Question

1. \\(\overline{pq} \parallel \overline{mn}\\) given
2. \\(\angle lmn \cong \angle lpq\\) corresponding angles are congruent
3. \\(\angle mnl \cong \angle pql\\) corresponding angles are congruent
4. \\(\triangle lmn \sim \triangle lpq\\) aa similarity criteria
5. \\(\frac{lm}{lp} = \frac{ln}{lq}\\) corresponding sides of similar triangles are proportional
6. \\(lm = lp + pm\\) and \\(ln = lq + qn\\) segment addition postulate

select the statement that explains how to correctly complete the proof.

solve the proportion in statement 5 for both \\(lm\\) and \\(ln\\) next, substitute these new proportions into statement 6 for \\(lm\\) and \\(ln\\) finally, solve both of the newly created equations for \\(lp\\) and set them equal to each other.

rewrite the proportion in statement 5 as equivalent cross products. next, use the equation created in statement 6 to substitute for \\(lm\\) and \\(ln\\) then, divide both sides of the equation by \\(lp\\).

use the equations created in statement 6 to substitute for the numerators in the proportion in statement 5. next, simplify each side of the proportion. then, subtract 1 from each side of the proportion.

Explanation:

To complete the proof, we analyze the given statements. Statement 5 gives a proportion \(\frac{LM}{LP}=\frac{LN}{LQ}\), and statement 6 provides \(LM = LP + PM\) and \(LN = LQ + QN\). The third option suggests substituting the numerators (\(LM\) and \(LN\)) in the proportion from statement 5 with the expressions from statement 6. After substitution, simplifying each side (e.g., \(\frac{LP + PM}{LP}=\frac{LQ + QN}{LQ}\) simplifies to \(1+\frac{PM}{LP}=1+\frac{QN}{LQ}\)) and then subtracting 1 from both sides would lead to \(\frac{PM}{LP}=\frac{QN}{LQ}\), which is a logical step in proving the relationship (likely for parallel lines and proportional segments). The first option's approach of solving for \(LM\) and \(LN\) and substituting into statement 6 is not the typical way to proceed. The second option's cross - product and dividing by \(LP\) does not align with the goal of relating \(PM\), \(LP\), \(QN\), and \(LQ\). So the third option is correct.

Answer:

The third option (Use the equations created in statement 6 to substitute for the numerators in the proportion in statement 5. Next, simplify each side of the proportion. Then, subtract 1 from each side of the proportion.)