Exercise 41 Page 162

Let's begin by reviewing the idea of a flow proof. Arrows show the logical connections between the statements. Reasons are written below the statements. We can begin our proof with the first given information, j ∥ k. Statement1 j ∥ k Given Let's see what we can conclude from this, keeping in mind that we also know that ∠ 12 ≅ ∠ 8.

As we can see, ∠ 9 and ∠ 3 are alternate interior angles. Recall that the Alternate Interior Angles Theorem tells us that if a transversal intersects two parallel lines, then alternate interior angles are congruent. Thus, by this theorem we conclude that ∠ 9 ≅ ∠ 3. Statement2 ∠ 9 ≅ ∠ 3 Alternate Interior Angles Theorem Notice that since these angles are congruent, then m ∠ 9 = m ∠ 3. The second piece of information we are given is that m ∠ 8 + m ∠ 9=180. We will use it to continue our proof. Statement3 m ∠ 8 + m ∠ 9=180 Given Let's summarize what we know from the two last statements. m ∠ 9= m ∠ 3 and m ∠ 8 + m ∠ 9=180 Now, by the Substitution Property of Equality we can substitute m ∠ 3 for m ∠ 9 in our equation. This gives us m ∠ 8 + m∠ 3 = 180. Statement4 m ∠ 8 + m∠ 3 = 180 Substitution Property of Equality Notice that this statement follows from Statements 2 and 3. Thus, we will have two arrows that lead to Statement 4 in our flow proof. Let's continue!

As we can see, ∠ 3 and ∠ 8 are same-side interior angles. We also proved that the sum of their measures is 180. Thus, by the Converse of the Same-Side Interior Angles Postulate we can conclude that l ∥ n. Statement5 l ∥ n Converse of the Same-Side Interior Angles Theorem

Final Proof

&Given:j ∥ k, m ∠ 8+ m∠ 9 =180 &Prove: l ∥ n Proof: