Review the idea of a flow proof. Remember that arrows show the logical connections between the statements.
Practice makes perfect
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:
