Question

given: ∠q ≅ ∠s, \\(\overline{pq} \parallel \overline{rs}\\)
prove: \\(\overline{qr} \parallel \overline{st}\\)

1. ∠q ≅ ∠s, \\(\overline{pq} \parallel \overline{rs}\\) \t1. given
2. click here to insert \t2. if two parallel lines are cut by a transversal, then their alternate interior angles are congruent.
3. click here to insert \t3. click here to insert

Explanation:

Answer:

Step 2:

To apply the alternate interior angles theorem for \(\overline{PQ} \parallel \overline{RS}\), the transversal here is \(\overline{QR}\). So the alternate interior angles would be \(\angle Q \cong \angle R\) (since \(PQ \parallel RS\) and \(QR\) is the transversal). So the statement for step 2 is \(\boldsymbol{\angle Q \cong \angle R}\).

Step 3:

We know from step 1 that \(\angle Q \cong \angle S\) and from step 2 that \(\angle Q \cong \angle R\). By the transitive property of congruence (if \(\angle Q \cong \angle R\) and \(\angle Q \cong \angle S\), then \(\angle R \cong \angle S\)). Now, \(\angle R\) and \(\angle S\) are alternate interior angles for lines \(\overline{QR}\) and \(\overline{ST}\) with transversal \(\overline{RS}\). So if alternate interior angles are congruent, then the lines are parallel. So first, we establish \(\boldsymbol{\angle R \cong \angle S}\) (using transitive property: since \(\angle Q \cong \angle R\) and \(\angle Q \cong \angle S\), then \(\angle R \cong \angle S\)), and then the reason for step 3 (to conclude \(QR \parallel ST\)) would be "If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel". But for the statement in step 3 (before the final conclusion), we have \(\angle R \cong \angle S\).

Final Proof Steps Summary:

1. \(\angle Q \cong \angle S\), \(\overline{PQ} \parallel \overline{RS}\) - Given
2. \(\angle Q \cong \angle R\) - If two parallel lines (\(PQ \parallel RS\)) are cut by a transversal (\(QR\)), then alternate interior angles are congruent.
3. \(\angle R \cong \angle S\) - Transitive property of congruence (\(\angle Q \cong \angle R\) and \(\angle Q \cong \angle S\) implies \(\angle R \cong \angle S\))
4. \(\overline{QR} \parallel \overline{ST}\) - If two lines (\(QR\) and \(ST\)) are cut by a transversal (\(RS\)) and alternate interior angles (\(\angle R\) and \(\angle S\)) are congruent, then the lines are parallel.

(Note: The problem seems to be missing step 4 in the given table, but based on the proof structure, the above steps complete the proof. For the given table with steps 1 - 3, step 2 statement is \(\angle Q \cong \angle R\), step 3 statement is \(\angle R \cong \angle S\) and reason is transitive property, and then step 4 (not shown) would be \(QR \parallel ST\) with the alternate interior angles converse as reason.)