Write a two-column proof of Theorem 6-2-3, which is: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
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Statement Reason
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1. Quadrilateral ABCD is a Given
parallelogram.
2. AB || DC and AD || BC. Definition of a parallelogram.
3. ∠ABC + ∠BCD = 180° Supplementary angles.
Proof:
We are given that quadrilateral ABCD is a parallelogram (Statement 1).
By definition of a parallelogram, we know that AB is parallel to DC and AD is parallel to BC (Statement 2).
We need to prove that the consecutive angles of ABCD, ∠ABC and ∠BCD, are supplementary, meaning they add up to 180°.
Since AB is parallel to DC, and AD is parallel to BC, we can conclude that ∠ABC and ∠BCD are corresponding angles (Alternate Interior Angles Theorem).
By the Alternate Interior Angles Theorem, corresponding angles formed by parallel lines are congruent (Theorem 4-3-1).
Therefore, ∠ABC ≅ ∠BCD.
Since congruent angles add up to 180°, we can say that ∠ABC + ∠BCD = 180° (Statement 3).
Hence, we have proved that if a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
To construct a two-column proof of Theorem 6-2-3, we need to provide the statements (facts or geometric properties) and the reasons (justifications or logical deductions) for each step. Here is the proof:
| Statements | Reasons |
| --- | --- |
| 1. ABCD is a parallelogram. | Given |
| 2. AB is parallel to DC and BC is parallel to AD. | Definition of a parallelogram |
| 3. ∠A and ∠D are corresponding angles. | Corresponding angles theorem |
| 4. ∠C and ∠B are corresponding angles. | Corresponding angles theorem |
| 5. ∠A ≅ ∠D and ∠C ≅ ∠B. | Definition of corresponding angles |
| 6. m∠A + m∠D = 180° and m∠C + m∠B = 180°. | Supplementary angles theorem |
| 7. ∠A and ∠D are consecutive angles. | Definition of consecutive angles |
| 8. ∠C and ∠B are consecutive angles. | Definition of consecutive angles |
| 9. ∠A and ∠D are supplementary angles. | Substitution (Step 6) |
| 10. ∠C and ∠B are supplementary angles. | Substitution (Step 6) |
Therefore, if a quadrilateral is a parallelogram, its consecutive angles are supplementary.