Exercise 16 Page 508

Draw a diagram, labeling a pair of consecutive interior angles and a third angle that we can relate to both of these angles.

See solution

Practice makes perfect

Let's draw a diagram, labeling a pair of consecutive interior angles — ∠ 1 and ∠ 2 — and a third angle that we can relate to both of these, ∠ 3.

From the diagram we see that ∠ 1 and ∠ 3 form a linear pair. By the Linear Pair Postulate, we know that these angles are supplementary, which means they sum to 180^(∘). m∠ 1+m∠ 3 = 180^(∘) We also see that ∠ 1 and ∠ 2 are alternate interior angles. Since p∥ q we know by the Alternate Interior Angles Theorem that ∠ 3 ≅ ∠ 2. Using the Substitution Property of Equality, we can prove that ∠ 1 and ∠ 2 are supplementary m∠ 1+ m∠ 2 = 180^(∘) Let's show this as a two-column proof.

Statement	Reason
1. p ∥ q	1. Given
2. ∠ 1 and ∠ 3 are a linear pair	2. Definition of linear pair as seen in the diagram
3. ∠ 1 and ∠ 3 are supplementary	3. Linear Pair Postulate
4. m∠ 1 +m∠ 3=180^(∘)	4. Definition of supplementary angles
5. ∠ 3 ≅ ∠ 2	5. Alternate Interior Angles Theorem
6. m∠ 3 = m∠ 2	6. Definition of congruent angles
7. m∠ 1 +m∠ 2=180^(∘)	7. Substitution Property of Equality
8. ∠ 1 and ∠ 2 are supplementary	8. Definition of supplementary angles

Hints