Question

In the diagram, a || b and e || f. Sylvia writes a proof to prove the opposite angles, ∠6 and ∠11, are congruent in the parallelogram. Drag and drop the statements and reasons into their correct locations in the two-column proof.

Statement 1:
1. a || b and e || f
Reason 1:
Given
Statement 2:
Blank
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∠13 + m∠6 = 180 degrees and m∠13 + m∠11 = 180 degrees
Reason 4:
Blank
Statement 5:
Blank
Reason 5:
Transitive of Equality
Statement 6:
m∠6 = m∠11
Reason 6:
Blank

Answers to the blanks (you have to figure out which answer goes into what blank)
Answer 1: m∠5 + m∠7 = 180 degrees and m∠13 + m∠14 = 180 degrees
Answer 2: CPCIC Theorem
Answer 3: Substitution Property of Equality
Answer 4: m∠5 = m∠13 and m∠9 = m∠13
Answer 5: m∠13 + m∠6 = m∠13 + m∠11
Answer 6: Subtraction Property of Equality

Answer 1

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.