Exercise 57 Page 569

Use both the Consecutive Interior Angles Theorem and the Converse Consecutive Interior Angles Theorem.

Statements	Reasons
1. r∥ t	1. Given
2. ∠ 5 and ∠ 4 are supplementary	2. Consecutive Interior Angles Theorem
3. m∠ 5 + m∠ 4 = 180^(∘)	3. Definition of supplementary angles
4. ∠ 5 ≅ ∠ 6	4. Given
5. m∠ 5=m∠ 6	5. Definition of congruent angles
6. m∠ 6 + m∠ 4 = 180^(∘)	6. Substitution
7. ∠ 4 and ∠ 6 are supplementary	7. Definition of supplementary angles
8. l ∥ m	8. Converse Consecutive Interior Angles Theorem

Practice makes perfect

We are given the diagram below, where r∥ t and ∠ 5 ≅ ∠ 6.

By the Consecutive Interior Angles Theorem we have that ∠ 5 and ∠ 4 are supplementary.

m ∠ 5 + m∠ 4 = 180^(∘) Since ∠ 5 ≅ ∠ 6, we have that m ∠ 5 = m ∠ 6. Let's substitute it above. m ∠ 6 + m∠ 4 = 180^(∘) From the above ∠ 6 and ∠ 4 are supplementary. Thus, the Converse Consecutive Interior Angles Theorem tells us that l ∥ m.

Two-Column Proof

Given: & r∥ t, ∠ 5 ≅ ∠ 6 Prove: & l ∥ m Let's summarize the proof we did above in the following two-column table.

Statements	Reasons
1. r∥ t	1. Given
2. ∠ 5 and ∠ 4 are supplementary	2. Consecutive Interior Angles Theorem
3. m∠ 5 + m∠ 4 = 180^(∘)	3. Definition of supplementary angles
4. ∠ 5 ≅ ∠ 6	4. Given
5. m∠ 5=m∠ 6	5. Definition of congruent angles
6. m∠ 6 + m∠ 4 = 180^(∘)	6. Substitution
7. ∠ 4 and ∠ 6 are supplementary	7. Definition of supplementary angles
8. l ∥ m	8. Converse Consecutive Interior Angles Theorem

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Two-Column Proof