Exercise 40 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, l ∥ n. Statement1 l ∥ n Given Let's see what we can conclude from this while keeping in mind that we also know that ∠ 12 ≅ ∠ 8.

As we can see, ∠ 4 and ∠ 8 are corresponding angles. Recall that the Corresponding Angles Theorem tells us that if a transversal intersects two parallel lines, then the corresponding angles are congruent. By this theorem, we can conclude that ∠ 4 ≅ ∠ 8. Statement2 ∠ 4 ≅ ∠ 8 Corresponding Angles Theorem The second piece of information we are given is that ∠ 12 ≅ ∠ 8. We will use it to continue our proof. Statement3 ∠ 12 ≅ ∠ 8 Given Let's summarize what we know from the two last statements. ∠ 4 ≅ ∠ 8 and ∠ 8 ≅ ∠ 12 Now, by the Transitive Property of Congruence, we can conclude that ∠ 4 ≅ ∠ 12. Statement4 ∠ 4 ≅ ∠ 12 Transitive Property of Congruence 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, ∠ 4 and ∠ 12 are corresponding angles. We also proved that they are congruent. By the Converse of the Corresponding Angles Theorem, we can conclude that j ∥ k. Statement5 j ∥ k Converse of the Corresponding Angles Theorem

Final Proof

Combining everything from above, we can write our final proof. &Given:l ∥ n, ∠ 12 ≅ ∠ 8 &Prove: j ∥ k Proof: