lines to establish the relationships between angles formed.
- Statements 3 and 4 further relate those degrees of angles to the angles that are being examined for similarity (specifically ∠6 and ∠11).
- Statements 5 and 6 ultimately bring you to the conclusion that ∠6 and ∠11 are congruent based on the equality established in previous steps.
This proof shows the logical deduction that leads to the conclusion about the angles in the parallelogram.
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